Estimating Effects of Long-Term Treatments

enumerate• (Missing treatments) At the moment of estimation, the experimenter has not conducted any treatment in the future periods. • (Missing observations) At the moment of estimation, the experimenter has not observed any outcome in the future periods.

assumption[Longitudinal Surrogacy] The treatment assignment at an earlier period is independent of the primary and intermediate outcomes at a later period, conditional on the intermediate outcomes at a middle period, i.e., there exists a subset of time indices $\mathbb{T} = \{t_1,t_2,...,t_K\} \subseteq [T]$, such that for any $i \in [N]$, any $t \in \mathbb{T}$, and any $t' > t$, \begin{align*} \left(Y_{it'}, \bm{S}_{it'}\right) \perp \!\!\! \perp \bm{W}_{i, 1:t} \vert \bm{S}_{it}, \bm{X}_i. \end{align*} Moreover, we refer to the intermediate outcomes at the time periods $t \in \mathbb{T}$ as surrogate outcomes, or, simply, surrogates.

definition[Longitudinal Surrogate Index] For any $t \in [T], \bm{s} \in \mathbb{S}, \bm{x} \in \mathbb{X}, \bm{w}_{1:t} \in \{\bm{0}_t, \bm{1}_t\}$, the surrogate index is the conditional expectation of the primary outcome at time $t$, given the surrogate outcomes at time $0$, the pre-treatment variables, and the treatment assignments, i.e., \begin{align*} h_t(\bm{s}, \bm{x}, \bm{w}_{1:t}) = \mathbb{E}_{\mathcal{F}} \left[ Y_{it} \vert \bm{S}_{i0} = \bm{s}, \bm{X}_i = \bm{x}, \bm{W}_{i,1:t} = \bm{w}_{1:t}\right], \end{align*} where the expectation is taken over $Y_{it}$.

definition[Pivot Index] For any $t \in [T]$, $\bm{s} \in \mathbb{S}$, $\bm{x} \in \mathbb{X}$, $\bm{w}_{1:t} \in \{\bm{0}_t, \bm{1}_t\}$, the pivot index is a vector of the conditional expectations of the surrogate outcomes at time $t$, given the surrogate outcomes at time $0$, the pre-treatment variables, and the treatment assignments, i.e., \begin{align*} \bm{g}_t(\bm{s}, \bm{x}, \bm{w}_{1:t}) = \mathbb{E}_{\mathcal{F}} \left[ \bm{S}_{it} \vert \bm{S}_{i0} = \bm{s}, \bm{X}_i = \bm{x}, \bm{W}_{i,1:t} = \bm{w}_{1:t}\right], \end{align*} where the expectation is taken over $\bm{S}_{it}$. Moreover, we denote the conditional surrogate outcomes at time $t$, given the surrogate outcomes at time $0$, the pre-treatment variables, and the treatment assignments to be, \begin{align*} \bm{G}_t(\bm{s}, \bm{x}, \bm{w}_{1:t}) \sim \bm{S}_{it} \vert \bm{S}_{i0} = \bm{s}, \bm{X}_i = \bm{x}, \bm{W}_{i,1:t} = \bm{w}_{1:t}. \end{align*}

assumption[Comparability] The primary and intermediate outcomes across different periods share the same support. The distribution of the primary and intermediate outcomes at a later period, conditional on the intermediate outcomes at an earlier period, on the treatment assignments during the earlier and later periods, and on the pre-treatment variables, is the same across different time periods, i.e., for any $t,t' \in [T]$, and any positive integer $\delta \in \mathbb{N}^+$, \begin{align*} (Y_{it}, \bm{S}_{it}) \vert \bm{S}_{i(t-\delta)}, \bm{W}_{i,t-\delta+1:t}, \bm{X}_i \ \sim \ (Y_{it'}, \bm{S}_{it'}) \vert \bm{S}_{i(t'-\delta)}, \bm{W}_{i,t'-\delta+1:t'}, \bm{X}_i. \end{align*}

lemmaConsider the special case when $T_E = T_F$. Under Assumptions (ref) -- (ref), where Assumption (ref) holds for $\mathbb{T} = \{T_E\}$, the average effect of long-term treatments on the primary outcome is equal to the following expression, \begin{align*} \tau_T = \mathbb{E}_{\mathcal{F}} \left[ h_{T_E}\left( \bm{G}_{T_E}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{T_E}), \bm{X}_i, \bm{1}_{T_E} \right) \right] - \mathbb{E}_{\mathcal{F}} \left[ h_{T_E}\left( \bm{G}_{T_E}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{T_E}), \bm{X}_i, \bm{0}_{T_E} \right) \right]. \end{align*}

theorem[Longitudinal Surrogate Model] Under Assumptions (ref) and (ref), where Assumption (ref) holds for $\mathbb{T} = \{t_1,t_2,...,t_K\}$, the average effect of long-term treatments on the primary outcome is equal to the following expression, \begin{multline*} \tau_T = \mathbb{E}_{\mathcal{F}} \left[ h_{\Delta t_{K+1}}\left( \bm{G}_{\Delta t_K}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K}), \bm{X}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \\ - \mathbb{E}_{\mathcal{F}} \left[ h_{\Delta t_{K+1}}\left( \bm{G}_{\Delta t_K}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{\Delta t_1}) ... ,\bm{X}_i, \bm{0}_{\Delta t_K}), \bm{X}_i, \bm{0}_{\Delta t_{K+1}} \right) \right], \end{multline*} where the expectation is taken over $\bm{S}_{i0}, \bm{X}_i$, as well as the conditional surrogate outcomes $\bm{G}_{\Delta t_1}, ..., \bm{G}_{\Delta t_K}$.

assumption[Linearity of Surrogates] \begin{enumerate} • The surrogate index function is linear with respect to the surrogates, i.e., there exists $\alpha_d(\bm{x}, \bm{w}_{1:t})$, $\forall d\in\{0,1,...,D\}, \bm{x}\in\mathbb{X}, \bm{w}_{1:t}\in\{\bm{0}_t, \bm{1}_t\}$, such that \begin{align} h_t(\bm{s}, \bm{x}, \bm{w}_{1:t}) = \alpha_0(\bm{x}, \bm{w}_{1:t}) + \sum_{d=1}^D s_d \cdot \alpha_d(\bm{x}, \bm{w}_{1:t}). \end{align} • The pivot index function is linear with respect to the surrogates, i.e., there exists $\beta_{d,d'}(\bm{x}, \bm{w}_{1:t})$, $\forall d\in[D], d'\in\{0,1,...,D\}, \bm{x}\in\mathbb{X}, \bm{w}_{1:t}\in\{\bm{0}_t, \bm{1}_t\}$, such that for each $d\in[D]$, \begin{align} g_{t,d}(\bm{s}, \bm{x}, \bm{w}_{1:t}) = \beta_{d,0}(\bm{x}, \bm{w}_{1:t}) + \sum_{d'=1}^D s_d \cdot \beta_{d,d'}(\bm{x}, \bm{w}_{1:t}), \end{align} where $g_{t,d}(\bm{s}, \bm{x}, \bm{w}_{1:t})$ stands for the $d$-th component of $\bm{g}_t(\bm{s}, \bm{x}, \bm{w}_{1:t})$ the pivot index. \end{enumerate}

theorem[Linear Surrogate Model] Under Assumptions (ref), (ref), and (ref), where Assumption (ref) holds for $\mathbb{T} = \{t_1,t_2,...,t_K\}$, the average effect of long-term treatments on the primary outcome is equal to the following expression, \begin{multline*} \tau_T = \mathbb{E}_{\mathcal{F}} \Big[ h_{\Delta t_{K+1}} \big( \bm{g}_{\Delta t_K}( ... \bm{g}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K}), \bm{X}_i, \bm{1}_{\Delta t_{K+1}} \big) \Big] \\ - \mathbb{E}_{\mathcal{F}} \Big[ h_{\Delta t_{K+1}}\left( \bm{g}_{\Delta t_K}( ... \bm{g}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{\Delta t_1}) ... ,\bm{X}_i, \bm{0}_{\Delta t_K}), \bm{X}_i, \bm{0}_{\Delta t_{K+1}} \right) \Big], \end{multline*} where the expectation is taken over $\bm{S}_{i0}, \bm{X}_i$.

itemize• Constant Extrapolation Baseline (CEB): We use the average treatment effects observed during the first $T_E$ weeks of the experiment to predict the treatment effects for the future period. Although obviously this approach cannot capture any increasing or decreasing trends in the treatment effects, this serves as a common industry practice. • Vector Autoregressive (VAR) Model: We employ a Vector Autoregressive (VAR) model with lag order \( p = T_E - 2 \) on the initial \( T_E \) weeks of the multivariate time series, using the average values of four surrogates and one primary outcome variable as input candidates. This allows the VAR model to forecast future outcomes based on past values of all included variables, though VAR is traditionally used for forecasting rather than causal inference stock2001vector.\footnote{The forecasted effect is calculated by taking the difference between the predicted average primary outcomes of the treatment and control groups at each future time point. The lag order \( p \) is selected as the largest feasible term to maintain model performance. When \( p < 5 \), we include the primary outcome and randomly select \( p-1 \) of the surrogates; otherwise, all five variables are included. For the edge case ($T_E = 2$), the result from constant extrapolation is used instead. We choose $p=T_E-2$ because this is the the largest possible term that can be selected, ensuring the VAR model's performance.}

APPENDICES\setcounter{figure}{0} \setcounter{table}{0} \centerline{ Online Appendix} \setcounter{page}{1} \section{Linear Additive Model} In addition to the linear surrogate model described in the main text, we introduce a linear additive model. This model requires slightly stronger assumptions. We first outline the theoretical framework and then show our empirical findings. Our results suggest that the assumption required for this model is only applicable to a few restrictive situations. Therefore, we advise against using this approach in common settings. \subsection{Model and Assumption} We introduce one additional assumption that helps establish the linear additive model. Note that this additional assumption is strong and may not hold in a number of scenarios. The linear additive model should thus be used with cautions. Combined with the previous three assumptions, this new set of four assumptions is the third level of assumptions. \begin{assumption}[Additive Treatment Effects] The average treatment effect of long-term treatments is linear additive to the treatments, i.e., there exists a subset of time indices $\mathbb{T} = \{t_1, t_2, ..., t_K\} \subseteq [T]$, such that for any $i \in [N]$, \begin{align*} \tau = \sum_{k=1}^{K+1} \left\{ \mathbb{E}_{\mathcal{F}}\bigg[ Y_{iT} (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}, \bm{0}_{t_k+1:T}) \bigg] - \mathbb{E}_{\mathcal{F}}\bigg[ Y_{iT}(\bm{0}_T) \bigg] \right\}. \end{align*} \end{assumption} Assumption (ref) suggests that the average effect of long-term treatments is the summation of several components. Each component is the effect of a subset of short-term treatments. Assumption (ref) holds when the effect of long-term treatments can be decomposed into the summation of a number of carryover effects. For example, when advertisements are sent to users regularly, each advertisement may marginally increase the average click-through rate where the effects of the earlier advertisements quickly decay with the time. This could be a context when Assumption (ref) holds. In contrast, Assumption (ref) does not hold in some other contexts, such as in estimating the “novelty effect,” where users click a button more frequently when its color is changed, but their clicks quickly decrease back to normal as they become familiar with the new color. Under Assumption (ref), we leverage multiple surrogate indices whose subscripts are different, and re-write the average effect of long-term treatments as follows. \begin{theorem}[Linear Additive Model] Under Assumptions (ref), (ref), and (ref), where Assumptions (ref) and (ref) hold for $\mathbb{T}=\{t_1,t_2,...,t_K\}$, the average effect of long-term treatments on the primary outcome is equal to the following expression, \begin{multline*} \tau_T = \sum_{k=1}^{K} \mathbb{E}_{\mathcal{F}} \bigg[ h_{T-t_k}\left( \bm{G}_{t_k}\left(\bm{S}_{i0},\bm{X}_i,(\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k})\right),\bm{X}_i,\bm{0}_{T-t_k} \right) \bigg] \\ + \mathbb{E}_{\mathcal{F}} \bigg[ h_{T}\left( \bm{S}_{i0},\bm{X}_i, (\bm{0}_{1:t_K},\bm{1}_{t_K+1:T}) \right) \bigg] - \sum_{k=1}^{K+1} \mathbb{E}_{\mathcal{F}}\left[h_{T}\left(\bm{S}_{i0},\bm{X}_i,\bm{0}_T\right)\right]. \end{multline*} \end{theorem} Theorem (ref) is similar to Lemma (ref), in the sense that for each $k$, each potential outcome $Y_{iT} (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}, \bm{0}_{t_k+1:T})$ is decomposed into two iterations using only one conditional surrogate outcomes $\bm{G}_{t_k}$. However, different from Lemma (ref), Theorem (ref) does not require the length of future periods $T_F$ to be equal to the length of experimental periods $T_E$. Compared with Theorem (ref), Theorem (ref) makes one additional Assumption (ref). This additional assumption enables Theorem (ref) to avoid iterating the pivots for multiple times and use summation instead. To fully leverage the benefits of the linear additive model, we assume the availability of historical data, which is collected during the observational periods without any intervention. During the observational periods $t \in \{0,-1,-2,...\}$, we can observe both the surrogate and primary outcomes $(\bm{S}_{it}, Y_{it})$. Compared with the number of treatment subjects in the experimental periods, there are many more subjects in the observational periods. But no subject is treated during the observational periods, i.e, $W_{it} = 0, \forall t \in \{0,-1,-2,...\}$. See Figure (ref) for an illustration. \begin{figure}[tb] \caption{An illustrator of the observational periods, the experimental periods and the future periods.} \end{figure} The linear additive model as described in Theorem (ref) allows us to estimate the surrogate indices $h_{T-t_k}(\cdot, \cdot, \bm{0}_{T-t_k}), \forall k \in [K]$ and $h_{T}(\cdot, \cdot, \bm{0}_{T})$ all from abundant historical data. This identification strategy does not suffer from the length of experimental periods $T_E$ being too small. \subsection{Estimators for the Linear Additive Model} Finally, we propose the third estimation strategy to estimate the surrogate index functions from the abundant data in the observational periods. Given the estimators of the surrogate index and estimators of the conditional surrogate outcomes, we follow Theorem (ref) and obtain the following plug-in estimator, \begin{multline} \widehat{\tau}_T = \frac{1}{N_1} \sum_{k=1}^{K} \sum_{i\in[N]} \mathbbm{1}\{\bm{W}_{i,1:T_E} = \bm{1}_{T_E}\} \mathbb{E}_{\widehat{\bm{G}}_{t_k}}\left[ \widehat{h}_{T-t_k}\left( \widehat{\bm{G}}_{t_k}\left(\bm{s}_{i0}, \bm{x}_i, (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k})\right), \bm{x}_i, \bm{0}_{T-t_k} \right) \right] \\ + \frac{1}{N_1} \sum_{i\in[N]} \mathbbm{1}\{\bm{W}_{i,1:T_E} = \bm{1}_{T_E}\} \widehat{h}_{T}\left( \bm{s}_{i0},\bm{x}_i, (\bm{0}_{1:t_K},\bm{1}_{t_K+1:T}) \right) \\ - (K+1) \frac{1}{N_0} \sum_{i\in[N]} \mathbbm{1}\{\bm{W}_{i,1:T_E} = \bm{0}_{T_E}\} \widehat{h}_T(\bm{s}_{i0}, \bm{x}_i, \bm{0}_T). \end{multline} We explain how to estimate the surrogate index functions in (ref). For any $t \in [T_E] \cup \{0,-1,-2,...\}$, $\bm{x} \in \mathbb{X}$, $\bm{s} \in \mathbb{S}$, one naive estimator of the surrogate index under consecutive controls is given by \begin{align*} \widehat{h}_{T-t_k}(\bm{s}, \bm{x}, \bm{0}_{T-t_k}) = \frac{\sum_{i \in [N]}Y_{it} \mathbbm{1}\{\bm{X}_i = \bm{x}, \bm{S}_{i(t-T+t_k)} = \bm{s}, \bm{W}_{i,t-T+t_k+1:t} = \bm{0}_{T-t_k}\}}{\sum_{i \in [N]} \mathbbm{1}\{\bm{X}_i = \bm{x}, \bm{S}_{i(t-T+t_k)} = \bm{s}, \bm{W}_{i,t-T+t_k+1:t} = \bm{0}_{T-t_k}\}}. \end{align*} Under Assumptions (ref), such an estimator is unbiased for the surrogate index function. To estimate the middle term of (ref), for any $\bm{x} \in \mathbb{X}$, $\bm{s} \in \mathbb{S}$, one naive estimator of the surrogate index under $t_K$ controls followed by $\Delta t_{K+1}$ treatments is given by \begin{align*} \widehat{h}_{T}\left( \bm{s}, \bm{x}, (\bm{0}_{1:t_K},\bm{1}_{t_K+1:T}) \right) = \frac{\sum_{i \in [N]}Y_{i\Delta t_{K+1}} \mathbbm{1}\{\bm{X}_i = \bm{x}, \bm{S}_{i(-t_K)} = \bm{s}, \bm{W}_{i,-t_K+1:\Delta t_{K+1}} = (\bm{0}_{1:t_K},\bm{1}_{t_K+1:T})\}}{\sum_{i \in [N]} \mathbbm{1}\{\bm{X}_i = \bm{x}, \bm{S}_{i(-t_K)} = \bm{s}, \bm{W}_{i,-t_K+1:\Delta t_{K+1}} = (\bm{0}_{1:t_K},\bm{1}_{t_K+1:T})\}}. \end{align*} Under Assumptions (ref), such an estimator is unbiased for the surrogate index function. Note that, such an estimator above uses data from the treatment subjects only, because of the part $\bm{W}_{i,-t_K+1:\Delta t_{K+1}} = (\bm{0}_{1:t_K},\bm{1}_{t_K+1:T})$ in the indicators. In words, $\bm{W}_{i,-t_K+1:\Delta t_{K+1}} = (\bm{0}_{1:t_K},\bm{1}_{t_K+1:T})$ refers to the $t_K$ treatment assignments right before the experimental periods (which are all controls), and the next $\Delta t_{K+1}$ treatment assignments in the experimental periods (which are all treatments). Such an estimator takes the advantage of the treatment subjects having received consecutive controls during the observational periods. Finally, we estimate the distributions of the conditional surrogate outcomes in (ref). Borrowing ideas from Markov Chains, for any $k\in[K]$, $\bm{x} \in \mathbb{X}$, $\bm{s} \in \mathbb{S}$, and $\mathcal{S} \subseteq \mathbb{S}$, one naive estimator of the transition kernel is given by \begin{align*} \widehat{p}_{t_k}(\bm{s}, \mathcal{S}) = \frac{\sum_{i \in [N]} \mathbbm{1}\{\bm{X}_i = \bm{x}, \bm{S}_{i(-t_{k-1})} = \bm{s}, \bm{W}_{i,-t_{k-1}+1:\Delta t_k} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}), \bm{S}_{it_k} \in \mathcal{S}\}}{\sum_{i \in [N]} \mathbbm{1}\{\bm{X}_i = \bm{x}, \bm{S}_{i(-t_{k-1})} = \bm{s}, \bm{W}_{i,-t_{k-1}+1:\Delta t_k} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k})\}}. \end{align*} Similar to the surrogate index estimator under $t_K$ controls followed by $\Delta t_{K+1}$ treatments, this estimator of the probability also takes the advantage of the treatment subjects having received consecutive controls during the observational periods. \subsection{Empirical Results} We apply the linear surrogate model in Experiment 1 to illustrate the practical performance. The result is shown in Figure (ref). We observe that the linear additive model tends to overestimate the treatment effect over extended periods, which indicates the potential violation of Assumption (ref) in this experiment. Therefore, we advise against using this approach in common settings. \begin{figure}[tb] \caption{Estimated effect for 7-week treatment with $T_E$ weeks observed data under the Linear Additive Model for Experiment 1} { Note: Grey dashed curves represent the true average treatment effect on search_uv from week 1 to week 7. Solid red curves represent the estimated effects with the linear additive model. Shadows indicate 95% confidence intervals. The three panels represent the scenarios when we use the first $T_E$ weeks as the experimental period and the last $T_F$ weeks as the future period.} \end{figure} \section{Missing Algorithms for Inference and Testing} In Section (ref) we discuss procedures to test the sharp null hypothesis in Eq. (ref) and calculate the $p$-value, and (2) calculate the variance and obtain confidence intervals. Now we provide the detailed description of our algorithms for conducting the permutation test, and conducting the bootstrap. \begin{algorithm}[!htb] \begin{algorithmic}[1] \caption{Algorithm for testing the sharp-null hypothesis Eq. (ref)} \REQUIRE Fix $M$, total number of samples drawn. \FOR{$m$ in $1:M$} \STATE For all subjects $i \in [N]$, sample treatment assignments $\bm{W}_{i,1:T}^{[m]}$ according to $\pi(\bm{X}_i)$. \STATE Hold $(Y_{it}, \bm{S}_{it})$ unchanged. Compute $\widehat{\tau}_T^{[m]}$ according to one of the estimation strategies (ref), (ref), or (ref). More specifically, estimate $\widehat{h}^{[m]}_t(\cdot, \cdot, \cdot)$, $\widehat{\bm{g}}^{[m]}_t(\cdot, \cdot, \cdot)$, and $\widehat{\bm{G}}^{[m]}_t(\cdot, \cdot, \cdot)$ using $\bm{W}_{i,1:T}^{[m]}$ instead of $\bm{W}_{i,1:T}$. \ENDFOR \STATE Compute $\widehat{p} = M^{-1} \sum_{m=1}^M \mathbbm{1}\{|\widehat{\tau}_T^{[m]}| > | \widehat{\tau}_T|\}$ \RETURN $\widehat{p}$, the estimated $p$-value. For large $M$, this is exact. \end{algorithmic} \end{algorithm} The permutation test depends on the fact that, under the sharp null hypothesis of no treatment effect (ref), any treatment assignment $\bm{w}^{[m]}_{i,1:T}$ leads to the same primary and surrogate outcomes, for any subject $i\in[N]$ at any time $t\in[T]$. In particular, in Step 3, we assume the observed primary and surrogate outcomes remain unchanged. \begin{algorithm}[!htb] \begin{algorithmic}[1] \caption{Algorithm for bootstrap} \REQUIRE Fix $M$, total number of samples drawn. \FOR{$m$ in $1:M$} \STATE For all subjects $i \in [N]$, sample treatment assignments $\bm{W}_{i,1:T}^{[m]}$ according to $\pi(\bm{X}_i)$. \STATE Hold $(Y_{it}, \bm{S}_{it})$ unchanged. Compute $\widehat{\tau}_T^{[m]}$ according to one of the estimation strategies (ref), (ref), or (ref). More specifically, estimate $\widehat{h}^{[m]}_t(\cdot, \cdot, \cdot)$, $\widehat{\bm{g}}^{[m]}_t(\cdot, \cdot, \cdot)$, and $\widehat{\bm{G}}^{[m]}_t(\cdot, \cdot, \cdot)$ using $\bm{W}_{i,1:T}^{[m]}$ instead of $\bm{W}_{i,1:T}$. \ENDFOR \STATE Compute sample variance $\widehat{\mathrm{Var}}(\widehat{\tau}_T) = (M-1)^{-1} \sum_{m=1}^M \Big(\widehat{\tau}_T^{[m]} - M^{-1} \sum_{l=1}^M \widehat{\tau}_T^{[l]}\Big)^2$. \RETURN $\widehat{\mathrm{Var}}(\widehat{\tau}_T)$, the estimated variance. \end{algorithmic} \end{algorithm} The bootstrap procedure depends on that we conduct a completely randomized experiment. We acknowledge that, in practice, obtaining confidence intervals through this approach could be computationally challenging. One could parallelize the computation for different values of $m$ to obtain the results more efficiently. \section{Missing Statements and Proofs} \subsection{Proofs of Lemma (ref) and Theorem (ref)} Since Lemma (ref) is a special case of Theorem (ref), the proof of Theorem (ref) applies to Lemma (ref). However, we prove Lemma (ref) first as a separate proof for easier understanding. \proof{Proof of Lemma (ref).