Semiparametric inference for impulse response functions using double/debiased machine learning

abstractWe introduce a double/debiased machine learning estimator for the impulse response function in settings where a time series of interest is subjected to multiple discrete treatments, assigned over time, which can have a causal effect on future outcomes. The proposed estimator can rely on fully nonparametric relations between treatment and outcome variables, opening up the possibility to use flexible machine learning approaches to estimate impulse response functions. To this end, we extend the theory of double machine learning from an i.i.d. to a time series setting and show that the proposed estimator is consistent and asymptotically normally distributed at the parametric rate, allowing for semiparametric inference for dynamic effects in a time series setting. The properties of the estimator are validated numerically in finite samples by applying it to learn the impulse response function in the presence of serial dependence in both the confounder and observation innovation processes. We also illustrate the methodology empirically by applying it to the estimation of the effects of macroeconomic shocks.

assumptionFor all $s,t \in \mathcal{T}$ and $h \in \mathbb{N}_0$, the following holds. \begin{enumerate} • The impulse response function is time-invariant, i.e. $\theta_0^{(h)} = \mathbb{E}[\mathbb{E}[Y_{t+h}|D_t = 1, X_t] - \mathbb{E}[Y_{t+h}|D_t = 0, X_t]] = \mathbb{E}[\mathbb{E}[Y_{s+h}|D_s = 1, X_s] - \mathbb{E}[Y_{s+h}| D_s = 0, X_s]]$. • The conditional first moments of $Y_{t+h}$ and $D_t$ are time-invariant, i.e. $\mu_0(d, x, h) = \mathbb{E}[Y_{t+h}|D_t=d, X_t=x] = \mathbb{E}[Y_{s+h}|D_s=d, X_s=x]$ and $e_0(x) =\Pr(D_t=1|X_t=x) = \Pr(D_s=1|X_s=x)$. \end{enumerate}

assumption[Angrist2018,Rambachan2021] For all $t \in \mathcal{T}$ and $h \in \mathbb{N}_0$ the following holds. \begin{enumerate} • The potential outcomes are conditionally independent of the treatment, i.e. \\$Y_{t+h}(1), Y_{t+h}(0) \perp D_t \vert X_t$. \item\label{ass:identification_sutva} The observed outcome is $Y_{t+h} = D_t Y_{t+h}(1) + (1-D_t) Y_{t+h}(0)$. \item\label{ass:identification_overlap} For all $x\in \mathcal{X}$ it holds that $\eta < e_0(x) < 1-\eta$, for some $0 < \eta < 1$. \end{enumerate}

theoremUnder Assumptions (ref) and (ref) the ATE is identified as \begin{equation*} \theta_0^{(h)} = \mathbb{E}\left[Y_{t+h}(1)-Y_{t+h}(0)\right]. \end{equation*}

assumptionFor some $\beta>2$ and all $h \in \mathbb{N}_0$, the following conditions hold. \begin{enumerate} • The stochastic processes $\mathcal{G}^{(h)}$ are weakly stationary. • The variance satisfies that $0 < V^{(h)}_0 = \lim_{T\to\infty} \mathrm{Var}\left[\frac{1}{\sqrt{T}} \sum_{t \in \mathcal{T}} g\left(Z_t^{(h)}, h; \Gamma_0\right)\right]$. • $\mathcal{G}^{(h)}$ is uniformly $L_\beta$-bounded, i.e. $\sup_{t\in\mathcal{T}} \mathbb{E}\left[\left\vert g\left(Z_t^{(h)}, h; \Gamma_0\right)\right\vert^\beta\right] < \infty$. • $\mathcal{G}^{(h)}$ is $\alpha$-mixing, with coefficients $\alpha(s)$, $s \in \mathbb{N}$, satisfying $\sum_{s=1}^\infty \alpha(s)^{(\beta-2)/\beta} < \infty$. \end{enumerate}