} We prove Lemma (ref) by definition. From the definition of the causal effect, \begin{align*} \tau = \mathbb{E}_{\mathcal{F}}\bigg[ Y_{iT}(\bm{1}_{T}) - Y_{iT}(\bm{0}_{T}) \bigg], \end{align*} we start from the first part $\mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T})\right]$. \begin{align} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T}) \right] = \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_T \right] = \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_T, \bm{x}_i, \bm{s}_{i0} \right] \Pr(\bm{x}_i, \bm{s}_{i0}). \end{align} Using the law of total expectation, \begin{multline*} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0} \right] \\ = \sum_{\bm{s} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{S}_{i T_E} = \bm{s}, \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0} \right] \cdot \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0}\right). \end{multline*} Note that, \begin{align*} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{S}_{i T_E} = \bm{s}, \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0} \right] = & \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{S}_{i T_E} = \bm{s}, \bm{W}_{i,T_E+1:T} = \bm{1}_{T_F}, \bm{x}_i \right] \\ = & \mathbb{E}_{\mathcal{F}}\left[Y_{i T_E} \bigg\vert \bm{S}_{i 0} = \bm{s}, \bm{W}_{i,1:T_E} = \bm{1}_{T_E}, \bm{x}_i \right], \end{align*} where the first equality is due to Assumption (ref) when $t=T_E, t'=T$; the second equality is due to Assumption (ref) when $t=T_E, t'=T, \delta=T_E=T_F$. Note also that, \begin{align*} \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0}\right) = \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,1:T_E} = \bm{1}_{T_E}, \bm{x}_i, \bm{s}_{i0}\right). \end{align*} Combining both parts, \begin{multline*} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0} \right] \\ = \sum_{\bm{s} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[Y_{i T_E} \bigg\vert \bm{S}_{i 0} = \bm{s}, \bm{W}_{i,1:T_E} = \bm{1}_{T_E}, \bm{x}_i \right] \cdot \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,1:T_E} = \bm{1}_{T_E}, \bm{x}_i, \bm{s}_{i0}\right). \end{multline*} Putting the above expression in (ref), \begin{align*} & \ \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T}) \right] \\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_T, \bm{x}_i, \bm{s}_{i0} \right] \Pr(\bm{x}_i, \bm{s}_{i0}) \\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[Y_{i T_E} \bigg\vert \bm{S}_{i 0} = \bm{s}, \bm{W}_{i,1:T_E} = \bm{1}_{T_E}, \bm{x}_i \right] \cdot \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,1:T_E} = \bm{1}_{T_E}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s} \in \mathbb{S}} h_{T_E}(\bm{s}, \bm{x}_i, \bm{1}_{T_E}) \cdot \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,1:T_E} = \bm{1}_{T_E}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[h_{T_E}\left(\bm{G}_{T_E}\left(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}\right), \bm{x}_i, \bm{1}_{T_E}\right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ = & \ \mathbb{E}_{\mathcal{F}} \left[ h_{T_E}\left( \bm{G}_{T_E}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{T_E}), \bm{X}_i, \bm{1}_{T_E} \right) \right] \end{align*} Next, we focus on the second part $\mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{0}_{T})\right]$. Similarly we have \begin{align*} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{0}_{T}) \right] = \mathbb{E}_{\mathcal{F}} \left[ h_{T_E}\left( \bm{G}_{T_E}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{T_E}), \bm{X}_i, \bm{0}_{T_E} \right) \right]. \end{align*} Combining both parts, \begin{align*} \tau_T = & \mathbb{E}_{\mathcal{F}} \left[ h_{T_E}\left( \bm{G}_{T_E}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{T_E}), \bm{X}_i, \bm{1}_{T_E} \right) \right] - \mathbb{E}_{\mathcal{F}} \left[ h_{T_E}\left( \bm{G}_{T_E}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{T_E}), \bm{X}_i, \bm{0}_{T_E} \right) \right], \end{align*} which finishes the proof. \halmos \endproof \proof{Proof of Theorem (ref).} We prove Theorem (ref) by definition. From the definition of the causal effect, \begin{align*} \tau = \mathbb{E}_{\mathcal{F}}\bigg[ Y_{iT}(\bm{1}_{T}) - Y_{iT}(\bm{0}_{T}) \bigg], \end{align*} and we start from the first part $\mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T})\right]$. \begin{align} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T}) \right] = \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_T \right] = \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_T, \bm{x}_i, \bm{s}_{i0} \right] \Pr(\bm{x}_i, \bm{s}_{i0}). \end{align} Using the law of total expectation, \begin{multline} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0} \right] \\ = \sum_{\bm{s}_{t_K} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{S}_{i t_K} = \bm{s}_{t_K}, \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0} \right] \cdot \Pr\left(\bm{S}_{i t_K} = \bm{s}_{t_K} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0}\right). \end{multline} Note that, \begin{align} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{S}_{i t_K} = \bm{s}_{t_K}, \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0} \right] = & \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{S}_{i t_K} = \bm{s}_{t_K}, \bm{W}_{i,t_K+1:T} = \bm{1}_{\Delta t_{K+1}}, \bm{x}_i \right]\nonumber \\ = & \mathbb{E}_{\mathcal{F}}\left[Y_{i \Delta t_{K+1}} \bigg\vert \bm{S}_{i 0} = \bm{s}_{t_K}, \bm{W}_{i,1:\Delta t_{K+1}} = \bm{1}_{\Delta t_{K+1}}, \bm{x}_i \right], \end{align} where the first equality is due to Assumption (ref) when $t=t_K, t'=T$; the second equality is due to Assumption (ref) when $t=t_K, t'=T, \delta=t_K$. Note also that, \begin{align} & \ \Pr\left(\bm{S}_{i t_K} = \bm{s}_{t_K} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0}\right) = \ \Pr\left(\bm{S}_{i t_K} = \bm{s}_{t_K} \bigg\vert \bm{W}_{i,1:t_K} = \bm{1}_{t_K}, \bm{x}_i, \bm{s}_{i0}\right) \nonumber \\ = & \sum_{\bm{s}_{t_{K-1}} \in \mathbb{S}} \Pr\left(\bm{S}_{i t_K} = \bm{s}_{t_K} \bigg\vert \bm{W}_{i,1:t_K} = \bm{1}_{t_K}, \bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr\left(\bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}} \bigg\vert \bm{W}_{i,1:t_{K-1}} = \bm{1}_{t_{K-1}}, \bm{x}_i, \bm{s}_{i0}\right) \nonumber \\ = & \ldots \nonumber \\ = & \sum_{\bm{s}_{t_{K-1}}, \bm{s}_{t_{K-2}}, ..., \bm{s}_{t_{2}}, \bm{s}_{t_1} \in \mathbb{S}} \Pr\left(\bm{S}_{i t_K} = \bm{s}_{t_K} \bigg\vert \bm{W}_{i,1:t_K} = \bm{1}_{t_K}, \bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \nonumber \\ & \Pr\left(\bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}} \bigg\vert \bm{W}_{i,1:t_{K-1}} = \bm{1}_{t_{K-1}}, \bm{S}_{i t_{K-2}} = \bm{s}_{t_{K-2}}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \nonumber \\ & \ldots \cdot \Pr\left(\bm{S}_{i t_2} = \bm{s}_{t_2} \bigg\vert \bm{W}_{i,1:t_2} = \bm{1}_{t_2}, \bm{S}_{i t_{1}} = \bm{s}_{t_{1}}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr\left(\bm{S}_{i t_{1}} = \bm{s}_{t_{1}} \bigg\vert \bm{W}_{i,1:t_{1}} = \bm{1}_{t_{1}}, \bm{x}_i, \bm{s}_{i0}\right) \nonumber \\ = & \sum_{\bm{s}_{t_{K-1}}, \bm{s}_{t_{K-2}}, ..., \bm{s}_{t_{2}}, \bm{s}_{t_1} \in \mathbb{S}} \Pr\left(\bm{S}_{i t_K} = \bm{s}_{t_K} \bigg\vert \bm{W}_{i,t_{K-1}+1:t_K} = \bm{1}_{\Delta t_K}, \bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}}, \bm{x}_i\right) \cdot \nonumber \\ & \Pr\left(\bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}} \bigg\vert \bm{W}_{i,t_{K-2}+1:t_{K-1}} = \bm{1}_{\Delta t_{K-1}}, \bm{S}_{i t_{K-2}} = \bm{s}_{t_{K-2}}, \bm{x}_i \right) \cdot \nonumber \\ & \ldots \cdot \Pr\left(\bm{S}_{i t_2} = \bm{s}_{t_2} \bigg\vert \bm{W}_{i,t_1+1:t_2} = \bm{1}_{\Delta t_2}, \bm{S}_{i t_{1}} = \bm{s}_{t_{1}}, \bm{x}_i \right) \cdot \Pr\left(\bm{S}_{i t_{1}} = \bm{s}_{t_{1}} \bigg\vert \bm{W}_{i,1:t_{1}} = \bm{1}_{\Delta t_{1}}, \bm{x}_i, \bm{s}_{i0}\right), \end{align} where the last equality is due to Assumption (ref). Putting the above expressions (ref)--(ref) into (ref), \begin{align*} & \ \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T}) \right] \\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_T, \bm{x}_i, \bm{s}_{i0} \right] \Pr(\bm{x}_i, \bm{s}_{i0}) \\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s}_{t_K} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[Y_{i \Delta t_{K+1}} \bigg\vert \bm{S}_{i 0} = \bm{s}_{t_K}, \bm{W}_{i,1:\Delta t_{K+1}} = \bm{1}_{\Delta t_{K+1}}, \bm{x}_i \right] \cdot \Pr\left(\bm{S}_{i t_K} = \bm{s}_{t_K} \bigg\vert \bm{W}_{i,1:t_K} = \bm{1}_{t_K}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s}_{t_K} \in \mathbb{S}} h_{\Delta t_{K+1}}(\bm{s}_{t_K}, \bm{x}_i, \bm{1}_{\Delta t_{K+1}}) \cdot \Pr\left(\bm{S}_{i t_K} = \bm{s}_{t_K} \bigg\vert \bm{W}_{i,1:t_K} = \bm{1}_{t_K}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s}_{t_K}, ..., \bm{s}_{t_1} \in \mathbb{S}} h_{\Delta t_{K+1}}(\bm{s}_{t_K}, \bm{x}_i, \bm{1}_{\Delta t_{K+1}}) \cdot \Pr\left(\bm{S}_{i t_K} = \bm{s}_{t_K} \bigg\vert \bm{W}_{i,t_{K-1}+1:t_K} = \bm{1}_{\Delta t_K}, \bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}}, \bm{x}_i\right) \cdot \\ & \Pr\left(\bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}} \bigg\vert \bm{W}_{i,t_{K-2}+1:t_{K-1}} = \bm{1}_{\Delta t_{K-1}}, \bm{S}_{i t_{K-2}} = \bm{s}_{t_{K-2}}, \bm{x}_i \right) \cdot \\ & ... \cdot \Pr\left(\bm{S}_{i t_{1}} = \bm{s}_{t_{1}} \bigg\vert \bm{W}_{i,1:t_{1}} = \bm{1}_{\Delta t_{1}}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h_{\Delta t_{K+1}}\left( \bm{G}_{\Delta t_K}( ... \bm{G}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \bigg\vert \bm{x}_i, \bm{s}_{i0} \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ = & \ \mathbb{E}_{\mathcal{F}} \left[ h_{\Delta t_{K+1}}\left( \bm{G}_{\Delta t_K}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K}), \bm{X}_i, \bm{1}_{\Delta t_{K+1}} \right) \right], \end{align*} where the second equality is plugging in (ref)--(ref); the third equality is using Definition (ref); the fourth equality is plugging in (ref). Next, we focus on the second part $\mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{0}_{T})\right]$. Similarly, we have \begin{align*} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{0}_{T}) \right] = \mathbb{E}_{\mathcal{F}} \left[ h_{\Delta t_{K+1}}\left( \bm{G}_{\Delta t_K}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{\Delta t_1}) ... ,\bm{X}_i, \bm{0}_{\Delta t_K}), \bm{X}_i, \bm{0}_{\Delta t_{K+1}} \right) \right]. \end{align*} Combining both parts, \begin{multline*} \tau_T = \mathbb{E}_{\mathcal{F}} \left[ h_{\Delta t_{K+1}}\left( \bm{G}_{\Delta t_K}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K}), \bm{X}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \\ - \mathbb{E}_{\mathcal{F}} \left[ h_{\Delta t_{K+1}}\left( \bm{G}_{\Delta t_K}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{\Delta t_1}) ... ,\bm{X}_i, \bm{0}_{\Delta t_K}), \bm{X}_i, \bm{0}_{\Delta t_{K+1}} \right) \right]. \end{multline*} which finishes the proof. \halmos \endproof \subsection{Proof of Theorem (ref)} \proof{Proof of Theorem (ref).} Under Assumptions (ref) and (ref), Theorem (ref) yields \begin{multline*} \tau_T = \mathbb{E}_{\mathcal{F}} \left[ h_{\Delta t_{K+1}}\left( \bm{G}_{\Delta t_K}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K}), \bm{X}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \\ - \mathbb{E}_{\mathcal{F}} \left[ h_{\Delta t_{K+1}}\left( \bm{G}_{\Delta t_K}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{\Delta t_1}) ... ,\bm{X}_i, \bm{0}_{\Delta t_K}), \bm{X}_i, \bm{0}_{\Delta t_{K+1}} \right) \right]. \end{multline*} We start from the first part \begin{align*} \mathbb{E}_{\mathcal{F}} \left[ h_{\Delta t_{K+1}}\left( \bm{G}_{\Delta t_K}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K}), \bm{X}_i, \bm{1}_{\Delta t_{K+1}} \right) \right]. \end{align*} Denote $\bm{G}_{\Delta t_K} = (G_{\Delta t_K, 1}, ..., G_{\Delta t_K, D})$ and $\bm{g}_{\Delta t_K} = (g_{\Delta t_K, 1}, ..., g_{\Delta t_K, D})$. \begin{align} & \mathbb{E}_{\mathcal{F}} \left[h_{\Delta t_{K+1}}(\bm{G}_{\Delta t_K}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K}), \bm{X}_i, \bm{1}_{\Delta t_{K+1}})\right] \nonumber \\ = & \mathbb{E}_{\mathcal{F}} \left[ \alpha_0(\bm{X}_i, \bm{1}_{\Delta t_{K+1}}) + \sum_{d=1}^D G_{\Delta t_K, d}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K}) \cdot \alpha_d(\bm{X}_i, \bm{1}_{\Delta t_{K+1}})\right] \nonumber \\ = & \alpha_0(\bm{X}_i, \bm{1}_{\Delta t_{K+1}}) + \sum_{d=1}^D \mathbb{E}_{\mathcal{F}} \left[ G_{\Delta t_K, d}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K})\right] \cdot \alpha_d(\bm{X}_i, \bm{1}_{\Delta t_{K+1}}) \nonumber \\ = & \alpha_0(\bm{X}_i, \bm{1}_{\Delta t_{K+1}}) + \sum_{d=1}^D \mathbb{E}_{\mathcal{F}} \left[ g_{\Delta t_K, d}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K})\right] \cdot \alpha_d(\bm{X}_i, \bm{1}_{\Delta t_{K+1}}) \nonumber \\ = & h_{\Delta t_{K+1}}\left(\mathbb{E}_{\mathcal{F}} \left[\bm{g}_{\Delta t_K}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K}), \bm{X}_i, \bm{1}_{\Delta t_{K+1}}\right]\right), \end{align} where the first equality is due to Assumption (ref); the second equality is due to linearity of expectation; in the third equality, we write the expectation of $G_{\Delta t_K, d}$ as $g_{\Delta t_K, d}$, but there is still uncertainty from $\bm{G}_{\Delta t_{K-1}, d}$, which keeps the expectation notion. Next, denote $\bm{G}_{\Delta t_{K-1}} = (G_{\Delta t_{K-1}, 1}, ..., G_{\Delta t_{K-1}, D})$ and $\bm{g}_{\Delta t_{K-1}} = (g_{\Delta t_{K-1}, 1}, ..., g_{\Delta t_{K-1}, D})$. For each $d \in [D]$, we focus on $g_{\Delta t_K, d}$ as follows, \begin{align*} & \mathbb{E}_{\mathcal{F}} \left[g_{\Delta t_{K},d}(\bm{G}_{\Delta t_{K-1}}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_{K-1}}), \bm{X}_i, \bm{1}_{\Delta t_K})\right] \\ = & \mathbb{E}_{\mathcal{F}} \left[ \beta_{d,0}(\bm{X}_i, \bm{1}_{\Delta t_{K}}) + \sum_{d'=1}^D G_{\Delta t_{K-1}, d'}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_{K-1}}) \cdot \beta_{d,d'}(\bm{X}_i, \bm{1}_{\Delta t_{K}})\right] \\ = & \beta_{d,0}(\bm{X}_i, \bm{1}_{\Delta t_{K}}) + \sum_{d'=1}^D \mathbb{E}_{\mathcal{F}} \left[G_{\Delta t_{K-1}, d'}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_{K-1}})\right] \cdot \beta_{d,d'}(\bm{X}_i, \bm{1}_{\Delta t_{K}}) \\ = & \beta_{d,0}(\bm{X}_i, \bm{1}_{\Delta t_{K}}) + \sum_{d'=1}^D \mathbb{E}_{\mathcal{F}} \left[g_{\Delta t_{K-1}, d'}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_{K-1}})\right] \cdot \beta_{d,d'}(\bm{X}_i, \bm{1}_{\Delta t_{K}}) \\ = & g_{\Delta t_{K}, d}\left(\mathbb{E}_{\mathcal{F}} \left[\bm{g}_{\Delta t_{K-1}}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_{K-1}}), \bm{X}_i, \bm{1}_{\Delta t_{K}}\right)\right]. \end{align*} Collecting the above equality by vector form, \begin{multline} \mathbb{E}_{\mathcal{F}} \left[\bm{g}_{\Delta t_{K}}(\bm{G}_{\Delta t_{K-1}}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_{K-1}}), \bm{X}_i, \bm{1}_{\Delta t_K})\right] = \\ \bm{g}_{\Delta t_{K}}\left(\mathbb{E}_{\mathcal{F}} \left[\bm{g}_{\Delta t_{K-1}}( ... \bm{G}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_{K-1}}), \bm{X}_i, \bm{1}_{\Delta t_{K}}\right)\right]. \end{multline} Iteratively applying (ref) and combining with (ref) finishes the proof. \halmos \endproof \subsection{Proof of Theorem (ref)} \proof{Proof of Theorem (ref).} For each $k \in \{1,2,...,K+1\}$, we start with the first part. \begin{align} & \mathbb{E}_{\mathcal{F}}\bigg[ Y_{iT} (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}, \bm{0}_{t_k+1:T})\bigg] \nonumber \\ = & \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}, \bm{0}_{t_k+1:T}) \right] \nonumber \\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}, \bm{0}_{t_k+1:T}), \bm{x}_i, \bm{s}_{i0} \right] \Pr(\bm{x}_i, \bm{s}_{i0}). \end{align} Using the law of total expectation, \begin{multline*} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}, \bm{0}_{t_k+1:T}), \bm{x}_i, \bm{s}_{i0} \right] \\ = \sum_{\bm{s} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{S}_{i T_E} = \bm{s}, \bm{W}_{i,1:T} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}, \bm{0}_{t_k+1:T}), \bm{x}_i, \bm{s}_{i0} \right] \\ \cdot \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,1:T} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}, \bm{0}_{t_k+1:T}), \bm{x}_i, \bm{s}_{i0}\right). \end{multline*} Note that, \begin{align*} & \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{S}_{i T_E} = \bm{s}, \bm{W}_{i,1:T} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}, \bm{0}_{t_k+1:T}), \bm{x}_i, \bm{s}_{i0} \right] \\ = & \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{S}_{i t_k} = \bm{s}, \bm{W}_{i,t_k+1:T} = \bm{0}_{T-t_k}, \bm{x}_i \right] \\ = & \mathbb{E}_{\mathcal{F}}\left[Y_{i (T-t_k)} \bigg\vert \bm{S}_{i 0} = \bm{s}, \bm{W}_{i,1:T-t_k} = \bm{0}_{T-t_k}, \bm{x}_i \right], \end{align*} where the first equality is due to Assumption (ref) when $t=t_k, t'=T$; the second equality is due to Assumption (ref) when $t=t_k, t'=T, \delta=t_k$. Also note that, \begin{align*} & \Pr\left(\bm{S}_{i t_k} = \bm{s} \bigg\vert \bm{W}_{i,1:T} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}, \bm{0}_{t_k+1:T}), \bm{x}_i, \bm{s}_{i0}\right) \\ = & \Pr\left(\bm{S}_{i t_k} = \bm{s} \bigg\vert \bm{W}_{i,1:t_k} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}), \bm{x}_i, \bm{s}_{i0}\right). \end{align*} Combining both parts, \begin{multline*} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}, \bm{0}_{t_k+1:T}), \bm{x}_i, \bm{s}_{i0} \right] \\ = \sum_{\bm{s} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[Y_{i (T-t_k)} \bigg\vert \bm{S}_{i 0} = \bm{s}, \bm{W}_{i,1:T-t_k} = \bm{0}_{T-t_k}, \bm{x}_i \right] \cdot \Pr\left(\bm{S}_{i t_k} = \bm{s} \bigg\vert \bm{W}_{i,1:t_k} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}), \bm{x}_i, \bm{s}_{i0}\right). \end{multline*} Putting the above expression in (ref), \begin{align*} & \ \mathbb{E}_{\mathcal{F}}\bigg[ Y_{iT} (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}, \bm{0}_{t_k+1:T})\bigg] \\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}, \bm{0}_{t_k+1:T}), \bm{x}_i, \bm{s}_{i0} \right] \Pr(\bm{x}_i, \bm{s}_{i0}) \\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[Y_{i (T-t_k)} \bigg\vert \bm{S}_{i 0} = \bm{s}, \bm{W}_{i,1:T-t_k} = \bm{0}_{T-t_k}, \bm{x}_i \right] \\ & \cdot \Pr\left(\bm{S}_{i t_k} = \bm{s} \bigg\vert \bm{W}_{i,1:t_k} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}), \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s} \in \mathbb{S}} h_{T-t_k}(\bm{s}, \bm{x}_i, \bm{0}_{T-t_k}) \cdot \Pr\left(\bm{S}_{i t_k} = \bm{s} \bigg\vert \bm{W}_{i,1:t_k} = (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k}), \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[h_{T-t_k}\left(\bm{G}_{t_k}\left(\bm{s}_{i0}, \bm{x}_i, (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k})\right), \bm{x}_i, \bm{0}_{T-t_k}\right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ = & \ \mathbb{E}_{\mathcal{F}} \left[ h_{T-t_k}\left( \bm{G}_{t_k}\left(\bm{S}_{i0}, \bm{X}_i, (\bm{0}_{1:t_{k-1}}, \bm{1}_{t_{k-1}+1:t_k})\right), \bm{X}_i, \bm{0}_{T-t_k} \right) \right] \end{align*} Next, we focus on the second part $\mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{0}_{T})\right]$. \begin{multline*} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{0}_{T}) \right] = \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = \bm{0}_T \right] \\ = \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = \bm{0}_T, \bm{x}_i, \bm{s}_{i0} \right] \Pr(\bm{x}_i, \bm{s}_{i0}) = \mathbb{E}_{\mathcal{F}}\left[h_{T}\left(\bm{S}_{i0},\bm{X}_i,\bm{0}_T\right)\right]. \end{multline*} Combining both parts we finish the proof. \halmos \endproof \subsection{Statement and Proof of Lemma (ref)} \begin{lemma} Under Assumption (ref), assume the coefficients $\widehat{\alpha}_{d}, \forall d \in \{0,1,...,D\}$ and $\widehat{\beta}_{d,d'}, \forall d \in [D], d' \in \{0,1,...,D\}$ are consistently estimated, that is, for any $\bm{x} \in \mathbb{X}, \bm{w}_{1:t} \in \{\bm{0}_t, \bm{1}_t\}$, \begin{align*} \Pr\Big( \lim_{N \to \infty} \widehat{\alpha}_{d}^{(N)}(\bm{x}, \bm{w}_{1:t}) = \alpha_{d}(\bm{x}, \bm{w}_{1:t}) \Big) & = 1, & \Pr\Big( \lim_{N \to \infty} \widehat{\beta}_{d,d'}^{(N)}(\bm{x}, \bm{w}_{1:t}) = \beta_{d,d'}(\bm{x}, \bm{w}_{1:t}) \Big) & = 1, \end{align*} where we use the superscript $(N)$ to stand for the dependence on sample size $N$. Then, $\widehat{\tau}_T$ as defined in (ref) is a consistent estimator of $\tau_T$ as defined in (ref), that is, \begin{align*} \Pr\Big( \lim_{N \to \infty} \widehat{\tau}_{T}^{(N)} = \tau_{T} \Big) & = 1. \end{align*} \end{lemma} \proof{Proof of Lemma (ref).} Note that, with probability one, $\lim_{N \to \infty} \widehat{\beta}_{d,d'}^{(N)}(\bm{x}, \bm{w}_{1:t}) = \beta_{d,d'}(\bm{x}, \bm{w}_{1:t})$. This implies that, for any $\bm{s} \in \mathbb{S}, \bm{x} \in \mathbb{X}, \bm{w}_{1:t} \in \{\bm{0}_t, \bm{1}_t\}$, and $d\in[D]$, with probability one, \begin{multline*} \lim_{N \to \infty} \widehat{g}^{(N)}_{t,d}(\bm{s}, \bm{x}, \bm{w}_{1:t}) = \lim_{N \to \infty} \Big( \widehat{\beta}^{(N)}_{d,0}(\bm{x}, \bm{w}_{1:t}) + \sum_{d'=1}^D s_d \cdot \widehat{\beta}^{(N)}_{d,d'}(\bm{x}, \bm{w}_{1:t}) \Big) \\ = \beta_{d,0}(\bm{x}, \bm{w}_{1:t}) + \sum_{d'=1}^D s_d \cdot \beta_{d,d'}(\bm{x}, \bm{w}_{1:t}) = g_{t,d}(\bm{s}, \bm{x}, \bm{w}_{1:t}). \end{multline*} Furthermore, for any $t$ and $t'$, with probability one, \begin{multline*} \lim_{N \to \infty} \widehat{g}^{(N)}_{t',d}(\widehat{g}^{(N)}_{t,d}(\bm{s}, \bm{x}, \bm{w}_{1:t}), \bm{x}, \bm{w}_{1:t'}) = \lim_{N \to \infty} \Big( \widehat{\beta}^{(N)}_{d,0}(\bm{x}, \bm{w}_{1:t'}) + \sum_{d'=1}^D \widehat{g}^{(N)}_{t,d}(\bm{s}, \bm{x}, \bm{w}_{1:t}) \cdot \widehat{\beta}^{(N)}_{d,d'}(\bm{x}, \bm{w}_{1:t'}) \Big) \\ = \beta_{d,0}(\bm{x}, \bm{w}_{1:t'}) + \sum_{d'=1}^D g_{t,d'}(\bm{s}, \bm{x}, \bm{w}_{1:t}) \cdot \beta_{d,d'}(\bm{x}, \bm{w}_{1:t'}) = g_{t',d}(\bm{g}_{t}(\bm{s}, \bm{x}, \bm{w}_{1:t}), \bm{x}, \bm{w}_{1:t'}). \end{multline*} where the second equality is due to Slutsky's theorem. Similarly, we can establish the above equality for surrogate index function $h_t(\cdot, \cdot, \cdot)$. In addition, using the fact that all units are i.i.d. sampled, we know that with probability one, \begin{align*} & \lim_{N \to \infty} \widehat{\tau}^{(N)}_T \\ = & \frac{1}{N_1} \lim_{N \to \infty} \sum_{i\in[N]} \mathbbm{1}\{\bm{W}_{i,1:T_E} = \bm{1}_{T_E}\} h_{\Delta t_{K+1}}\left( \bm{g}_{\Delta t_K}( ... \bm{g}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K}), \bm{X}_i, \bm{1}_{\Delta t_{K+1}} \right) \\ & - \frac{1}{N_0} \lim_{N \to \infty} \sum_{i\in[N]} \mathbbm{1}\{\bm{W}_{i,1:T_E} = \bm{0}_{T_E}\} h_{\Delta t_{K+1}}\left( \bm{g}_{\Delta t_K}( ... \bm{g}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{\Delta t_1}) ... ,\bm{X}_i, \bm{0}_{\Delta t_K}), \bm{X}_i, \bm{0}_{\Delta t_{K+1}} \right) \\ = & \mathbb{E}_\mathcal{F}\left[ h_{\Delta t_{K+1}}\left( \bm{g}_{\Delta t_K}( ... \bm{g}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K}), \bm{X}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \\ & - \mathbb{E}_\mathcal{F}\left[ h_{\Delta t_{K+1}}\left( \bm{g}_{\Delta t_K}( ... \bm{g}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{\Delta t_1}) ... ,\bm{X}_i, \bm{0}_{\Delta t_K}), \bm{X}_i, \bm{0}_{\Delta t_{K+1}} \right) \right] \\ = & \tau_T, \end{align*} where the second equality is due to Theorem (ref). \halmos \endproof \section{Practical Guidelines} In this section, we provide additional insights on how to apply our methodology in real-world settings—specifically, the practice of A/B testing in companies—more effectively. First, choosing surrogates is the critical decision to be made in applying our method. In practice, even hundreds of metrics are measured and collected for each experiment. Therefore, how to choose surrogates that better satisfy Assumption (ref) --- the longitudinal surrogacy assumption --- can be a challenge. In particular, the assumption is not testable. In the selection of surrogates for longitudinal studies, it is essential to choose metrics that are highly responsive to treatments and reflect the diverse aspects of user behavior that lead to variations in primary outcomes over time duan2021online,deng2013improving. Surrogates should capture both up-stream behaviors (like search queries and navigation patterns) and down-stream behaviors (such as post-click actions including purchases or sharing), as these can robustly predict future primary outcomes due to their close alignment with user intentions and deeper engagement respectively. In our real-world experiments, we mostly use down-stream behaviors as surrogates and demonstrated their effectiveness. The current primary outcome itself also serves as a strong predictive surrogate. To streamline surrogate selection in A/B testing, creating a library of surrogates based on metrics from prior experiments can considerably enhance efficiency and consistency across studies. This repository approach not only saves time and effort but also ensures the reliability of surrogate effectiveness across different contexts. However, selecting an appropriate number of surrogates is crucial; too few may not adequately capture the causal links necessary for reliable estimates, leading to potential biases, while too many can cause overfitting due to the curse of dimensionality. Techniques such as elastic net regularization are recommended to manage the number of surrogates by reducing dimensionality and focusing on the most predictive metrics, thus balancing the need for comprehensive data representation against the risks of overfitting. See detailed discussion in Appendix (ref). Additionally, in our main framework, the linear surrogate model, which extends the basic longitudinal surrogate model by incorporating a linearity assumption, generally outperforms non-linear models such as nearest neighbors, random forests, or neural networks in predicting future treatment effects. Our empirical tests, including those using the $k$-nearest neighbors (kNN) model, reveal that while non-linear models can fit within the confidence bounds of the treatment's actual long-term effect, they tend to produce unaffordably large variance and consequently, larger mean squared errors (MSE) compared to linear models. This is further evidenced by our tests with a relatively large $k$ (e.g., $k=20$) in the kNN model, which, despite its computational efficiency in finding close matches in the surrogate space, still results in higher variance and MSE. Hence, we recommend using linear models over non-linear approaches for treatment effect estimation, as they provide more reliable and precise estimates with smaller confidence intervals, as outlined in Appendix (ref). \section{More Details for Empirical Experiment Results} Here we provide a number of additional experimental results mentioned in the main text. \subsection{Description of Surrogates} The detailed descriptions of surrogates and primary outcome we used in both empirical experiments is illustrated in Table (ref). \begin{table}[h] \scriptsize{ \caption{The detailed descriptions of surrogates and primary outcome in Experiments 1 and 2} \begin{tabularx}{\textwidth}{@llX@} \toprule \multicolumn{1}{l}{Variable}&\multicolumn{1}{l}{\textbf{Role}}&\multicolumn{1}{l}{\textbf{Description}}\\ \midrule \textit{search_uv} & {Primary outcome, Surrogate} & A float number representing the average number of days that the user has searched in a week, normalized by being divided by 7\\ \textit{search_qv} & Surrogate & A float number representing the average number of times that the user has searched in a week, normalized by being divided by 7\\ \textit{recall_qv} & Surrogate & A float number representing the average number of times of search with results that the user has made in a week, normalized by being divided by 7.\\ \textit{expose_qv} & Surrogate & A float number representing the average number of times of search with results that have been exposed to the user in a week, normalized by being divided by 7.\\ \textit{click_qv} & Surrogate & A float number representing the average number of times of search with results that have been clicked by the user in a week, normalized by being divided by 7.\\ \bottomrule \end{tabularx}} \end{table} \subsection{Randomization Check} To illustrate the sample balance in our empirical experiments, we conducted the Sample Ratio Mismatch (SRM) fabijan2019diagnosing test between the treatment and control groups and provided summary statistics on pre-treatment variables for both experiments. In Experiment 1, the treatment group includes 667,206 users, while the control group includes 665,830 users. Given the expected ratio of 50% for both groups, the experiment passes the chi-squared test with a statistic of 1.420 and a $p$-value of 0.233. This result indicates no significant difference between the observed and expected group sizes, suggesting no sample ratio mismatch problems. In Experiment 2, the treatment group includes 1,807,335 users, while the control group includes 1,803,675 users, with the expected ratio again being 50% for both groups. The chi-squared test result for Experiment 2 shows a statistic of 3.710 and a $p$-value of 0.054, also implying no sample ratio mismatch problems. Table (ref) presents the summary statistics for several pre-treatment variables across the treatment and control groups for both empirical experiments. These pre-treatment variables hold the same meaning as the surrogate variables used in the model but are collected before the experiment started. For Experiment 1, each variable records its average value over the 7 weeks prior to the experiment, while for Experiment 2, the average value is recorded over the 20 weeks before the experiment began. The table shows that the treatment and control groups are essentially balanced on these pre-treatment variables, indicating the randomness of both empirical experiments. \begin{table}[htbp] \caption{Summary statistics for two empirical experiments across treatment and control groups}\tiny \begin{tabular}{cccccccccccccc} \toprule & & \multicolumn{5}{c}{\textbf{Treatment}} & \multicolumn{5}{c}{\textbf{Control}} & \multicolumn{2}{c}{\textbf{Compare Means}}\\ \cmidrule(lr){3-7} \cmidrule(lr){8-12} \cmidrule(lr){13-14} \textbf{Expt.} & \shortstack{\textbf{Pre-treatment} \\ \textbf{Variable}} & \textbf{Count} & \textbf{Mean} & \textbf{Std.} & \textbf{Max} & \textbf{Min} & \textbf{Count} & \textbf{Mean} & \textbf{Std.} & \textbf{Max} & \textbf{Min} & \textbf{t-statistic} & \textbf{p-value}\\ \midrule \multirow{5}{*}{Exp 1} & \textit{search_uv} & \multirow{5}{*}{667,206} & 0.146 & 0.161 & 1.000 & 0.000 & \multirow{5}{*}{665,830} & 0.146 & 0.161 & 1.000 & 0.000 & 0.288 & 0.773\\ & \textit{search_qv} & & 0.369 & 0.936 & 127.918 & 0.000 & & 0.370 & 0.941 & 135.020 & 0.000 & 0.918 & 0.359\\ & \textit{recall_qv} & & 0.363 & 0.926 & 127.653 & 0.000 & & 0.364 & 0.932 & 131.612 & 0.000 & 0.954 & 0.340\\ & \textit{expose_qv} & & 0.347 & 0.915 & 127.306 & 0.000 & & 0.349 & 0.921 & 130.878 & 0.000 & 1.018 & 0.309\\ & \textit{click_qv} & & 0.226 & 0.629 & 75.612 & 0.000 & & 0.228 & 0.654 & 108.735 & 0.000 & 1.123 & 0.261\\ \midrule \multirow{5}{*}{Exp 2} & \textit{search_uv} & \multirow{5}{*}{1,807,335} & 0.117 & 0.134 & 0.979 & 0.000 & \multirow{5}{*}{1,803,675} & 0.117 & 0.135 & 0.979 & 0.000 & 0.366 & 0.714\\ & \textit{search_qv} & & 0.282 & 0.760 & 119.421 & 0.000 & & 0.283 & 0.766 & 162.084 & 0.000 & 0.660 & 0.509\\ & \textit{recall_qv} & & 0.277 & 0.749 & 119.214 & 0.000 & & 0.277 & 0.757 & 161.941 & 0.000 & 0.674 & 0.500\\ & \textit{expose_qv} & & 0.257 & 0.734 & 117.788 & 0.000 & & 0.258 & 0.742 & 161.427 & 0.000 & 0.737 & 0.461\\ & \textit{click_qv} & & 0.171 & 0.526 & 81.371 & 0.000 & & 0.172 & 0.535 & 122.720 & 0.000 & 1.025 & 0.305\\ \bottomrule \end{tabular} \end{table} \subsection{Different Choices of $T_E$ in Experiment 2} In the main text, for space reasons we only present the results with $T_E=8,9,10$. Here we provide additional results for the performance of the linear surrogate model given different choices for the duration of the experimental periods $T_E$. \begin{figure}[h!] \caption{Long-term effect estimation for Experiment 2 with $T_E=11,12,13$ for the linear surrogate model } { \textit{Note}: Grey dashed curves represent the true average treatment effect on \textit{search_uv} from week 1 to week 20. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The three panels represent the scenarios when we use the first $T_E$ weeks as the experimental period and the last $T_F$ weeks as the future period.} \end{figure} Here, we select $T_E=11, 12, 13$ and present the corresponding results in Figure (ref). As shown in the figures, our estimates still consistently outperform those of the baseline and effectively capture the decreasing trend of the treatment effects. \subsection{athey2019surrogate} As discussed in the main text, athey2019surrogate addresses a fundamentally different problem from the one we explore. The detailed comparison is listed in Table (ref). \begin{table}[ht] \scriptsize \begin{tabular}{p{0.15\linewidth}p{0.4\linewidth}p{0.4\linewidth}} \hline & \textbf{Our Work} & \textbf{Athey et al.} \\ \hline \textbf{Focus} & Long-term effects of \textbf{long-term treatments} & Long-term effects of \textbf{short-term treatments} \\[1em] \textbf{Motivation} & Evaluating the impact of persistent interventions over extended periods, common in A/B testing scenarios & Estimating long-term impacts when outcomes are not immediately observable \\[1em] \textbf{Technical Framework} & Sequential longitudinal framework accounting for interventions, surrogates, and outcomes at each period & Surrogate index built on historical data without consideration of multiple time periods \\[1em] \textbf{Estimation Approach} & Proposed three novel estimation strategies for long-term treatments & Surrogate index method using observed short-term outcomes to predict long-term effects \\[1em] \textbf{Type of Treatment} & Continuous or ongoing treatments & One-shot or short-term treatments \\[1em] \textbf{Applications} & Continuous healthcare management, ongoing software updates, etc. & Short-lived campaigns such as promotional events, educational workshops, marketing campaigns, policy changes \\[1em] \hline \end{tabular} \caption{Comparative Analysis between Our Work and Athey et al.} \end{table} In a nutshell, our paper focuses on the persistent long-term effects of treatments that extend beyond the observed experimental period into future periods. This involves analyzing both the “carryover effect”—the impact of treatments during the experimental periods as measured in subsequent future periods—and the “direct effect”—the impact of treatments during future periods measured concurrently. In contrast, the methodology presented by athey2019surrogate is specifically designed to capture the “carryover effect,” but does not account for the effects of treatments administered in future periods. This limitation arises from the method's inability to identify a set of surrogate variables that can capture the effects of future treatments from the perspective of current experimental periods. In other words, the approach described in athey2019surrogate is not appropriate for estimating these long-term effects due to its failure to meet the necessary surrogacy assumptions. Our paper addresses this gap by establishing a longitudinal surrogate model premised on the longitudinal surrogacy assumption. That said, we provide the estimation results of applying athey2019surrogate to Experiment 1 as an illustration. We present the estimation results in Figure (ref). We observe that the estimates by athey2019surrogate consistently underestimate the treatment effects, yielding values much lower than the ground truth. This discrepancy can be explained by the approach's exclusion of the direct treatment effects in the future period $T_F$. In this experiment, the effects during $T_F$ are positive, and their approach, which only considers the carryover effects from the previous treatment, tends to underestimate the treatment effects in $T_F$. \begin{figure}[h!] \caption{Estimated effect for 7-week treatment with $T_E$ weeks observed data under the traditional surrogate model (athey2019surrogate) for Experiment 1} { \textit{Note}: Grey dashed curves represent the true average treatment effect on \textit{search_uv} from week 1 to week 7. Solid blue curves represent the estimated effects with the traditional surrogate model introduced in athey2019surrogate. Shadows indicate 95% confidence intervals. The three panels represent the scenarios when we use the first $T_E$ weeks as the experimental period and the last $T_F$ weeks as the future period.} \end{figure} We include the following toy example to further illustrate the difference between our method and Athey's method. \begin{example}[Athey's method cannot handle consistent treatment effect] Consider a simple setting with no covariate $\bm{X_i}$ for subject $i$ involved. The objective is to estimate the treatment effects on the primary outcome $Y_i$, where the total treatment duration spans $T = T_E + T_F$ time periods, with data collected from periods $1$ to $T_E$. Suppose that $T_E = 2\cdot T_F$, and the relationship between the primary outcome $Y_i$ and the surrogate variable $S_i$ can be functionally expressed as \begin{equation*} \begin{cases} & Y_{iT}(\bm{1}_{T_E}, \bm{0}_{T_F}) = \beta_0 + \beta_1\cdot S_{iT_E}\\ & Y_{iT}(\bm{0}_{T_E}, \bm{1}_{T_F}) = \alpha\\ & Y_{iT}(\bm{1}_{T}) = \alpha + \beta_0 + \beta_1\cdot S_{iT_E}\\ & Y_{iT}(\bm{0}_{T}) = 0 \end{cases} \end{equation*} It is evident that $S_i$ fully captures the treatment effects on $Y_i$ during periods $1$ to $T_E$. Following the instruction proposed in athey2019surrogate, Athey’s method constructs a surrogate index $\hat{Y}_{iT}$ by leveraging historical data spanning $T$ periods based on the observed surrogate variable $S_{iT_E}$. By estimating a linear regression model using OLS, it derives $$\tilde{Y}_{iT} = \beta_0 + \beta_1\cdot S_{iT_E} + \epsilon_i $$ and the estimate of the treatment effect based on the surrogate index is then calculated as $$\hat{\tau}_T = \frac{1}{N_1}\sum^{N}_{i=1}\tilde{Y}_{iT}\bm{W}_i - \frac{1}{N_0}\sum^{N}_{i=1}\tilde{Y}_{iT}(1-\bm{W}_i)$$ which systematically underestimates the true effect with a bias of magnitude approximately $\alpha$. On the other hand, our method first discretize the $T$ periods into three intervals, $\Delta t_1,\Delta t_2$ and $\Delta t_3$, with each spanning $T_F$ periods. Leveraging data from the first $T_E$ ($\Delta t_1$ and $\Delta t_2$) periods, we construct a longitudinal surrogate model by fitting the following linear regression $$\hat{Y}_{iT_E} = \alpha + \beta_0 + \beta_1\cdot S_{i\Delta t_1} + e_i $$ Plug-in the estimation of $S_{iT_E}$ (the process is omitted) the $\hat{Y}_{iT}$ can be obtained as $$\hat{Y}_{iT} = \alpha + \beta_0 + \beta_1\cdot \hat{S}_{iT_E} + u_i$$ Using the same estimator of the treatment effect $\hat{\tau}_T$ stated above, we can see that the estimation with our longitudinal surrogate model is unbiased. \end{example} In summary, this result showcases that it is inappropriate to directly apply athey2019surrogate to our distinct setting. It is necessary to employ a longitudinal perspective and develop the assumptions and theoretical conclusions applied in our study. \subsection{Robustness Check} \subsubsection{Model Performance on Heterogeneous Groups} Although we showcase two experiments in our main results, we can partition the sample into several subgroups based on covariates and examine the performance of our approach when applied to each subgroup. Note that due to confidentiality concerns, we cannot leverage the demographic information of users involved in the experiment. Instead, we used the value of pre-treatment variables, (i.e., \textit{search_uv}), to categorize sample into different groups. We divided the users into five groups based on the [20, 40, 60, 80] percentiles of the average \textit{search_uv} value from seven days before the experiment began. Figures (ref) and (ref) present the estimated long-term treatment effect for each heterogeneous group in Experiment 1 and Experiment 2 respectively. The value in the last column, denoted as $X$, represents a subgroup ($X=0,1,...,4$ corresponding to 0-20%, 20-40%, 40-60%, 60-80%, and 80-100%). A larger value signifies a higher average of \textit{search_uv} during the pre-treatment period, indicating a more active user group. We also present the biases and MSEs for each subgroups in Tables (ref) and (ref). From these figures and tables, we observe that the estimation result closely aligns with the true effect across multiple groups, suggesting that our method exhibits considerable robustness. \begin{figure}[tb] \caption{Estimated effect using the linear surrogate model on heterogeneous groups for Experiment 1} { \textit{Note}: Grey dashed curves represent the true average treatment effect on \textit{search_uv} from week 1 to week 7. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The five rows correspond to specific heterogeneous user groups, with a larger value indicating a more active user group. The three panels represent the scenarios when we use the first $T_E$ weeks as the experimental period and the last $T_F$ weeks as the future period. } \end{figure} \begin{figure}[tb] \caption{Estimated effect using the linear surrogate model on heterogeneous groups for Experiment 2} { \textit{Note}: Grey dashed curves represent the true average treatment effect on \textit{search_uv} from week 1 to week 20. Solid red curves represent the estimated effects with the linear surrogate model. The five rows correspond to specific heterogeneous user groups, with a larger value indicating a more active user group. The three panels represent the scenarios when we use the first $T_E$ weeks as the experimental period and the last $T_F$ weeks as the future period. } \end{figure} \begin{table}[h!] \caption{Comparison result between different heterogeneous groups for Bias and MSE for Experiment 1} \begin{tabular}{c c c ccc ccc} \toprule \multirow{2}{*}{\textbf{Expt.}}&\multirow{2}{*}{\textbf{Groups}}&\multirow{2}{*}{\textbf{Method}}&\multicolumn{3}{c}{\ \textbf{Bias}}&\multicolumn{3}{c}{\textbf{MSE}}\\ & & & {$\mathbf{T_E=2}$} & {$\mathbf{T_E=3}$} & {$\mathbf{T_E=4}$} & {$\mathbf{T_E=2}$} & {$\mathbf{T_E=3}$} & {$\mathbf{T_E=4}$} \\ \midrule \multirow{10}{*}{Exp 1} & \multirow{2}{*}{$X=0$} & Linear Surrogate Model & 0.286 & 0.096 & 0.188 & 0.253 & 0.192 & 0.220\\ & & VAR Model (Baseline) & 0.351 & 0.065 & 0.126 & 0.210 & 0.579 & 0.348\\ & \multirow{2}{*}{$X=1$} & Linear Surrogate Model & 0.222 & 0.177 & 0.305 & 0.332 & 0.390 & 0.481\\ & & VAR Model (Baseline) & 0.418 & 0.182 & 0.220 & 0.308 & 1.113 & 0.534\\ & \multirow{2}{*}{$X=2$} & Linear Surrogate Model & 0.252 & 0.314 & 0.500 & 0.566 & 0.605 & 0.911\\ & & VAR Model (Baseline) & 0.438 & 0.432 & 0.387 & 0.415 & 2.195 & 0.919\\ & \multirow{2}{*}{$X=3$} & Linear Surrogate Model & 0.285 & 0.187 & 0.271 & 0.610 & 1.159 & 0.868\\ & & VAR Model (Baseline) & 0.534 & 0.205 & 0.124 & 0.551 & 4.998 & 1.106\\ & \multirow{2}{*}{$X=4$} & Linear Surrogate Model & 0.347 & 0.283 & 0.080 & 2.516 & 3.002 & 2.206\\ & & VAR Model (Baseline) & 0.458 & 0.171 & 0.273 & 0.984 & 1.763 & 9.570\\ \bottomrule \end{tabular} \end{table} \begin{table}[h!] \caption{Comparison result between different heterogeneous groups for Bias and MSE for Experiment 2} \begin{tabular}{c c c ccc ccc} \toprule \multirow{2}{*}{\textbf{Expt.}}&\multirow{2}{*}{\textbf{Groups}}&\multirow{2}{*}{\textbf{Method}}&\multicolumn{3}{c}{\ \textbf{Bias}}&\multicolumn{3}{c}{\textbf{MSE}}\\ & & & {$\mathbf{T_E=8}$} & {$\mathbf{T_E=9}$} & {$\mathbf{T_E=10}$} & {$\mathbf{T_E=8}$} & {$\mathbf{T_E=9}$} & {$\mathbf{T_E=10}$} \\ \midrule \multirow{10}{*}{Exp 2} & \multirow{2}{*}{$X=0$} & Linear Surrogate Model & 0.035 & 0.060 & 0.117 & 0.123 & 0.121 & 0.136 \\ & & VAR Model (Baseline) & 0.035 & 0.046 & 0.087 & 0.134 & 0.128 & 0.135 \\ & \multirow{2}{*}{$X=1$} & Linear Surrogate Model & 0.060 & 0.112 & 0.239 & 0.133 & 0.154 & 0.225 \\ & & VAR Model (Baseline) & 0.134 & 0.119 & 0.172 & 0.156 & 0.200 & 0.626 \\ & \multirow{2}{*}{$X=2$} & Linear Surrogate Model & 0.062 & 0.245 & 0.084 & 0.302 & 0.311 & 0.273 \\ & & VAR Model (Baseline) & 0.336 & 0.395 & 0.553 & 0.767 & 0.841 & 40.037 \\ & \multirow{2}{*}{$X=3$} & Linear Surrogate Model & 0.290 & 0.250 & 0.108 & 0.470 & 0.495 & 0.379 \\ & & VAR Model (Baseline) & 0.807 & 0.105 & 0.434 & 18.614 & 3.558 & 37.813 \\ & \multirow{2}{*}{$X=4$} & Linear Surrogate Model & 0.283 & 0.356 & 0.668 & 1.523 & 0.957 & 1.522 \\ & & VAR Model (Baseline) & 0.369 & 0.289 & 1.055 & 5.243 & 5.949 & 5.270 \\ \bottomrule \end{tabular} \end{table} \subsubsection{Linear Surrogate Model with Regularization} In the main context, we employ a linear surrogate model to generate the estimation of long-term effect in both two empirical studies. In reality, one potential challenge of this surrogate model is the curse of dimensionality. Given that in total $\#S$ surrogates are applied, there will be $\#S\times (T_E-1)$ features in the prediction models, and overfitting becomes a concern. To tackle this, we suggest employing the elastic net regularization zou2005regularization. This method helps mitigate the effects of irrelevant dimensions and manage multicollinearity, common issues associated with the curse of dimensionality. In the following analysis, we would like to show that the estimation result is still robust given that the regularization method is applied. To effectively use elastic net regularization, we must tune two hyperparameters. To do this, we uniformly select 100 potential values within the range of $[0,1]$ for these hyperparameters. We then optimize the two hyperparameters using five-fold cross-validation to identify their most suitable values for the model. This approach allows us to find the best configuration for the linear surrogate model and improve its predictive performance. Figure (ref) and Figure (ref) present the result of the estimated long-term effect with the linear surrogate model with the regularization in the two empirical experiments. Comparing them to the estimation results shown in Figure (ref) and Figure (ref) respectively, we can observe that the estimates are very similar, which demonstrate the robustness of our method. \begin{figure}[tb] \caption{Effects of Long-term Treatment using Pure Linear Surrogate Model for Experiment 1} { \textit{Note}: Grey dashed curves represent the true average treatment effect on \textit{search_uv} from week 1 to week 7. Solid red curves represent the estimated effects with the linear surrogate model without any regularization term. Shadows indicate 95% confidence intervals. The three panels represent the scenarios when we use the first $T_E$ weeks as the experimental period and the last $T_F$ weeks as the future period.} \end{figure} \begin{figure}[tb] \caption{Effects of Long-term Treatment using Pure Linear Surrogate Model for Experiment 2} { \textit{Note}: Grey dashed curves represent the true average treatment effect on \textit{search_uv} from week 1 to week 20. Solid red curves represent the estimated effects with the linear surrogate model without any regularization term. Shadows indicate 95% confidence intervals. The three panels represent the scenarios when we use the first $T_E$ weeks as the experimental period and the last $T_F$ weeks as the future period.} \end{figure} \subsection{Choosing surrogates. } In practical applications, Assumption (ref)—the longitudinal surrogacy assumption—is not testable. Past research indicates that bias can arise if this assumption is violated athey2019surrogate. Additionally, previous studies have provided guidelines for selecting surrogates duan2021online. In general, we recommend selecting intermediary metrics that are highly responsive to the treatment and meanwhile capture the diverse facets of user behavior leading to variations in primary outcomes during future periods. Both up-stream and down-stream behaviors during experimental periods can influence future primary outcomes. For instance, in the context where the click-through rates of products in an online marketplace serve as the primary outcome, up-stream behaviors might encompass actions such as users' search queries, their navigation patterns through categories, and the related content they peruse before ultimately clicking on a product. On the other hand, down-stream behaviors refer to the subsequent actions taken after clicking on a product, like reading reviews, adding the item to a cart, initiating a purchase, or even sharing the product with others. These up-stream behaviors are reflective of the users' intentions to click on products, and if a treatment can move these initial behaviors, it is plausible that it might impact down-stream outcomes in the future. Conversely, because down-stream behaviors represent deeper outcomes that indicate the realization of the primary outcome, they inherently have a strong predictive power for that primary outcome. In our real-world experiments, we mostly use down-stream behaviors as surrogates and demonstrate their effectiveness. Another potential surrogate to consider is simply the current primary outcome itself, which often exhibits a strong predictive power for the future primary outcome deng2013improving. To improve the efficiency of selecting surrogates for A/B tests, experimenters in companies can create a library of surrogates for treatments and primary outcomes based on the relevant metrics used in previous experiments. By using similar surrogate groups across related experiments, we can streamline the surrogate selection process and obtain estimates easily. This approach can save time and effort in identifying appropriate surrogates for each new experiment, making the process more efficient. Choosing either an excessive number of surrogates or too few can pose challenges. Choosing too few might not saturate the causal links between treatment and future primary outcomes and fulfill the longitudinal surrogacy assumption needed for reliable estimates, resulting in biased outcomes. Conversely, selecting an excessive number can lead to the “curse of dimensionality," increasing the risk of overfitting. For practitioners, we advise starting with as many relevant metrics as feasible and then employing techniques to reduce dimensionality. In our study, we utilized methods such as the elastic net zou2005regularization to eliminate or compress irrelevant dimensions with low predictive power.\footnote{One may also use auto-encoders rumelhart1986learning, vincent2008extracting, a common neural network based dimensionality reduction approach. However, when there are only a moderate number of surrogates, auto-encoders may omit useful information and present less satisfactory performance than the elastic net. We thus only recommend using the elastic net for dimensionality reduction. } \subsection{Adopting Non-linear Models} In our main framework, the linear surrogate model is an extension of the longitudinal surrogate model with an additional linearity assumption. Although it is intuitive to additionally assume non-linear relationships between surrogates and future surrogates or outcomes (i.e., $\bm{G}$ or $h$ are non-linear functions), its empirical performance is not as satisfactory as linear models. Our exploration with practitioners reveals that common non-linear machine learning such as nearest neighbors, random forests, or neural networks, cannot well predict future treatment effects. In addition, due to the non-deterministic nature and the requirement of a large sample size of these models, the model variance is often unaffordably large and cannot produce reasonably small confidence intervals. As an illustration, we provide estimation results that are based on the $k$-nearest neighbors ($k$NN) model. Empirically, instead of assuming linear function forms for $\bm{g}_t$ and $h_t$, we obtain $\hat{\bm{{g}}}_t(\bm{s}, \bm{x}, \bm{w}_{1:t})$ and ${\hat{h}}_t(\bm{s}, \bm{x}, \bm{w}_{1:t})$ as the averages of the $k$ nearest neighbors in terms of $\bm{s}$ in the training set. We then take the averages of their primary outcomes ($Y_{it}$) and intermediate metrics ($\bm{S}_{it}$) to obtain $\hat{\bm{{g}}}_t$ and ${\hat{h}}_t$, respectively. We apply the $k$-nearest neighbors model with $k=20$ to Experiment 1 as an illustrative example. As shown in Figure (ref), while the $k$NN based model seems to have confidence intervals that cover the true effects, it has unaffordably wide confidence intervals. The explanation is that $k$NN-based estimation is more vulnerable than linear models to local random noises: that is, if there exists a large degree of variation and stochastic in the local nearest neighborhood, estimates of each future primary or intermediate outcomes would exhibit large variance consequently. This would lead to very wide confidence intervals in practice. For instance, the $k$NN-based estimator exhibits an average MSE value that is eight times greater than that of the constant extrapolation baseline (CEB) in Experiment 1. This conclusion also holds when $k=10$ or $40$. Note that this issue does not only persist in $k$NN, but also in many other machine learning models such as random forests or neural networks. Therefore, we recommend using linear models instead throughout this work. \begin{figure}[tb] \caption{Estimated effect for 7-week treatment with $T_E$ weeks observed data using $k$NN model for Experiment 1} { \textit{Note}: Grey dashed curves represent the true average treatment effect on \textit{search_uv} from week 1 to week 7. Solid red curves represent the estimated effects with the $k$NN Model. Shadows indicate 95% confidence intervals. The three panels represent the scenarios when we use the first $T_E$ weeks as the experimental period and the last $T_F$ weeks as the future period.} \end{figure} \subsection{Estimation with Pre-treatment Variables} In the main text, we present the estimation results using a linear surrogate model on two empirical experiments, highlighting our approach's performance in terms of bias and MSE. To ensure a fair comparison with baseline models, we only use surrogate variables to construct all models mentioned. Additionally, we discover that including pre-treatment variables can reduce estimation variance. We construct an enhanced linear surrogate model that includes both the surrogate variables used in the main text and a pre-treatment variable: the value of \textit{search_uv} from a week before the experiment started, to estimate the long-term treatment effect. Tables (ref) and (ref) provide a comparison of the enhanced linear surrogate model with the previous linear surrogate model. We observe that incorporating pre-treatment variables effectively reduces variance and MSE compared to the results from not incorporating them. Bias is largely similar as we include pre-treatment variables in experimental data. \begin{table}[h] \caption{Comparison result between different methods in terms of Bias and MSE for Experiment 1} \scriptsize \begin{tabular}{c ccc ccc} \toprule \multirow{2}{*}{\textbf{Method}}&\multicolumn{3}{c}{\textbf{Bias}}&\multicolumn{3}{c}{\textbf{MSE}}\\ & {$\mathbf{T_E=2}$} & {$\mathbf{T_E=3}$} & {$\mathbf{T_E=4}$} & {$\mathbf{T_E=2}$} & {$\mathbf{T_E=3}$} & {$\mathbf{T_E=4}$} \\ \midrule \makecell{Linear Surrogate Model\\ (with pre-treatment variables)} & 0.105 & 0.068 & 0.205 & 0.228 & 0.216 & 0.294\\ \makecell{Linear Surrogate Model\\ (without pre-treatment variables)}& 0.087 & 0.199 & 0.165 & 0.327 & 0.324 & 0.314 \\ \bottomrule \end{tabular} \end{table} \begin{table}[h] \caption{Comparison result between different methods in terms of Bias and MSE for Experiment 2}\scriptsize \begin{tabular}{c ccc ccc} \toprule \multirow{2}{*}{\textbf{Method}}&\multicolumn{3}{c}{\textbf{Bias}}&\multicolumn{3}{c}{\textbf{MSE}}\\ & {$\mathbf{T_E=8}$} & {$\mathbf{T_E=9}$} & {$\mathbf{T_E=10}$} & {$\mathbf{T_E=8}$} & {$\mathbf{T_E=9}$} & {$\mathbf{T_E=10}$} \\ \midrule \makecell{Linear Surrogate Model\\ (with pre-treatment variables)} & 0.110 & 0.047 & 0.146 & 0.212 & 0.127 & 0.186 \\ \makecell{Linear Surrogate Model\\ (without pre-treatment variables)} & 0.098 & 0.048 & 0.158 & 0.233 & 0.136 & 0.201 \\ \bottomrule \end{tabular} \end{table} \subsection{Bootstrap with Full Sample} In the main analysis, we perform statistical inference with a bootstrap procedure that resamples 50% of users (with replacement) in each replicate—striking a pragmatic balance between accuracy and computational cost. To verify robustness, we also apply a full‑sample bootstrap that resamples the entire user set with replacement. The corresponding treatment‑effect estimates for the two empirical experiments are shown in Figures (ref) and (ref). The full‑sample results closely match those from the subsampling bootstrap, indicating that our findings are largely insensitive to the specific resampling scheme. Subsampling only half of the users therefore offers substantial computational savings without compromising inference reliability. \begin{figure}[h] \caption{Estimated Effects of Long-term Treatment with full sample bootstrap or Experiment 1} { \textit{Note}: Grey dashed curves represent the true average treatment effect on \textit{search_uv} from week 1 to week 7. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The three panels represent the scenarios when we use the first $T_E$ weeks as the experimental period and the last $T_F$ weeks as the future period.} \end{figure} \begin{figure}[h] \caption{Estimated Effects of Long-term Treatment with full sample bootstrap or Experiment 2} { \textit{Note}: Grey dashed curves represent the true average treatment effect on \textit{search_uv} from week 1 to week 20. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The three panels represent the scenarios when we use the first $T_E$ weeks as the experimental period and the last $T_F$ weeks as the future period.} \end{figure} \section{Validation of Comparability Assumption} \subsection{Testing results for Assumption (ref)} We provide example results on directly testing Assumption (ref) in both Empirical Experiment 1 and Empirical Experiment 2. The detailed approach is outlined in Section (ref). For each experiment, we use the same observable experimental time period \(T_E\) from the main text, testing on various combinations of \(t\) and \(t'\). We set \(\delta=1\), matching the criteria on five surrogate variables one period before \(t\) and \(t'\). The “$\bm{\#}$Tests" column indicates the number of tests on the possible sets of values in $S_{i,t - \delta}$ ($S_{i,t' - \delta}$). In Experiment 1, the average number of tests conducted for each stratum is around six thousand, while in Experiment 2 it's over nine thousand. Among these tests, some of them only include a small number of matched pairs. Across tests under all conditions, about 7% have a p-value less than 0.05, and about 11% have a p-value less than 0.1 in both experiments. The result suggests that there is no strong evidence that Assumption (ref) is violated, though stronger support for this claim is still needed. \begin{table}[h] \caption{Testing results of the comparability assumption for two empirical experiments}\scriptsize \begin{tabular}{cccccccccc} \toprule \textbf{Experiment} & \textbf{Groups} & $\bm{t}$ & $\bm{t'}$ & $\bm{\#}$\textbf{Tests} & $\bm{\# p<0.1}$ & $\bm{\# p<0.05}$ & $\bm{p<0.1 (\%)}$ & $\bm{p<0.05 (\%)}$ \\ \midrule \multirow{6}{*}{Experiment 1} & \multirow{3}{*}{Control} & 2 & 3 & 6,115 & 693 & 456 & 11.33 & 7.46 \\ & & 2 & 4 & 5,994 & 673 & 440 & 11.23 & 7.34 \\ & & 3 & 4 & 6,015 & 663 & 413 & 11.02 & 6.87 \\ \cmidrule{3-9} & \multirow{3}{*}{Treatment} & 2 & 3 & 6,219 & 710 & 454 & 11.42 & 7.30 \\ & & 2 & 4 & 6,253 & 753 & 472 & 12.04 & 7.55 \\ & & 3 & 4 & 6,215 & 724 & 466 & 11.65 & 7.50 \\ \midrule \multirow{12}{*}{Experiment 2} & \multirow{6}{*}{Control} & 7 & 8 & 9,085 & 997 & 617 & 10.97 & 6.79 \\ & & 7 & 9 & 9,335 & 1,112 & 699 & 11.91 & 7.49 \\ & & 7 & 10 & 9,393 & 1,048 & 661 & 11.16 & 7.04 \\ & & 8 & 9 & 9,314 & 990 & 612 & 10.63 & 6.57 \\ & & 8 & 10 & 9,376 & 1,026 & 629 & 10.94 & 6.71 \\ & & 9 & 10 & 9,504 & 1,078 & 687 & 11.34 & 7.23 \\ \cmidrule{3-9} & \multirow{6}{*}{Treatment} & 7 & 8 & 9,550 & 1,070 & 675 & 11.20 & 7.07 \\ & & 7 & 9 & 9,536 & 1,076 & 708 & 11.28 & 7.42 \\ & & 7 & 10 & 9,617 & 1,086 & 679 & 11.29 & 7.06 \\ & & 8 & 9 & 9,605 & 1,016 & 638 & 10.58 & 6.64 \\ & & 8 & 10 & 9,665 & 1,116 & 695 & 11.55 & 7.19 \\ & & 9 & 10 & 9,655 & 1,126 & 704 & 11.66 & 7.29 \\ \bottomrule \end{tabular} \end{table} Considering the dynamic nature of real-world environments, including market fluctuations, economic shocks, and product iterations, maintaining the comparability assumption can sometimes be challenging. However, the robustness of our method may still render it applicable. In practice, instead of strictly testing the comparability assumption, we suggest using weaker and alternative criteria to determine the feasibility of our approach. We explore such an alternative in the subsequent sections. \subsection{Parallel Trends Assumption} Given that we would like to generalize the comparability assumption, we need to define the surrogate index at different time periods. For any positive integer $\delta \in \mathbb{N}^+$, we denote a generalized version of the longitudinal surrogate index and the pivot index at time $t$ as follows. \begin{definitionp}{(ref)'}[Generalized Longitudinal Surrogate Index] For any $t \in [T]$, $\delta \in \mathbb{N}^+$, $\bm{s} \in \mathbb{S}$, $\bm{x} \in \mathbb{X}$, $\bm{w}_{t-\delta+1:t} \in \{\bm{0}_{\delta}, \bm{1}_{\delta}\}$, the generalized surrogate index is the conditional expectation of the primary outcome at time $t$, given the surrogate outcomes at time $t-\delta$, the pre-treatment variables, and the treatment assignments, i.e., \begin{align*} h^{\delta}_t(\bm{s}, \bm{x}, \bm{w}_{t-\delta+1:t}) = \mathbb{E}_{\mathcal{F}} \left[ Y_{it} \vert \bm{S}_{i(t-\delta)} = \bm{s}, \bm{X}_i = \bm{x}, \bm{W}_{i,t-\delta+1:t} = \bm{w}_{t-\delta+1:t}\right], \end{align*} where the expectation is taken over $Y_{it}$. \end{definitionp} \begin{definitionp}{(ref)'}[Generalized Pivot Index] For any $t \in [T]$, $\delta \in \mathbb{N}^+$, $\bm{s} \in \mathbb{S}$, $\bm{x} \in \mathbb{X}$, $\bm{w}_{t-\delta+1:t} \in \{\bm{0}_{\delta}, \bm{1}_{\delta}\}$, the generalized pivot index is a vector of the conditional expectations of the surrogate outcomes at time $t$, given the surrogate outcomes at time $t-\delta$, the pre-treatment variables, and the treatment assignments, i.e., \begin{align*} \bm{g}^{\delta}_t(\bm{s}, \bm{x}, \bm{w}_{t-\delta+1:t}) = \mathbb{E}_{\mathcal{F}} \left[ \bm{S}_{it} \vert \bm{S}_{i(t-\delta)} = \bm{s}, \bm{X}_i = \bm{x}, \bm{W}_{i,t-\delta+1:t} = \bm{w}_{t-\delta+1:t}\right], \end{align*} where the expectation is taken over $\bm{S}_{it}$. Moreover, we denote the conditional surrogate outcomes at time $t$, given the surrogate outcomes at time $t-\delta$, the pre-treatment variables, and the treatment assignments, to be \begin{align*} \bm{G}^{\delta}_t(\bm{s}, \bm{x}, \bm{w}_{1:t}) \sim \bm{S}_{it} \vert \bm{S}_{i(t-\delta)} = \bm{s}, \bm{X}_i = \bm{x}, \bm{W}_{i,t-\delta+1:t} = \bm{w}_{t-\delta+1:t}. \end{align*} \end{definitionp} The generalized longitudinal surrogate index and the generalized pivot index represent a broader and more flexible range of surrogate index functions. When $\delta = t$, the generalized longitudinal surrogate index function (Definition (ref)) reduces to the surrogate index function (Definition (ref)), and the generalized pivot index function (Definition (ref)) reduces to the pivot index function (Definition (ref)), that is, $h_t(\bm{s}, \bm{x}, \bm{w}_{1:t}) = h^{t}_t(\bm{s}, \bm{x}, \bm{w}_{1:t})$ and $\bm{g}_t(\bm{s}, \bm{x}, \bm{w}_{t-\delta+1:t}) = \bm{g}^{t}_t(\bm{s}, \bm{x}, \bm{w}_{t-\delta+1:t})$. Using Definitions (ref) and (ref) , we introduce an assumption that relaxes the comparability assumption (Assumption (ref)). \begin{assumptionp}{(ref)'}[Parallel Trends] The difference in the conditional expectation of the primary outcomes across time periods under the treatment condition is equal to that under the control condition. For any $t,t' \in [T]$ and any positive integer $\delta \in \mathbb{N}^+$, \begin{align*} h^{\delta}_t(\bm{s}_{1}, \bm{x}_i, \bm{1}_{\delta}) - h^{\delta}_{t'}(\bm{s}_{1}, \bm{x}_i, \bm{1}_{\delta}) = h^{\delta}_t(\bm{s}_{0}, \bm{x}_i, \bm{0}_{\delta}) - h^{\delta}_{t'}(\bm{s}_{0}, \bm{x}_i, \bm{0}_{\delta}). \end{align*} \end{assumptionp} \begin{assumptionp}{(ref)”}[Extended Parallel Trends] For any $t,t', u_0, ..., u_{K-1}\in [T]$ and any positive integer $\delta \in \mathbb{N}^+$, \begin{multline*} h_{u_0}(\bm{g}^{\delta}_t(\bm{s}_{1}, \bm{x}_i, \bm{1}_{\delta}), \bm{x}_i, \bm{1}_{u_0}) - h_{u_0}(\bm{g}^{\delta}_{t'}(\bm{s}_{1}, \bm{x}_i, \bm{1}_{\delta}), \bm{x}_i, \bm{1}_{u_0}) \\ = h_{u_0}(\bm{g}^{\delta}_t(\bm{s}_{0}, \bm{x}_i, \bm{0}_{\delta}), \bm{x}_i, \bm{0}_{u_0}) - h_{u_0}(\bm{g}^{\delta}_{t'}(\bm{s}_{0}, \bm{x}_i, \bm{0}_{\delta}), \bm{x}_i, \bm{0}_{u_0}), \end{multline*} \begin{multline*} h_{u_0}(\bm{g}_{u_1}(\bm{g}^{\delta}_t(\bm{s}_{1}, \bm{x}_i, \bm{1}_{\delta}), \bm{x}_i, \bm{1}_{u_1}), \bm{x}_i, \bm{1}_{u_0}) - h_{u_0}(\bm{g}_{u_1}(\bm{g}^{\delta}_{t'}(\bm{s}_{1}, \bm{x}_i, \bm{1}_{\delta}), \bm{x}_i, \bm{1}_{u_1}), \bm{x}_i, \bm{1}_{u_0}) \\ = h_{u_0}(\bm{g}_{u_1}(\bm{g}^{\delta}_t(\bm{s}_{0}, \bm{x}_i, \bm{0}_{\delta}), \bm{x}_i, \bm{0}_{u_1}), \bm{x}_i, \bm{0}_{u_0}) - h_{u_0}(\bm{g}_{u_1}(\bm{g}^{\delta}_{t'}(\bm{s}_{0}, \bm{x}_i, \bm{0}_{\delta}), \bm{x}_i, \bm{0}_{u_1}), \bm{x}_i, \bm{0}_{u_0}), \end{multline*} \begin{multline*} h_{u_0}(\bm{g}_{u_1}( ... \bm{g}_{u_{K-1}}(\bm{g}^{\delta}_t(\bm{s}_{1}, \bm{x}_i, \bm{1}_{\delta}), \bm{x}_i, \bm{1}_{u_{K-1}}) ..., \bm{x}_i, \bm{1}_{u_1}), \bm{x}_i, \bm{1}_{u_0}) \\ \qquad \qquad - h_{u_0}(\bm{g}_{u_1}( ... \bm{g}_{u_{K-1}}(\bm{g}^{\delta}_{t'}(\bm{s}_{1}, \bm{x}_i, \bm{1}_{\delta}), \bm{x}_i, \bm{1}_{u_{K-1}}) ..., \bm{x}_i, \bm{1}_{u_1}), \bm{x}_i, \bm{1}_{u_0}) \\ = h_{u_0}(\bm{g}_{u_1}( ... \bm{g}_{u_{K-1}}(\bm{g}^{\delta}_t(\bm{s}_{0}, \bm{x}_i, \bm{0}_{\delta}), \bm{x}_i, \bm{0}_{u_{K-1}}) ..., \bm{x}_i, \bm{0}_{u_1}), \bm{x}_i, \bm{0}_{u_0}) \qquad \qquad \\ - h_{u_0}(\bm{g}_{u_1}( ... \bm{g}_{u_{K-1}}(\bm{g}^{\delta}_{t'}(\bm{s}_{0}, \bm{x}_i, \bm{0}_{\delta}), \bm{x}_i, \bm{0}_{u_{K-1}}) ..., \bm{x}_i, \bm{0}_{u_1}), \bm{x}_i, \bm{0}_{u_0}). \end{multline*} \end{assumptionp} We first explore the relationship between the parallel trends assumption and the comparability assumption. \begin{proposition} Assumption (ref) is a sufficient condition of Assumption (ref). \end{proposition} \proof{Proof of Proposition (ref).} For any $t,t' \in [T]$, $\delta \in \mathbb{N}^+$, $\bm{s}_0,\bm{s}_1 \in \mathbb{S}$, $\bm{x} \in \mathbb{X}$, $\bm{w}_{t-\delta+1:t} \in \{\bm{0}_{\delta}, \bm{1}_{\delta}\}$, Assumption (ref) implies that \begin{align*} h^{\delta}_t(\bm{s}_{1}, \bm{x}_i, \bm{1}_{\delta}) = h^{\delta}_{t'}(\bm{s}_{1}, \bm{x}_i, \bm{1}_{\delta}),\ h^{\delta}_t(\bm{s}_{0}, \bm{x}_i, \bm{0}_{\delta}) = h^{\delta}_{t'}(\bm{s}_{0}, \bm{x}_i, \bm{0}_{\delta}). \end{align*} Therefore, \begin{align*} h^{\delta}_t(\bm{s}_{1}, \bm{x}_i, \bm{1}_{\delta}) - h^{\delta}_{t'}(\bm{s}_{1}, \bm{x}_i, \bm{1}_{\delta}) = 0 = h^{\delta}_t(\bm{s}_{0}, \bm{x}_i, \bm{0}_{\delta}) - h^{\delta}_{t'}(\bm{s}_{0}, \bm{x}_i, \bm{0}_{\delta}), \end{align*} which leads to Assumption (ref). \halmos \endproof We have proved Theorem (ref) under Assumptions (ref), (ref), and (ref) in the previous section. Here we show that without the comparability assumption, the conclusion of the Theorem (ref) continue to hold given that the generalized surrogate index function is linear and the parallel trends assumption is met. We first rewrite the linearity of surrogates assumption with the generalized longitudinal surrogate index. \begin{assumptionp}{(ref)'}[Extended Linearity of Surrogates] \begin{enumerate} • The generalized surrogate index function is linear with respect to the surrogates, i.e., $\forall t \in [T],\ \delta \in \mathbb{N}^+,\ d\in\{0,1,...,D\},\ \bm{x}\in\mathbb{X},\ \bm{w}_{t-\delta+1:t}\in\{\bm{0}_{\delta}, \bm{1}_{\delta}\}$, there exists $\alpha^{\delta}_{d,t}(\bm{x}, \bm{w}_{t-\delta+1:t})$, such that \begin{align*} h^{\delta}_t(\bm{s}, \bm{x}, \bm{w}_{t-\delta+1:t}) = \alpha^{\delta}_{0,t}(\bm{x}, \bm{w}_{t-\delta+1:t}) + \sum_{d=1}^D s_d \cdot \alpha^{\delta}_{d,t}(\bm{x}, \bm{w}_{t-\delta+1:t}). \end{align*} • The generalized pivot index function is linear with respect to the surrogates, i.e., $\forall t \in [T],\ \delta \in \mathbb{N}^+,\ d\in\{0,1,...,D\},\ \bm{x}\in\mathbb{X},\ \bm{w}_{t-\delta+1:t}\in\{\bm{0}_{\delta}, \bm{1}_{\delta}\}$, there exists $\beta^{\delta}_{d,d',t}(\bm{x}, \bm{w}_{t-\delta+1:t})$, such that for each $d\in[D]$, \begin{align*} g^{\delta}_{t,d}(\bm{s}, \bm{x}, \bm{w}_{t-\delta+1:t}) = \beta^{\delta}_{d,0,t}(\bm{x}, \bm{w}_{t-\delta+1:t}) + \sum_{d'=1}^D s_d \cdot \beta^{\delta}_{d,d',t}(\bm{x}, \bm{w}_{t-\delta+1:t}), \end{align*} where $g^{\delta}_{t,d}(\bm{s}, \bm{x}, \bm{w}_{t-\delta+1:t})$ stands for the $d$-th component of $\bm{g}^{\delta}_t(\bm{s}, \bm{x}, \bm{w}_{t-\delta+1:t})$ the pivot index. \end{enumerate} \end{assumptionp} With the above alternative assumptions, we can rewrite the Theorem (ref) as follows. \begin{theoremp}{(ref)'}[Linear Surrogate Model] When $T_E = T_F$, assume Assumption (ref); when $T_E < T_F$, assume Assumptions (ref) and (ref). In addition, assume Assumptions (ref) and (ref), where Assumption (ref) holds for $\mathbb{T} = \{t_1,t_2,...,t_K\}$. Then, the average effect of long-term treatments on the primary outcome is equal to the following expression, \begin{multline*} \tau_T = \mathbb{E}_{\mathcal{F}} \Big[ h_{\Delta t_{K+1}} \big( \bm{g}_{\Delta t_K}( ... \bm{g}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K}), \bm{X}_i, \bm{1}_{\Delta t_{K+1}} \big) \Big] \\ - \mathbb{E}_{\mathcal{F}} \Big[ h_{\Delta t_{K+1}}\left( \bm{g}_{\Delta t_K}( ... \bm{g}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{\Delta t_1}) ... ,\bm{X}_i, \bm{0}_{\Delta t_K}), \bm{X}_i, \bm{0}_{\Delta t_{K+1}} \right) \Big], \end{multline*} where the expectation is taken over $\bm{S}_{i0}, \bm{X}_i$. \end{theoremp} Prior to proving Theorem (ref), we first re-state the first half of Theorem (ref) in the following Lemma (ref). Lemma (ref) illustrates the main idea of the proof for easier understanding. \begin{lemma} Consider the special case when $T_E = T_F$. Under Assumptions (ref), (ref), and (ref)-(i), where Assumption (ref) holds for $\mathbb{T} = \{T_E\}$, the average effect of long-term treatments on the primary outcome is equal to the following expression, \begin{align*} \tau_T = \mathbb{E}_{\mathcal{F}} \Big[h_{T_E}\left( \bm{g}_{T_E}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{T_E}), \bm{X}_i, \bm{1}_{T_E} \right)\Big] - \mathbb{E}_{\mathcal{F}} \Big[h_{T_E}\left( \bm{g}_{T_E}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{T_E}), \bm{X}_i, \bm{0}_{T_E} \right)\Big], \end{align*} where the expectation is taken over $\bm{S}_{i0}, \bm{X}_i$. \end{lemma} \subsection{Proof of Lemma (ref) with Parallel Trends Assumption} \proof{Proof of Lemma (ref).} From the definition of the causal effect, \begin{align*} \tau_T = \mathbb{E}_{\mathcal{F}}\bigg[ Y_{iT}(\bm{1}_{T}) - Y_{iT}(\bm{0}_{T}) \bigg], \end{align*} we start with the first part $\mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T})\right]$. \begin{align} & \ \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T}) \right] \nonumber\\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_T, \bm{x}_i, \bm{s}_{i0} \right] \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber\\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{S}_{i T_E} = \bm{s}, \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0} \right] \cdot \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \end{align} Note that, for any $\bm{s}, \bm{x}_i$, \begin{align} & \ \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{S}_{i T_E} = \bm{s}, \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0} \right] \nonumber\\ = & \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{S}_{i T_E} = \bm{s}, \bm{W}_{i,T_E+1:T} = \bm{1}_{T_F}, \bm{x}_i \right] \nonumber\\ = & \mathbb{E}_{\mathcal{F}}\left[Y_{i T_E} \bigg\vert \bm{S}_{i 0} = \bm{s}, \bm{W}_{i,1:T_E} = \bm{1}_{T_E}, \bm{x}_i \right] \nonumber\\ & + \left(\mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{S}_{i T_E} = \bm{s}, \bm{W}_{i,T_E+1:T} = \bm{1}_{T_F}, \bm{x}_i \right] - \mathbb{E}_{\mathcal{F}}\left[Y_{i T_E} \bigg\vert \bm{S}_{i 0} = \bm{s}, \bm{W}_{i,1:T_E} = \bm{1}_{T_E}, \bm{x}_i \right]\right) \nonumber\\ = & h^{T_E}_{T_E}(\bm{s}, \bm{x}_i, \bm{1}_{T_E}) + \left( h^{T_F}_{T}\Big(\bm{s}, \bm{x}_i, \bm{1}_{T_F}\Big) - h^{T_E}_{T_E}\Big(\bm{s}, \bm{x}_i, \bm{1}_{T_E}\Big)\right), \end{align} where the first equality is due to Assumption (ref) when $t=T_E, t'=T$; the second equality is because we add and subtract the same term; the third equality uses short-hand notation for the generalized longitudinal surrogate index. Putting (ref) back into (ref), we have \begin{align} & \ \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T}) \right] \nonumber\\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s} \in \mathbb{S}} h^{T_E}_{T_E}(\bm{s}, \bm{x}_i, \bm{1}_{T_E}) \cdot \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber\\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s} \in \mathbb{S}} \left( h^{T_F}_{T}\Big(\bm{s}, \bm{x}_i, \bm{1}_{T_F}\Big) - h^{T_E}_{T_E}\Big(\bm{s}, \bm{x}_i, \bm{1}_{T_E}\Big)\right) \cdot \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \end{align} Note that, the treatment assignment remains the same throughout the entire horizon, which means that, \begin{align*} \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0}\right) = \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,1:T_E} = \bm{1}_{T_E}, \bm{x}_i, \bm{s}_{i0}\right). \end{align*} Using the above expression, (ref) can be rewritten as \begin{align} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T}) \right] = & \ \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s} \in \mathbb{S}} h^{T_E}_{T_E}\Big(\bm{s}, \bm{x}_i, \bm{1}_{T_E}\Big) \cdot \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,1:T_E} = \bm{1}_{T_E}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber\\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s} \in \mathbb{S}} h^{T_F}_{T}\Big(\bm{s}, \bm{x}_i, \bm{1}_{T_F}\Big) \cdot \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,T_E+1:T} = \bm{1}_{T_F}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber\\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s} \in \mathbb{S}} h^{T_E}_{T_E}\Big(\bm{s}, \bm{x}_i, \bm{1}_{T_E}\Big) \cdot \Pr\left(\bm{S}_{i T_E} = \bm{s} \bigg\vert \bm{W}_{i,1:T_E} = \bm{1}_{T_E}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber\\ = & \ \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[h^{T_E}_{T_E}(\bm{G}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}), \bm{x}_i, \bm{1}_{T_E})\right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber\\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[h^{T_F}_{T}(\bm{G}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}), \bm{x}_i, \bm{1}_{T_F})\right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber\\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[h^{T_E}_{T_E}(\bm{G}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}), \bm{x}_i, \bm{1}_{T_E})\right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \end{align} Denote $\bm{G}_{T_E} = (G_{T_E, 1}, ..., G_{T_E, D})$ and $\bm{g}_{T_E} = (g_{T_E, 1}, ..., g_{T_E, D})$. Then for any pre-treatment variables and surrogates $\bm{x}_i, \bm{s}_{i0}$, we can write $\mathbb{E}_{\mathcal{F}}\left[h^{T_F}_{T}(\bm{G}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}), \bm{x}_i, \bm{1}_{T_F})\right]$ as \begin{align*} \mathbb{E}_{\mathcal{F}}\left[h^{T_F}_{T}(\bm{G}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}), \bm{x}_i, \bm{1}_{T_F})\right] = &\ \mathbb{E}_{\mathcal{F}}\left[ \alpha^{T_F}_{0,t}(\bm{x}_i, \bm{1}_{T_F}) + \sum_{d=1}^D G_{T_E, d}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}) \cdot \alpha^{T_F}_{d,t}(\bm{x}_i, \bm{1}_{T_F}) \right] \\ = &\ \alpha^{T_F}_{0,t}(\bm{x}_i, \bm{1}_{T_F}) + \sum_{d=1}^D \mathbb{E}_{\mathcal{F}}\left[G_{T_E, d}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E})\right] \cdot \alpha^{T_F}_{d,t}(\bm{x}_i, \bm{1}_{T_F}) \\ = &\ \alpha^{T_F}_{0,t}(\bm{x}_i, \bm{1}_{T_F}) + \sum_{d=1}^D g_{T_E, d}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}) \cdot \alpha^{T_F}_{d,t}(\bm{x}_i, \bm{1}_{T_F}) \\ = &\ h^{T_F}_{T}(\bm{g}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}), \bm{x}_i, \bm{1}_{T_F}), \end{align*} where the first equality is due to Assumption (ref); the second equality is due to linearity of expectation; the third equality is due to Definition (ref). Similarly, we have \begin{align*} \mathbb{E}_{\mathcal{F}}\left[h^{T_E}_{T_E}(\bm{G}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}), \bm{x}_i, \bm{1}_{T_E})\right] = h^{T_E}_{T_E}(\bm{g}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}), \bm{x}_i, \bm{1}_{T_E}) \end{align*} Putting the above expressions in (ref), and noting that $h^{T_E}_{T_E}(\cdot, \cdot, \cdot) = h_{T_E}(\cdot, \cdot, \cdot)$, we have \begin{multline*} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T}) \right] = \mathbb{E}_{\mathcal{F}} \Big[h_{T_E}\left( \bm{g}_{T_E}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{T_E}), \bm{X}_i, \bm{1}_{T_E} \right)\Big] \\ + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \Big(h^{T_F}_{T}(\bm{g}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}), \bm{x}_i, \bm{1}_{T_F}) - h^{T_E}_{T_E}(\bm{g}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}), \bm{x}_i, \bm{1}_{T_E})\Big) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \end{multline*} Similarly, \begin{multline*} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{0}_{T}) \right] = \mathbb{E}_{\mathcal{F}} \Big[h_{T_E}\left( \bm{g}_{T_E}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{T_E}), \bm{X}_i, \bm{0}_{T_E} \right)\Big] \\ + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \Big(h^{T_F}_{T}(\bm{g}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{T_E}), \bm{x}_i, \bm{0}_{T_F}) - h^{T_E}_{T_E}(\bm{g}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{T_E}), \bm{x}_i, \bm{0}_{T_E})\Big) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \end{multline*} Combining both parts, \begin{align*} \tau_T = &\ \mathbb{E}_{\mathcal{F}} \Big[h_{T_E}\left( \bm{g}_{T_E}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{T_E}), \bm{X}_i, \bm{1}_{T_E} \right)\Big] - \mathbb{E}_{\mathcal{F}} \Big[h_{T_E}\left( \bm{g}_{T_E}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{T_E}), \bm{X}_i, \bm{0}_{T_E} \right)\Big] \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \Big(h^{T_F}_{T}(\bm{g}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}), \bm{x}_i, \bm{1}_{T_F}) - h^{T_E}_{T_E}(\bm{g}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E}), \bm{x}_i, \bm{1}_{T_E})\Big) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \Big(h^{T_F}_{T}(\bm{g}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{T_E}), \bm{x}_i, \bm{0}_{T_F}) - h^{T_E}_{T_E}(\bm{g}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{T_E}), \bm{x}_i, \bm{0}_{T_E})\Big) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ = &\ \mathbb{E}_{\mathcal{F}} \Big[h_{T_E}\left( \bm{g}_{T_E}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{T_E}), \bm{X}_i, \bm{1}_{T_E} \right)\Big] - \mathbb{E}_{\mathcal{F}} \Big[h_{T_E}\left( \bm{g}_{T_E}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{T_E}), \bm{X}_i, \bm{0}_{T_E} \right)\Big], \end{align*} where the equality is due to Assumption (ref) when $t=T,\ t'=T_E, \delta=T_E=T_F$, and we view $\bm{s}_1 = \bm{g}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{T_E})$ and $\bm{s}_0 = \bm{g}_{T_E}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{T_E})$. \halmos \endproof \subsection{Proof of Theorem (ref) with Parallel Trends Assumption} Lemma (ref) is a special case of Theorem (ref) when $T_E=T_F$. Below we iteratively employ the same approach to prove Theorem (ref). \proof{Proof of Theorem (ref).} From the definition, \begin{align*} \tau_T = \mathbb{E}_{\mathcal{F}}\bigg[ Y_{iT}(\bm{1}_{T}) - Y_{iT}(\bm{0}_{T}) \bigg], \end{align*} We start from the first part $\mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T})\right]$, which can be expressed as \begin{align} & \ \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T}) \right] \nonumber\\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_T, \bm{x}_i, \bm{s}_{i0} \right] \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber\\ = & \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s}_{t_K} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}}\left[ Y_{iT} \bigg\vert \bm{S}_{i t_K} = \bm{s}_{t_K}, \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0} \right] \cdot \Pr\left(\bm{S}_{i t_K} = \bm{s}_{t_K} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \end{align} We divide the following proof into two stages. First, we reformulate the term $\mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T})\right]$ by iteratively adding and subtracting the same term. Second, we use Assumption (ref) (i.e., the linearity assumption) to further simplify the expression of $\mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T})\right]$, and apply Assumptions (ref) and (ref) (i.e., the parallel trends assumption) to conclude the proof. \textbf{\textit{Step 1.}} Note that, \begin{align} & \ \Pr\left(\bm{S}_{i t_K} = \bm{s}_{t_K} \bigg\vert \bm{W}_{i,1:T} = \bm{1}_{T}, \bm{x}_i, \bm{s}_{i0}\right) \nonumber \\ = & \sum_{\bm{s}_{t_{K-1}}, \bm{s}_{t_{K-2}}, ..., \bm{s}_{t_{2}}, \bm{s}_{t_1} \in \mathbb{S}} \Pr\left(\bm{S}_{i t_K} = \bm{s}_{t_K} \bigg\vert \bm{W}_{i,1:t_K} = \bm{1}_{t_K}, \bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \nonumber \\ & \Pr\left(\bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}} \bigg\vert \bm{W}_{i,1:t_{K-1}} = \bm{1}_{t_{K-1}}, \bm{S}_{i t_{K-2}} = \bm{s}_{t_{K-2}}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \nonumber \\ & \ldots \cdot \Pr\left(\bm{S}_{i t_2} = \bm{s}_{t_2} \bigg\vert \bm{W}_{i,1:t_2} = \bm{1}_{t_2}, \bm{S}_{i t_{1}} = \bm{s}_{t_{1}}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr\left(\bm{S}_{i t_{1}} = \bm{s}_{t_{1}} \bigg\vert \bm{W}_{i,1:t_{1}} = \bm{1}_{t_{1}}, \bm{x}_i, \bm{s}_{i0}\right) \nonumber \\ = & \sum_{\bm{s}_{t_{K-1}}, \bm{s}_{t_{K-2}}, ..., \bm{s}_{t_{2}}, \bm{s}_{t_1} \in \mathbb{S}} \Pr\left(\bm{S}_{i t_K} = \bm{s}_{t_K} \bigg\vert \bm{W}_{i,t_{K-1}+1:t_K} = \bm{1}_{\Delta t_K}, \bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}}, \bm{x}_i\right) \cdot \nonumber \\ & \Pr\left(\bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}} \bigg\vert \bm{W}_{i,t_{K-2}+1:t_{K-1}} = \bm{1}_{\Delta t_{K-1}}, \bm{S}_{i t_{K-2}} = \bm{s}_{t_{K-2}}, \bm{x}_i \right) \cdot \nonumber \\ & \ldots \cdot \Pr\left(\bm{S}_{i t_2} = \bm{s}_{t_2} \bigg\vert \bm{W}_{i,t_1+1:t_2} = \bm{1}_{\Delta t_2}, \bm{S}_{i t_{1}} = \bm{s}_{t_{1}}, \bm{x}_i \right) \cdot \Pr\left(\bm{S}_{i t_{1}} = \bm{s}_{t_{1}} \bigg\vert \bm{W}_{i,1:t_{1}} = \bm{1}_{\Delta t_{1}}, \bm{x}_i, \bm{s}_{i0}\right), \end{align} where the first equality is using the law of total probability; the second is because when any unit is assigned to the treatment group, it is always in the treatment group, that is, \begin{align*} \Pr\left(\bm{W}_{i,1:t_{k}} = \bm{1}_{t_{k}} \bigg\vert\bm{W}_{i,t_{k}+1:t_{k+1}} = \bm{1}_{\Delta t_{k+1}}\right) = 1. \end{align*} Putting the above expression (ref) into (ref), we have \begin{align} & \ \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T}) \right] \nonumber \\ = & \ \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s}_{t_K}, ..., \bm{s}_{t_1} \in \mathbb{S}} h^{\Delta t_{K+1}}_{T}(\bm{s}_{t_K}, \bm{x}_i, \bm{1}_{\Delta t_{K+1}}) \cdot \Pr\left(\bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}} \bigg\vert \bm{W}_{i,t_{K-1}+1:t_K} = \bm{1}_{\Delta t_K}, \bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}}, \bm{x}_i\right) \nonumber \\ & \cdot \Pr\left(\bm{S}_{i t_{K-2}} = \bm{s}_{t_{K-2}} \bigg\vert \bm{W}_{i,t_{K-2}+1:t_{K-1}} = \bm{1}_{\Delta t_{K-1}}, \bm{S}_{i t_{K-2}} = \bm{s}_{t_{K-2}}, \bm{x}_i \right) \nonumber \\ & ... \cdot \Pr\left(\bm{S}_{i t_{1}} = \bm{s}_{t_{1}} \bigg\vert \bm{W}_{i,1:t_1} = \bm{1}_{\Delta t_{1}}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ = & \ \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{T}\left( \bm{G}^{\Delta t_K}_{t_K}( ... \bm{G}^{\Delta t_2}_{t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ = & \ \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{T}\left( \bm{G}^{\Delta t_K}_{t_K}( ... \bm{G}^{\Delta t_2}_{t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{t_K}( ... \bm{G}^{\Delta t_2}_{t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{t_K}( ... \bm{G}^{\Delta t_2}_{t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{\Delta t_K}( \bm{G}^{\Delta t_{K-1}}_{ t_{K-1}}( ... \bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}) ... ,\bm{x}_i, \bm{1}_{\Delta t_{K-1}}),\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{\Delta t_K}( \bm{G}^{\Delta t_{K-1}}_{ t_{K-1}}( ... \bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}) ... ,\bm{x}_i, \bm{1}_{\Delta t_{K-1}}),\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & \ldots \nonumber \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{\Delta t_K}( ... \bm{G}^{\Delta t_2}_{\Delta t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{\Delta t_K}( ... \bm{G}^{\Delta t_2}_{\Delta t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{\Delta t_K}( ... \bm{G}^{\Delta t_2}_{\Delta t_2}(\bm{G}^{\Delta t_1}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{\Delta t_K}( ... \bm{G}^{\Delta t_2}_{\Delta t_2}(\bm{G}^{\Delta t_1}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}), \end{align} where the second equality is due to Definition (ref); the last equality is adding and subtracting the same $K$ quantities (there are $2K$ additional terms). Similarly, we have \begin{align} & \ \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{0}_{T}) \right] \nonumber \\ = & \ \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \sum_{\bm{s}_{t_K}, ..., \bm{s}_{t_1} \in \mathbb{S}} h^{\Delta t_{K+1}}_{T}(\bm{s}_{t_K}, \bm{x}_i, \bm{0}_{\Delta t_{K+1}}) \cdot \Pr\left(\bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}} \bigg\vert \bm{W}_{i,t_{K-1}+1:t_K} = \bm{0}_{\Delta t_K}, \bm{S}_{i t_{K-1}} = \bm{s}_{t_{K-1}}, \bm{x}_i\right) \nonumber \\ & \cdot \Pr\left(\bm{S}_{i t_{K-2}} = \bm{s}_{t_{K-2}} \bigg\vert \bm{W}_{i,t_{K-2}+1:t_{K-1}} = \bm{0}_{\Delta t_{K-1}}, \bm{S}_{i t_{K-2}} = \bm{s}_{t_{K-2}}, \bm{x}_i \right) \nonumber \\ & ... \cdot \Pr\left(\bm{S}_{i t_{1}} = \bm{s}_{t_{1}} \bigg\vert \bm{W}_{i,1:t_1} = \bm{0}_{\Delta t_{1}}, \bm{x}_i, \bm{s}_{i0}\right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ = & \ \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{T}\left( \bm{G}^{\Delta t_K}_{t_K}( ... \bm{G}^{\Delta t_2}_{t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ = & \ \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{T}\left( \bm{G}^{\Delta t_K}_{t_K}( ... \bm{G}^{\Delta t_2}_{t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{t_K}( ... \bm{G}^{\Delta t_2}_{t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{t_K}( ... \bm{G}^{\Delta t_2}_{t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{\Delta t_K}( \bm{G}^{\Delta t_{K-1}}_{ t_{K-1}}( ... \bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}) ... ,\bm{x}_i, \bm{0}_{\Delta t_{K-1}}),\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{\Delta t_K}( \bm{G}^{\Delta t_{K-1}}_{ t_{K-1}}( ... \bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}) ... ,\bm{x}_i, \bm{0}_{\Delta t_{K-1}}),\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & \ldots \nonumber \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{\Delta t_K}( ... \bm{G}^{\Delta t_2}_{\Delta t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{\Delta t_K}( ... \bm{G}^{\Delta t_2}_{\Delta t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{\Delta t_K}( ... \bm{G}^{\Delta t_2}_{\Delta t_2}(\bm{G}^{\Delta t_1}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{\Delta t_K}( ... \bm{G}^{\Delta t_2}_{\Delta t_2}(\bm{G}^{\Delta t_1}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \right] \cdot \Pr(\bm{x}_i, \bm{s}_{i0}), \end{align} \textbf{\textit{Step 2.}} Next, similar to the proof of Lemma (ref), we apply the extended linearity assumption to replace the random intermediate outcomes $\bm{G}$ by their expectations $\bm{g}$. Denote, for any $t$ and $\delta$, $\bm{G}^\delta_t = (G^\delta_{t, 1}, ..., G^\delta_{t, D})$ and $\bm{g}^\delta_t = (g^\delta_{t, 1}, ..., g^\delta_{t, D})$. We then have \begin{align*} & \ \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{T}\left( \bm{G}^{\Delta t_K}_{t_K}( ... \bm{G}^{\Delta t_2}_{t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \\ = & \mathbb{E}_{\mathcal{F}} \left[ \alpha^{\Delta t_{K+1}}_{0,T}(\bm{x}_i, \bm{1}_{\Delta t_{K+1}}) + \sum_{d=1}^D G^{\Delta t_K}_{t_K, d}( ... \bm{G}^{\Delta t_2}_{t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}) \cdot \alpha^{\Delta t_{K+1}}_{d,T}(\bm{x}_i, \bm{1}_{\Delta t_{K+1}})\right] \\ = & \alpha^{\Delta t_{K+1}}_{0,T}(\bm{x}_i, \bm{1}_{\Delta t_{K+1}}) \\ & + \sum_{d=1}^D \mathbb{E}_{\mathcal{F}} \left[g^{\Delta t_K}_{t_K, d}(\bm{G}^{\Delta t_{K-1}}_{t_{K-1}}( ... \bm{G}^{\Delta t_2}_{t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_{K-1}}), \bm{x}_i, \bm{1}_{\Delta t_K})\right] \cdot \alpha^{\Delta t_{K+1}}_{d,T}(\bm{x}_i, \bm{1}_{\Delta t_{K+1}}) \\ = & \alpha^{\Delta t_{K+1}}_{0,T}(\bm{x}_i, \bm{1}_{\Delta t_{K+1}}) \\ & + \sum_{d=1}^D \mathbb{E}_{\mathcal{F}} \left[ \beta^{\Delta t_K}_{d,0,t_K}(\bm{x}_i, \bm{1}_{\Delta t_{K}}) + \sum_{d'=1}^D G^{\Delta t_{K-1}}_{t_{K-1}, d'}( ... \bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}) ... ,\bm{x}_i, \bm{1}_{\Delta t_{K-1}}) \cdot \beta^{\Delta t_K}_{d,d',t_K}(\bm{x}_i, \bm{1}_{\Delta t_{K}})\right] \cdot \alpha^{\Delta t_{K+1}}_{d,T}(\bm{x}_i, \bm{1}_{\Delta t_{K+1}}) \\ = & \alpha^{\Delta t_{K+1}}_{0,T}(\bm{x}_i, \bm{1}_{\Delta t_{K+1}}) \\ & + \sum_{d=1}^D \left(\beta^{\Delta t_K}_{d,0,t_K}(\bm{x}_i, \bm{1}_{\Delta t_{K}}) + \sum_{d'=1}^D \mathbb{E}_{\mathcal{F}} \left[g^{\Delta t_{K-1}}_{t_{K-1}, d'}( ... \bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}) ... ,\bm{x}_i, \bm{1}_{\Delta t_{K-1}})\right] \cdot \beta^{\Delta t_K}_{d,d',t_K}(\bm{x}_i, \bm{1}_{\Delta t_{K}})\right) \cdot \alpha^{\Delta t_{K+1}}_{d,T}(\bm{x}_i, \bm{1}_{\Delta t_{K+1}}) \\ = & \ldots \\ = & h^{\Delta t_{K+1}}_{T}\left( \bm{g}^{\Delta t_K}_{t_K}( ... \bm{g}^{\Delta t_2}_{t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}),\bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \end{align*} where the first equality is due to Assumption (ref); the second and forth equality is due to linearity of expectation; the third equality is due to Assumption (ref), with which the equation can be iteratively expressed into a linear form, and the position of the expectation notion can be adjusted inside. Similarly, \begin{align*} & \ \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}}\left( \bm{G}^{\Delta t_K}_{t_K}( ... \bm{G}^{\Delta t_2}_{t_2}(\bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \\ = & \alpha^{\Delta t_{K+1}}_{0,\Delta t_{K+1}}(\bm{x}_i, \bm{1}_{\Delta t_{K+1}}) \\ & + \sum_{d=1}^D \left(\beta^{\Delta t_K}_{d,0,t_K}(\bm{x}_i, \bm{1}_{\Delta t_{K}}) + \sum_{d'=1}^D \mathbb{E}_{\mathcal{F}} \left[g^{\Delta t_{K-1}}_{t_{K-1}, d'}( ... \bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}) ... ,\bm{x}_i, \bm{1}_{\Delta t_{K-1}})\right] \cdot \beta^{\Delta t_K}_{d,d',t_K}(\bm{x}_i, \bm{1}_{\Delta t_{K}})\right) \cdot \alpha^{\Delta t_{K+1}}_{d,\Delta t_{K+1}}(\bm{x}_i, \bm{1}_{\Delta t_{K+1}}) \\ = & \ldots \\ = & h^{\Delta t_{K+1}}_{\Delta t_{K+1}}\left( \bm{g}^{\Delta t_K}_{t_K}( ... \bm{g}^{\Delta t_2}_{t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}),\bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \end{align*} Similarly, \begin{align*} & \ \mathbb{E}_{\mathcal{F}} \left[ h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{G}^{\Delta t_K}_{\Delta t_K}( \bm{G}^{\Delta t_{K-1}}_{ t_{K-1}}( ... \bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}) ... ,\bm{x}_i, \bm{1}_{\Delta t_{K-1}}),\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \right] \\ = & \alpha^{\Delta t_{K+1}}_{0,\Delta t_{K+1}}(\bm{x}_i, \bm{1}_{\Delta t_{K+1}}) \\ & + \sum_{d=1}^D \Bigg(\beta^{\Delta t_K}_{d,0,\Delta t_K}(\bm{x}_i, \bm{1}_{\Delta t_{K}}) + \sum_{d'=1}^D \Big(\beta^{\Delta t_{K-1}}_{d',0,\Delta t_{K-1}}(\bm{x}_i, \bm{1}_{\Delta t_{K-1}}) \\ & + \sum_{d”=1}^D \mathbb{E}_{\mathcal{F}} \left[ ... \bm{G}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}) ... \right] \cdot \beta^{\Delta t_{K-1}}_{d',d,”\Delta t_{K-1}}(\bm{x}_i, \bm{1}_{\Delta t_{K-1}})\Big) \cdot \beta^{\Delta t_K}_{d,d',\Delta t_K}(\bm{x}_i, \bm{1}_{\Delta t_{K}})\Bigg) \cdot \alpha^{\Delta t_{K+1}}_{d,\Delta t_{K+1}}(\bm{x}_i, \bm{1}_{\Delta t_{K+1}}) \\ = & \ldots \\ = & h^{\Delta t_{K+1}}_{\Delta t_{K+1}}\left( \bm{g}^{\Delta t_K}_{\Delta t_K}( \bm{g}^{\Delta t_{K-1}}_{ t_{K-1}}( ... \bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}) ... ,\bm{x}_i, \bm{1}_{\Delta t_{K-1}}),\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \end{align*} Iteratively applying the above action, we are able to rewrite equation (ref) into the following expression: \begin{align*} & \ \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{1}_{T}) \right] \\ = & \ \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{T}\left( \bm{g}^{\Delta t_K}_{t_K}( ... \bm{g}^{\Delta t_2}_{t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{t_K}( ... \bm{g}^{\Delta t_2}_{t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{t_K}( ... \bm{g}^{\Delta t_2}_{t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( \bm{g}^{\Delta t_{K-1}}_{ t_{K-1}}( ... \bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}) ... ,\bm{x}_i, \bm{1}_{\Delta t_{K-1}}),\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( \bm{g}^{\Delta t_{K-1}}_{ t_{K-1}}( ... \bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}) ... ,\bm{x}_i, \bm{1}_{\Delta t_{K-1}}),\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & \ldots \nonumber \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \end{align*} Similarly, equation (ref) can be rewritten as \begin{align*} & \ \mathbb{E}_{\mathcal{F}}\left[ Y_{iT}(\bm{0}_{T}) \right] \\ = & \ \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{T}\left( \bm{g}^{\Delta t_K}_{t_K}( ... \bm{g}^{\Delta t_2}_{t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{t_K}( ... \bm{g}^{\Delta t_2}_{t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{t_K}( ... \bm{g}^{\Delta t_2}_{t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( \bm{g}^{\Delta t_{K-1}}_{ t_{K-1}}( ... \bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}) ... ,\bm{x}_i, \bm{0}_{\Delta t_{K-1}}),\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( \bm{g}^{\Delta t_{K-1}}_{ t_{K-1}}( ... \bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}) ... ,\bm{x}_i, \bm{0}_{\Delta t_{K-1}}),\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & \ldots \nonumber \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right)\cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \end{align*} Combining both parts, \begin{align} \tau_T = &\ \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \Bigg(h^{\Delta t_{K+1}}_{T}\left( \bm{g}^{\Delta t_K}_{t_K}( ... \bm{g}^{\Delta t_2}_{t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \nonumber\\ & - h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{t_K}( ... \bm{g}^{\Delta t_2}_{t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right)\Bigg) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \Bigg(h^{\Delta t_{K+1}}_{T}\left( \bm{g}^{\Delta t_K}_{t_K}( ... \bm{g}^{\Delta t_2}_{t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \nonumber \\ & - h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{t_K}( ... \bm{g}^{\Delta t_2}_{t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right)\Bigg) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \Bigg(h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{t_K}( ... \bm{g}^{\Delta t_2}_{t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \nonumber \\ & - h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( \bm{g}^{\Delta t_{K-1}}_{ t_{K-1}}( ... \bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}) ... ,\bm{x}_i, \bm{1}_{\Delta t_{K-1}}),\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right)\Bigg) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \Bigg(h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{t_K}( ... \bm{g}^{\Delta t_2}_{t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \nonumber \\ & - h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( \bm{g}^{\Delta t_{K-1}}_{ t_{K-1}}( ... \bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}) ... ,\bm{x}_i, \bm{0}_{\Delta t_{K-1}}),\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right)\Bigg) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & +\ \cdots\ -\ \cdots \nonumber\\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \Bigg(h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \nonumber \\ & - h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right)\Bigg) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ & - \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \Bigg(h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \nonumber \\ & - h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right) \Bigg) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \\ & + \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \Bigg(h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \nonumber \\ & - h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right)\Bigg) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ = &\ \sum_{\bm{x}_i \in \mathbb{X}, \bm{s}_{i0} \in \mathbb{S}} \Bigg(h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{1}_{\Delta t_1}), \bm{x}_i, \bm{1}_{\Delta t_2}) ... ,\bm{x}_i, \bm{1}_{\Delta t_K}), \bm{x}_i, \bm{1}_{\Delta t_{K+1}} \right) \nonumber \\ & - h^{\Delta t_{K+1}}_{\Delta t_{K+1}} \left( \bm{g}^{\Delta t_K}_{\Delta t_K}( ... \bm{g}^{\Delta t_2}_{\Delta t_2}(\bm{g}^{\Delta t_1}_{\Delta t_1}(\bm{s}_{i0}, \bm{x}_i, \bm{0}_{\Delta t_1}), \bm{x}_i, \bm{0}_{\Delta t_2}) ... ,\bm{x}_i, \bm{0}_{\Delta t_K}), \bm{x}_i, \bm{0}_{\Delta t_{K+1}} \right)\Bigg) \cdot \Pr(\bm{x}_i, \bm{s}_{i0}) \nonumber \\ = &\ \mathbb{E}_{\mathcal{F}}\left[h_{\Delta t_{K+1}}\left( \bm{g}_{\Delta t_K}( ... \bm{g}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{1}_{\Delta t_1}) ... ,\bm{X}_i, \bm{1}_{\Delta t_K}), \bm{X}_i, \bm{1}_{\Delta t_{K+1}} \right)\right] \nonumber \\ & - \mathbb{E}_{\mathcal{F}}\left[h_{\Delta t_{K+1}}\left( \bm{g}_{\Delta t_K}( ... \bm{g}_{\Delta t_1}(\bm{S}_{i0}, \bm{X}_i, \bm{0}_{\Delta t_1}) ... ,\bm{X}_i, \bm{0}_{\Delta t_K}), \bm{X}_i, \bm{0}_{\Delta t_{K+1}} \right)\right] \nonumber \end{align} where in the first equality, equation (ref) is equal to zero due to Assumption (ref) when $t=T,\ t'=\Delta t_{K+1},\ \delta=\Delta t_{K+1}$; equation (ref) is equal to zero due to Assumption (ref) when $u_0=\Delta t_{K+1},\ t=t_{K},\ t'=\Delta t_{K},\ \delta=\Delta t_{K}$; ... ; equation (ref) is equal to zero due to Assumption (ref) when $(u_0, ..., u_{K-1})=(\Delta t_{K+1},...,\Delta t_{2}),\ t=t_{1},\ t'=\Delta t_{1},\ \delta=\Delta t_{1}$. This concludes the proof. \halmos \endproof \subsection{Testing Parallel Trends Assumption} We further introduce a statistical test for whether the parallel trends assumption holds. This test focuses on two time periods, \(t\) and \(t'\), and a specified positive integer \(\delta\). For each unit $i$ characterized by the values of any observations of \(\bm{S}_{i,t-\delta}\) and \(\bm{X}_i\) at period $t$ in the treatment group (where \(\bm{W}_{i,t-\delta+1:t}=\bm{1}_{\delta}\)), we search for an exact match at time period \(t'\) based on the same values of \(\bm{S}_{i,t'-\delta}\), \(\bm{X}_i\) and \(\bm{W}_{i,t'-\delta+1:t'} = \bm{1}_{\delta}\). If an exact match is found, one match in period $t'$ is randomly selected for comparison; then we have a pair of ($Y_{it}$ vs $Y_{i't'}$ and $\bm{S}_{it}$ vs $\bm{S}_{i't'}$) for the treatment group. In cases where no match is found, the observation is not used in this evaluation. Similarly, we repeat the procedure for observations in the control group (where \(\bm{W}_{i,t-\delta+1:t}=\bm{0}_{\delta}\)), and obtain pairs of units at period $t$ and $t'$. The above matching process ensures the paired observations in both treatment and control groups are conditioned on the same distribution of pre-period surrogates and pre-treatment variables. By pooling these paired observations together, we are able to perform a test based on a difference-in-difference type regression to evaluate if the Assumption (ref) holds. Consider the following regression: $$Y_i = \beta_0 + \beta_1\cdot \mathbbm{1}\{\bm{w}_{i,\delta}=\bm{1}_{\delta}\} + \beta_2\cdot \mathbbm{1}\{period = t\} + \beta_3\cdot \mathbbm{1}\{\bm{w}_{i,\delta}=\bm{1}_{\delta}\}\cdot\mathbbm{1}\{period = t\} + \epsilon_i$$ where variable \(period\) controls the time period of the observation, \(\bm{w}_{i,\delta}\) controls the treatment assignment condition. We conduct the regression, and apply a t-test for the null \(H_{0}: \beta_3=0\). We say that Assumption (ref) is not significantly violated if the null hypothesis can not be rejected. Table (ref) presents the results of statistical testing for various combinations of $t$, $t'$, and $\delta$ across both Empirical Experiment 1 and Empirical Experiment 2. These tests are conducted on data collected during the observable experimental periods, which is week two to week four for Experiment 1, and week eight to week ten for Experiment 2. We observe that p-values for all of tests are greater than 0.05, and none of the tests were rejected, providing evidence that there is no significant violation of Assumption (ref) in either experiment. \begin{table}[htbp] \caption{Testing results of the parallel trends assumption for two empirical experiments}\scriptsize \begin{tabular}{cccccccc} \toprule \textbf{Experiment} &$\bm{t}$ & $\bm{t'}$ & $\bm{\delta}$ & $\widehat{\beta_3}$ & \textbf{t-statistic} & \textbf{p-value} & \textbf{Reject?} \\ \midrule \multirow{4}{*}{Experiment 1} & 2 & 3 & 1 & -0.0009 & -1.057 & 0.290 & No \\ & 2 & 4 & 1 & -0.0011 & -1.295 & 0.195 & No \\ & 3 & 4 & 1 & 0.0001 & 0.064 & 0.949 & No \\ & 3 & 4 & 2 & -0.0008 & -0.923 & 0.356 & No \\ \midrule \multirow{16}{*}{Experiment 2} & 8 & 9 & 6 & 0.00002 & 0.026 & 0.979 & No\\ & 8 & 9 & 7 & 0.0009 & 0.884 & 0.377 & No \\ &8 & 10 & 6 & 0.0003 & 0.284 & 0.776 & No \\ &8 & 10 & 7 & 0.0005 & 0.532 & 0.594 & No\\ &8 & 11 & 6 & -0.0010 & -1.122 & 0.261 & No\\ &8 & 11 & 7 & -0.0002 & -0.153 & 0.879 & No\\ &9 & 10 & 6 & 0.0003 & 0.333 & 0.739 & No\\ &9 & 10 & 7 & -0.0006 & -0.667 & 0.505 & No\\ &9 & 10 & 8 & 0.0005 & 0.452 & 0.651 & No\\ &9 & 11 & 6 & 0.0002 & 0.249 & 0.804 & No\\ &9 & 11 & 7 & -0.0006 & -0.663 & 0.507 & No\\ &9 & 11 & 8 & 0.0010 & 0.986 & 0.324 & No\\ &10 & 11 & 6 & 0.0008 & 0.827 & 0.408 & No\\ &10 & 11 & 7 & 0.0006 & 0.612 & 0.540 & No\\ &10 & 11 & 8 & 0.00001 & 0.006 & 0.996 & No\\ &10 & 11 & 9 & -0.0008 & -0.752 & 0.452 & No\\ \bottomrule \end{tabular} \end{table} \section{Sensitivity Analysis on Longitudinal Surrogacy Assumption} The surrogacy assumption necessitates that the chosen surrogate variables encapsulate the entire causal path from the previous treatment to the future primary outcome. A illustrator of the typical violation of Assumption (ref) is shown in Figure (ref). Here we show that our method's estimation remains robust even when the longitudinal surrogacy assumption is moderately violated (i.e., there exists a causal path not blocked by surrogates over time) through both synthetic and empirical experiments. \begin{figure}[h] \caption{An illustrator of a scenario where Assumption (ref) is violated.} { \textit{Note}: In this illustrator, each solid line represents a causal link. The treatment assignment at period $T_E+1:T_F$ impacts the primary outcomes at period $T_E+1:T_F$; The treatment assignment at period $T_E+1:T_F$ impact the primary outcome at period $T_E+1:T_F$ through both surrogate outcomes and omitted surrogates at period $1:T_E$. } \end{figure} \subsection{Sensitivity Analysis with Subsets of Surrogate} \subsubsection{Evidence from the synthetic experiments} We revisit the two synthetic experiments discussed in the main context. Both simulations initialize with four surrogates, denoted as \(\bm{S}_i = (S^1_{i0},S^2_{i0},S^3_{i0},S^4_{i0})\). Each surrogate follows a normal distribution: \(S^d_{i0}\sim\mathcal{N}(\mu_d,\sigma_d)\), where \(\mu_d \sim \mathcal{N}(2,1)\) and \(\sigma_d \sim \mathcal{N}(2,1)\) for \(d\in \{1,2,3,4\}\); if negative value is sampled we flip the sign. It is evident that these four surrogates are independent of each other. Throughout the experiment, the distribution of the surrogates remains unchanged in the control group, i.e. \(S^d_{it} \sim\mathcal{N}(\mu_d,\sigma_d)\) for \(d\in \{1,2,3,4\}\) and each period $t$. However, in the treatment group, the surrogate values decay over time governed by decay factors $\kappa_d$ of \([0.8,0.6,0.4,0.2]\) respectively, i.e. \(S^{d}_{i,t+1} = \kappa_d\cdot S^d_{t}\) for \(d\in \{1,2,3,4\}\) and each period $t$. The primary outcome, \(Y\), is designed as \(Y_{i,t+1} = -(0.1S^1_{it} + 0.1S^2_{it} + 0.4S^3_{it} + 0.4S^4_{it})\) in the first synthetic experiment, and \(Y_{i,t+1} = 0.1S^1_{it} + 0.1S^2_{it} + 0.4S^3_{it} + 0.4S^4_{it}\) in the second synthetic experiment. In the following analysis, we adhere to the same data generation process described above. However, we proceed as if we only observe a subset of the four underlying surrogates. Therefore, if we only use a subset as surrogates in our method, the longitudinal surrogacy assumption is violated. Figures (ref) and (ref) showcase the results for the estimated long-term treatment effect, utilizing only subsets of the full surrogates in the two synthetic experiments. The specific set of surrogates used for prediction is indicated in the far-right column of the figures. It is evident that our linear surrogate model performs better with a more comprehensive set of surrogates. Furthermore, the influence of surrogacy violation varies based on the length of the experimental period data. Given the same set of surrogates, the estimation approach demonstrates greater robustness with a longer experimental period as opposed to a shorter one. \begin{figure}[tb] \caption{Estimated effect with violated surrogacy assumption in the Synthetic Experiment 1} { \textit{Note}: Grey dashed curves represent the true average treatment effect on $Y$ from periods 1 to periods 10. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The four rows represent the specific set of surrogates used for prediction, where a smaller set signifies a more significant violation of the surrogacy assumption. The three panels represent the scenarios when we use the first $T_E$ weeks as the experimental period and the last $T_F$ weeks as the future period.} \end{figure} \begin{figure}[tb] \caption{Estimated effect with violated surrogacy assumption in the Synthetic Experiment 2} { \textit{Note}: Grey dashed curves represent the true average treatment effect on $Y$ from periods 1 to periods 10. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The four rows represent the specific set of surrogates used for prediction, where a smaller set signifies a more significant violation of the surrogacy assumption. The three panels represent the scenarios when we use the first $T_E$ weeks as the experimental period and the last $T_F$ weeks as the future period.} \end{figure} \subsubsection{Evidence from the empirical experiments} Similar to what has been done in the synthetic experiments, we attempt to use only a subset of surrogates to build the longitudinal surrogate model, and compare the estimation from which with the true effect in two empirical experiments. Figure (ref) and Figure (ref) present the estimation results obtained from the linear surrogate model using different subsets of the full surrogates for both empirical experiments. The specific surrogates employed for prediction are listed in the far-right column of the figures. In the real-world study, we can still observe an overall tendency that richer surrogates result in a more precise estimation. \begin{figure}[h] \caption{Estimated effect with violated surrogacy assumption in Experiment 1} { \textit{Note}: Grey dashed curves represent the true average treatment effect on \textit{search_uv} from week 1 to week 7. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The five rows represent the specific set of surrogates used for prediction, where a smaller set signifies a more significant violation of the surrogacy assumption. The three panels represent the scenarios when we use the first $T_E$ weeks as the experimental period and the last $T_F$ weeks as the future period.} \end{figure} \begin{figure}[h] \caption{Estimated effect with violated surrogacy assumption in Experiment 2} { \textit{Note}: Grey dashed curves represent the true average treatment effect on \textit{search_uv} from week 1 to week 20. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The five rows represent the specific set of surrogates used for prediction, where a smaller set signifies a more significant violation of the surrogacy assumption. The three panels represent the scenarios when we use the first $T_E$ weeks as the experimental period and the last $T_F$ weeks as the future period.} \end{figure} \subsection{Sensitivity Analysis of Omitted Surrogates} In real-world experiments, it is hard to quantify how much the surrogacy assumption is violated because we can't be certain if current surrogates fully satisfy this assumption. This challenge is similar to that faced in the literature regarding the validity of instrumental variables (IVs). Drawing on sensitivity analysis techniques used for IVs baiocchi2014instrumental, we carry out sensitivity analysis on the both empirical experiments introduced in the main text to demonstrate further the robustness of our method when the surrogacy assumption is relaxed in real-world scenarios. Suppose for any $i \in [N]$ and any $t \in \mathbb{T}$, the treatment assignment affects the primary outcome not only through the identified surrogates, but also via an unobserved variable $\zeta_{it}$ which follows normal distribution with mean zero, and variance equals to the average variance of the primary outcome during the experimental period. We manually introduce an additional causal path between the treatment assignment and the primary outcome through variable $\zeta_{it}$ as follows: \begin{align*} \tilde{Y}_{it}(\bm{w}_{1:t}) = Y_{it}(\bm{w}_{1:t}) + \theta \cdot \mathbbm{1}[\bm{w}_{1:t}=\bm{1}_{1:t}] \cdot \zeta_{it} \end{align*} In this way, if we use $\tilde{Y}$ instead of $Y$ as the outcome variable and consider only the existing surrogates, a larger $\theta$ indicates that our longitudinal surrogacy assumption is violated to a greater extent. Our objective is to illustrate that our model is relatively robust against this potential violation. Figure (ref) presents a comparative analysis of bias and root mean squared error (RMSE) for Experiment 1 and Experiment 2, with varying values of $\theta$. We apply the same experimental periods from the main text, calculating the average bias and RMSE for each $\theta$. Specifically, for Experiment 1, this is averaged over $T_E = 2, 3, 4$, and for Experiment 2, over $T_E = 8, 9, 10$. The analysis indicates that, compared to our main model estimates (where $\theta = 0$), the bias and RMSE remain similar when $\theta$ is relatively small, demonstrating the robustness of the estimation. \begin{figure}[h] \caption{Performance of Linear Surrogate Model in terms of Bias and RMSE under varying values of $\theta$ for Experiment 1 and Experiment 2} \end{figure} \section{Additional Results for Synthetic Experiments} \subsection{Results for synthetic experiments with stabilized treatment effect} Here we present the detailed results for the synthetic experiments discussed in Section (ref) of the main text. Figures (ref) and (ref) graphically illustrate our model alongside baseline methods. As shown in both figures, our model effectively captures the decreasing and stabilizing trend of long-term treatment effects, a pattern that baseline models fail to capture. Numerical comparisons provided in Tables (ref) and (ref) further demonstrate that our approach outperforms baseline models in terms of both bias and MSE. \begin{figure}[h] \caption{Effects of Long-term Treatment using Linear Surrogate Model for the Synthetic Experiment 1} { \textit{Note}: Grey dashed curves represent the true average treatment effect on $Y$ from periods 1 to periods 10. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The three panels represent the scenarios when we use the first $T_E$ periods as the experimental period and the last $T_F$ periods as the future period.} \end{figure} \begin{figure}[h] \caption{Effects of Long-term Treatment using Linear Surrogate Model for the Synthetic Experiment 1} { \textit{Note}: Grey dashed curves represent the true average treatment effect on $Y$ from periods 1 to periods 10. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The three panels represent the scenarios when we use the first $T_E$ periods as the experimental period and the last $T_F$ periods as the future period.} \end{figure} \begin{table}[h!] \caption{Comparison result between different methods in terms of bias and MSE for synthetic experiment 1}\scriptsize \begin{tabular}{c ccc ccc} \toprule \multirow{2}{*}{\textbf{Method}}&\multicolumn{3}{c}{\textbf{Bias}}&\multicolumn{3}{c}{\textbf{MSE}}\\ & {$\mathbf{T_E=2}$} & {$\mathbf{T_E=3}$} & {$\mathbf{T_E=4}$} & {$\mathbf{T_E=2}$} & {$\mathbf{T_E=3}$} & {$\mathbf{T_E=4}$} \\ \midrule Linear surrogate model & $0.026$ & $0.029$ & $0.028$ & $0.003$ & $0.003$ & $0.004$ \\ CEB & $1.661$ & $1.244$ & $0.992$ & $2.778$ & $1.567$ & $0.989$ \\ VAR model & $1.661$ & $0.785$ & $0.392$ & $2.778$ & $1.073$ & $0.459$ \\ \bottomrule \end{tabular} \end{table} \begin{table}[h!] \caption{Comparison result between different methods in terms of Bias and MSE for synthetic experiment 2}\scriptsize \begin{tabular}{c ccc ccc} \toprule \multirow{2}{*}{\textbf{Method}}&\multicolumn{3}{c}{\ \textbf{Bias}}&\multicolumn{3}{c}{\textbf{MSE}}\\ & {$\mathbf{T_E=2}$} & {$\mathbf{T_E=3}$} & {$\mathbf{T_E=4}$} & {$\mathbf{T_E=2}$} & {$\mathbf{T_E=3}$} & {$\mathbf{T_E=4}$} \\ \midrule Linear surrogate model & $0.015$ & $0.016$ & $0.015$ & $0.001$ & $0.001$ & $0.001$ \\ CEB & $1.338$ & $1.002$ & $0.800$ & $1.802$ & $1.007$ & $0.642$ \\ VAR model & $1.338$ & $0.048$ & $0.057$ & $1.802$ & $0.010$ & $0.007$ \\ \bottomrule \end{tabular} \end{table} \subsection{Violation of Comparability} In order to better illustrate the necessity of satisfying Assumption (ref) (Comparability Assumption), or alternatively, Assumption (ref) (Parallel Trends Assumption), we conduct following simulations where the primary outcome varies in how it breaches the comparability assumption. In the initial simulations, we consider two surrogates, $S_{i0,1}, S_{i0,2}$, each following a normal distribution: \(S_{i0,1} \sim\mathcal{N}(\mu_1,\sigma_1^2)\) \(S_{i0,2} \sim\mathcal{N}(\mu_2,\sigma_2^2)\), where \(\mu_1,\mu_2,\sigma_1,\sigma_2 \sim \mathcal{N}(2,1)\) before the experiment starts. For the control group, the distribution of surrogates remains unchanged throughout the experiment, i.e. \(S_{it,1} \sim\mathcal{N}(\mu_1,\sigma_1^2)\) \(S_{it,2}\sim\mathcal{N}(\mu_2,\sigma_2^2)\) for each period $t$. In contrast, for the treatment group, the values of the surrogates decrease over time, influenced by decay factors \([0.8,0.6]\) respectively, which means that $S_{i,t+1,1} = 0.8\cdot S_{i,t,1}$ and $S_{i,t+1,2} = 0.6\cdot S_{i,t,1}$ for each period $t$. We define the primary outcome \(Y_{it}\) as a linear combination of these two surrogates, formulated as \(Y_{it} = -(0.1S_{it,1} + 0.4S_{it,2})\) at each period $t$, except for $t=2$. At $t=2$, we introduce an external shock for the treatment group, where the primary outcome \(Y_{i2}\) is formulated as \(Y_{i2} = -\gamma *(0.1S_{i2,1} + 0.4S_{i2,2})\) for subjects in the treatment group, to simulate the possible scenario where a festival amplifies the effect of the treatment. For the control group, the primary outcome remains unchanged. We set $\gamma$ to be \([1,1.5,2,2.5,3]\), with larger values indicating a more significant violation of the comparability assumption (no violation when $\gamma=1$). We focus on the scenario where \(t=2\), \(t'=3\) and \(\delta=1\), comparing the distribution of the primary outcome at time period two, conditional on surrogates from one period before, with the distribution at time period three under the same conditions. A violation of the comparability assumption will result in a discrepancy between these distributions. We expect that both the comparability assumption test and the parallel trends assumption test we proposed can detect this violation. We first conduct the comparability assumption test for both the treatment and control groups under different values of \(\gamma\), as shown in Table (ref).\footnote{Note that the total number of tests remains the same across different \(\gamma\) values in the treatment group, since the condition, i.e., the value of surrogates, is unchanged and unrelated to \(\gamma\).} We observe that Assumption (ref) is most likely satisfied in the control group and in the treatment group when \(\gamma=1\). However, it is clearly violated in the treatment group as \(\gamma\) increases, with strong evidence. Table (ref), which presents the results for the parallel trends assumption test, reinforces this conclusion, as the t-test is rejected for \(\gamma > 1.5\). For the test under \(\gamma=1.5\), it is not rejected but shows a relatively small p-value, indicating that a slight violation of the parallel trends assumption may not severely impact the estimation due to the robustness of our approach. Figure (ref) displays the estimation results of our linear surrogate model under different \(\gamma\) values. We observe that as the degree of violation increases (\(\gamma\) becomes larger), the estimation becomes more biased. \begin{table}[h] \caption{Testing results of the comparability assumption with different values of $\gamma$}\scriptsize \begin{tabular}{cccccccc} \toprule $\bm{\gamma}$ & \textbf{Groups} & $\bm{\#}$\textbf{Tests} & $\bm{\# p<0.1}$ & $\bm{\# p<0.05}$ & $\bm{p<0.1 (\%)}$ & $\bm{p<0.05 (\%)}$ \\ \midrule 1.0 & \multirow{5}{*}{Treatment} & 217 & 17 & 10 & 7.83 & 4.61 \\ 1.5 & & 217 & 189 & 162 & 87.10 & 74.65 \\ 2.0 & & 217 & 201 & 180 & 92.63 & 82.95 \\ 2.5 & & 217 & 203 & 187 & 93.55 & 86.18 \\ 3.0 & & 217 & 207 & 195 & 95.39 & 89.86 \\ \cmidrule{3-7} / & Control & 62 & 7 & 4 & 11.29 & 6.45 \\ \bottomrule \end{tabular} \end{table} \begin{table}[htbp] \caption{Testing results of the parallel trends assumption with different values of $\gamma$}\scriptsize \begin{tabular}{ccccc} \toprule $\bm{\gamma}$ & $\widehat{\beta_3}$ & \textbf{t-statistic} & \textbf{p-value} & \textbf{Reject?} \\ \midrule 1.0 & -0.001 & -0.009 & 0.993 & No \\ 1.5 & -0.190 & -1.601 & 0.111 & No \\ 2.0 & 0.440 & -3.263 & 0.001 & Yes \\ 2.5 & -0.634 & -4.552 & 0.000 & Yes \\ 3.0 & -0.793 & -5.589 & 0.000 & Yes \\ \bottomrule \end{tabular} \end{table} \begin{figure}[h] \caption{Effect estimation in synthetic experiments when the comparability assumption is not satisfied.} { \textit{Note}: Grey dashed curves represent the true average treatment effect on $Y$ from periods 1 to periods 10. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The five columns represent a different degree of violation of the comparability assumption, with a larger $\gamma$ indicating a more severe violation (no violation when $\gamma=1$). We use the first three periods as the experimental period and the last seven periods as the future period.} \end{figure} \subsection{Non-linear} The previous synthetic experiments have had primary outcomes $Y_{it}$ as a linear combination of surrogates. As data distributions in real-world experiments may go beyond this linear formulation, we perturb the outcome functions in Section (ref) from linear to non-linear to probe the sensitivity of our approach. This shows the robustness of our approach under non-linear scenarios when Assumption (ref) is not satisfied. The simulations are initialized with two surrogate variables, denoted as $S_{i0,1}, S_{i0,2}$, each following a normal distribution: \(S_{i0,1} \sim\mathcal{N}(\mu_1,\sigma_1^2)\) \(S_{i0,2} \sim\mathcal{N}(\mu_2,\sigma_2^2)\), where \(\mu_1,\mu_2,\sigma_1,\sigma_2 \sim \mathcal{N}(2,1)\) before the experiment starts. Once the experiment begins, the surrogates in the control group continue to follow the same distribution as in the pre-treatment period, i.e. \(S_{it,1} \sim\mathcal{N}(\mu_1,\sigma_1^2)\) \(S_{it,2}\sim\mathcal{N}(\mu_2,\sigma_2^2)\) for each period $t$. By contrast, the values of surrogates in the treatment group decrease over time, influenced by decay factors of \([0.8,0.6]\) respectively, which means that $S_{i,t+1,1} = 0.8\cdot S_{i,t,1}$ and $S_{i,t+1,2} = 0.6\cdot S_{i,t,1}$ for each period $t$. In the first synthetic experiment, the primary outcome $Y$ is designed as a non-linear relationship, $Y_{i,t+1} = -(S_{it,1} + \theta\cdot e^{S_{it,2}})$ at each period t, where $\theta$ controls the magnitude of the exponential term. With this configuration, the average treatment effect of the long-term treatment initially increases and eventually converges to $\lim_{t\to\infty}\mathbb{E}[Y_{i,t}(1)-Y_{i,t}(0)] = \mu_1+\theta\cdot e^{\mu_2+\sigma_2^2/2}-1$ at time $t$ approaches infinity. The exponential term introduces nonlinearity to the function, thereby violating the linearity assumption when estimating the treatment effect with a linear surrogate index function. We present the results of our estimation in Figure (ref). We observe that the estimation is relatively stable and valid regardless of the magnitude of $\theta$, which illustrates the robustness of our approach. \begin{figure}[tb] \caption{Robust effect estimation in the first synthetic experiment when the linearity assumption is not satisfied.} { \textit{Note}: Grey dashed curves represent the true average treatment effect on $Y$ from periods 1 to periods 10. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The three rows represent a different degree of violation of the linearity assumption, with a larger $\theta$ indicating a more severe violation. The three panels represent the scenarios when we use the first $T_E$ periods as the experimental period and the last $T_F$ periods as the future period.} \end{figure} In the second synthetic experiment, we similarly design the primary outcome $Y$ as $Y_{i,t+1} = S_{it,1} + \theta\cdot e^{S_{it,2}}$, where $\theta$ determines the magnitude of the exponential term. In this scenario, the average treatment effect of the long-term treatment initially decreases and then stabilizes around a certain level, which is $1-\mu_1-\theta\cdot e^{\mu_2+\sigma_2^2/2}$. Similarly, as shown in Figure (ref), the estimation remains reasonably accurate under these conditions, further demonstrating the robustness of our method. \begin{figure}[tb] \caption{Robust effect estimation in the second synthetic experiment when the linearity assumption is not satisfied.} { \textit{Note}: Grey dashed curves represent the true average treatment effect on $Y$ from periods 1 to periods 10. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The three rows represent a different degree of violation of the linearity assumption, with a larger $\theta$ indicating a more severe violation. The three panels represent the scenarios when we use the first $T_E$ periods as the experimental period and the last $T_F$ periods as the future period.} \end{figure} \subsection{No Long-Term Treatment Effect} So far our synthetic experiments have focused on estimating the effect of a long-term treatment that has a significantly positive impact. We present an additional synthetic experiment to demonstrate that our methodology remains effective even when the long-term treatment effect eventually diminishes to zero. The initialization of the simulation is the same as in previous settings, which started with four surrogate variables, \(\bm{S}= [S_{i0,1},S_{i0,2},S_{i0,3},S_{i0,4}]\), each adhering to a normal distribution: \(S_{i0,d}\sim\mathcal{N}(\mu_d,\sigma_d^2)\), where \(\mu_d \sim \mathcal{N}(2,1)\) and \(\sigma_d \sim \mathcal{N}(2,1)\) for \(d\in \{1,2,3,4\}\) before the start of the experiment. Different from the previous settings, the surrogates in both the control group and the treatment group maintain the pre-treatment distribution throughout the experiment, i.e. \(S_{it,d} \sim\mathcal{N}(\mu_d,\sigma_d^2)\) for \(d\in \{1,2,3,4\}\) and each period $t$. The primary outcome $Y$ in the control group is formulated as \(Y_{i,t+1} = -(0.1S_{it,1}+ 0.1S_{it,2} + 0.4S_{it,3} + 0.4S_{it,4})\), while the $Y$ in the treatment group is formulated as \(Y_{t+1} = -(0.1S_{it,1}+ 0.1S_{it,2} + 0.4S_{it,3} + 0.4S_{it,4}) + \frac{(-1)^t}{(t+2)^3}\). The additional term for treatment group's $Y$ controls the volatility of the treatment effect. Given that $\frac{(-1)^t}{(t+2)^3}$ converges to zero as $t$ approaches infinity, the expectation of the average treatment effect in the long term is \textit{zero}. The estimated effect in the short term may exhibit significant fluctuations due to the disturbance term, which adds complexity to the prediction of the long-term treatment effect. However, as shown in Figure (ref), our approach can anticipate the eventual convergence level using only short-term experimental data, illustrating the capability of our method under various conditions. \begin{figure}[tb] \caption{Estimated effect using the linear surrogate model for a treatment with no long-term effect} { \textit{Note}: Grey dashed curves represent the true average treatment effect on $Y$ from periods 1 to periods 10. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The three panels represent the scenarios when we use the first $T_E$ periods as the experimental period and the last $T_F$ periods as the future period.} \end{figure} \subsection{Limitations of the Method} The additional synthetic experiments and sensitivity analyses demonstrate the robustness of our estimator from multiple angles. However, its predictive performance can deteriorate sharply when the underlying assumptions are grossly violated. To make these risks transparent, we examine representative failure modes associated with each of the three core assumptions—\emph{Surrogacy}, \emph{Comparability}, and \emph{Linearity}. \subsubsection{Violation of Assumption (ref)} In Section (ref) we show that our estimator remains partially informative when Assumption (ref) is only mildly violated---the resulting bias grows roughly in proportion to the severity of the violation. Here we explore two stark departures that can render the estimator unreliable. (i) \emph{Omitted surrogates}: the candidate set excludes key variables that carry most of the causal pathway from treatment to outcome. (ii) \emph{Irrelevant surrogates}: the set contains variables unrelated to the treatment effect, which inject noise and may aggravate any violation of the surrogacy condition. Consider the following simulation setup: The simulations begin with two variables, \(S_{i0,1}\) and \(S_{i0,2}\), each drawn from a normal distribution \(S_{i0,1} \sim\mathcal{N}(\mu_1,\sigma_1^2)\) \(S_{i0,2} \sim\mathcal{N}(\mu_2,\sigma_2^2)\), where \(\mu_1,\mu_2,\sigma_1,\sigma_2\) are themselves sampled from \(\mathcal{N}(2, 1)\) prior to the start of the experiment. For the control group, the variables remain unchanged throughout the experimental period, continuing to follow their initial distributions, i.e. \(S_{it,1} \sim\mathcal{N}(\mu_1,\sigma_1^2)\) \(S_{it,2}\sim\mathcal{N}(\mu_2,\sigma_2^2)\) for each period $t$. In the treatment group, the variables fluctuate periodically, influenced by factors \([1-(-1)^t/5,1-(-1)^{t+1}/5]\) respectively. Specifically, for \(S_{i,t+1,1}\), \(S_{i,t+1,1} = 1.2 \cdot S_{i,t,1}\) when \(t\) is odd and \(S_{i,t+1,1} = 0.8 \cdot S_{i,t,1}\) when \(t\) is even; Similarly, \(S_{i,t+1,2} = 0.8 \cdot S_{i,t,2}\) when \(t\) is odd and \(S_{i,t+1,2} = 1.2 \cdot S_{i,t,2}\) when \(t\) is even for each period $t$. The primary outcome \(Y\) in period $t$ is formulated as \(Y_{it} = -(0.5\cdot S_{it,1} + 0.5\cdot S_{it,2})\). Based on the described data generation process, we apply our method using different surrogate variable sets to assess the impact of surrogate selection on estimation. First, we consider a complete set of surrogate variables, \([S_1, S_2, Y]\), which fully satisfies Assumption (ref), capturing all key variables necessary to explain the causal relationship between the treatment and outcomes. Second, we use only the primary outcome, \([Y]\), as the surrogate variable. This omission of \(S_1\) and \(S_2\), which are critical variables, reduces the model's ability to explain the causality. Finally, we introduce an irrelevant variable, \(S'\), which follows a normal distribution in the control group, \(S'_{it} \sim \mathcal{N}(\mu', \sigma'^2)\), where \(\mu'\) and \(\sigma'\) are sampled from \(\mathcal{N}(2, 1)\), and evolves dynamically in the treatment group as \(S'_{i,t+1} = 2 \cdot S'_{i,t}\). We apply our method with the surrogate set \([Y, S']\) to demonstrate the impact of including misleading variables to the estimation process. Figure (ref) illustrates the estimation results of our method with these three surrogate sets given the experimental period \(T_E = 2\). The results show that only the first case produces an accurate estimation compared to the true effect, while the second and third cases fail to estimate the long-term treatment effect precisely. This analysis emphasizes the importance of selecting appropriate surrogate variables to ensure accurate and reliable results. \begin{figure}[h] \caption{Estimated effect using the linear surrogate model with different set of surrogates} { \textit{Note}: Grey dashed curves represent the true average treatment effect on $Y$ from periods 1 to periods 10. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The three panels represent the scenarios where we apply the linear surrogate model with different set of surrogates. We use the first 2 periods as the experimental period and the last 8 periods as the future period.} \end{figure} \subsubsection{Violation of Assumption (ref)} We now construct a scenario that explicitly violates the \emph{comparability} assumption to illustrate how our estimator can fail when Assumption (ref) is ignored. The simulated data is generated similarly to the synthetic experiment mentioned in Section (ref), with one key difference: the distribution of surrogate variables in the treatment group shift before and after period \(t = 3\). For \(t \leq 3\), the decaying factors \(\gamma_d\) for each \(S_{it,d}\) are set as \(\gamma_d = 1 - (d + 1) \cdot (-1)^t / 10\), causing fluctuations in the primary outcome \(Y\) during the first three periods. For \(t > 3\), the decaying factors remain consistent with the settings, and all other parameters are identical to those described in the previous synthetic experiments. Apparently, the comparability assumption is violated in this scenario, as the conditional distribution of the primary outcome changes before and after \(t = 3\). We apply our method to this modified synthetic experiment, and the effect estimation results are presented in Figure (ref). The poor performance of our method underscores the critical importance of verifying that the comparability assumption is satisfied before applying the method. \begin{figure}[h] \caption{Estimated effect using the linear surrogate model for a treatment violates the comparability assumption} { \textit{Note}: Grey dashed curves represent the true average treatment effect on $Y$ from periods 1 to periods 10. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The three panels represent the scenarios when we use the first $T_E$ periods as the experimental period and the last $T_F$ periods as the future period.} \end{figure} \subsubsection{Violation of Assumption (ref)} Section (ref) shows that our estimator remains reliable when Assumption (ref) is mildly relaxed: moderate nonlinearities can often be well approximated by linear functions once a rich set of surrogates is included. Nevertheless, when the data‐generating process is \emph{strongly} nonlinear, the linearity assumption becomes pivotal and our method can break down. We illustrate this extreme scenario through a synthetic experiment, adopting a setup analogous to the nonlinear evaluation in Section (ref) but with two critical modifications. First, the primary outcome function is defined as \(Y_{i,t+1} = \sin(\theta\cdot S_{it,1}) + \cos(\theta\cdot S_{it,2})\). Second, we vary \(\theta\) across \([2, 3, 4]\), which controls the minimum positive period of the sine and cosine functions. The results, illustrated in Figure (ref), reveal that our linear surrogate model struggles to accurately estimate the true treatment effects under this setting, particularly failing to capture the long-term trends of the true effect. \begin{figure}[h] \caption{Estimated effect using the linear surrogate model for a treatment violates the linearity assumption} { \textit{Note}: Grey dashed curves represent the true average treatment effect on $Y$ from periods 1 to periods 10. Solid red curves represent the estimated effects with the linear surrogate model. Shadows indicate 95% confidence intervals. The three rows represent a different type of violation of the linearity assumption. The three panels represent the scenarios when we use the first $T_E$ periods as the experimental period and the last $T_F$ periods as the future period.} \end{figure}