theoremLet the oracle IRF estimator be given by \begin{equation*} \tilde\theta^{(h)} = \frac{1}{T} \sum_{t\in\mathcal{T}} g\left(Z_t^{(h)}, h; \Gamma_0\right). \end{equation*} Under Assumptions (ref) and (ref), we have that \begin{equation*} \sqrt{T} (\tilde\theta^{(h)} - \theta_0^{(h)}) \overset{\mathrm{d}}{\to} \mathcal{N}\left(0, V^{(h)}_0 \right), \end{equation*} with $V^{(h)}_0 = \sum_{s=-\infty}^{+\infty} \mathrm{Cov}\left[g\left(Z_t^{(h)}, h; \Gamma_0\right), g\left(Z_{t-s}^{(h)}, h; \Gamma_0\right)\right]$.

procedure\caption{DML estimator for the IRF with cross-fitting on multiple independent stochastic processes} For each forecast horizon $h$, follow the subsequent procedure. \begin{enumerate} • For each $i=1, ..., K$ \begin{enumerate} • Fit appropriate machine learners $\hat\Gamma_{S_{-i}^{(h)}} = (\hat{\mu}_{S_{-i}^{(h)}}(d, x,h)$, $\hat{e}_{S_{-i}^{(h)}}(x))$ on the sample $S_{-i}^{(h)} = \bigcup_{j = 1, j\neq i}^K S_j^{(h)}$. • Compute the average of $g\left(z, h; \hat\Gamma_{S_{-i}^{(h)}}\right)$ on $S_i^{(h)}$ as \begin{equation*} \hat{\theta}^{(h)}_{S_i^{(h)}} = \frac{1}{|\mathcal{T}_i|} \sum_{t\in\mathcal{T}_i} g\left(Z_t^{(h)}, h; \hat\Gamma_{S_{-i}^{(h)}}\right). \end{equation*} \end{enumerate} • Compute the IRF estimator at horizon $h$ as \begin{equation*} \hat{\theta}^{(h)} = \sum_{i=1}^{K} \frac{|\mathcal{T}_i|}{T} \hat{\theta}^{(h)}_{S_i^{(h)}}. \end{equation*} \end{enumerate}

assumptionLet the realization set be $\Xi_{T}^{(h)}$, which is a shrinking neighborhood of the true nuisance functions $\Gamma_0 = (\mu_0(d,x,h), e_0(x))$. Let $\{\Delta_T\}_{T\geq1}$ and $\{\delta_T\}_{T\geq1}$ be sequences of positive constants converging to zero. Define the statistical rates $r_{\mu,T} = \sup_{t\in\mathcal{T}}\sup_{\mu \in \Xi_{T}^{(h)}} \left\lVert\mu(D_t,X_t, h) - \mu_0(D_t,X_t, h)\right\rVert_2$ and $r_{e,T} = \sup_{t\in\mathcal{T}}\sup_{e \in \Xi_{T}^{(h)}} \left\lVerte(X_t) - e_0(X_t)\right\rVert_2$. Let $C$ be a fixed strictly positive constant. For all $i=1, ..., K$ and $h\in\mathbb{N}_0$, the following conditions hold. \begin{enumerate} • The nuisance function estimators $\hat\Gamma_{S_{-i}^{(h)}}$ belong to $\Xi_{T}^{(h)}$ with probability at least $1-\Delta_T$. • For $q>2$, we have $\sup_{t\in\mathcal{T}}\sup_{\mu \in \Xi_{T}^{(h)}} \left\lVert\mu(D_t,X_t, h) - \mu_0(D_t,X_t, h)\right\rVert_q \leq C < \infty$ and $\sup_{t\in\mathcal{T}}\sup_{e \in \Xi_{T}^{(h)}} \left\lVerte(X_t) - e_0(X_t)\right\rVert_q \leq C < \infty$. • $r_{\mu,T} \leq \delta_T$, $r_{e,T} \leq \delta_T$ and $r_{\mu,T} \cdot r_{e,T} \leq T^{-1/2} \delta_T$. • $\sup_{t\in\mathcal{T}}\sup_{e \in \Xi_{T}^{(h)}}\left\lVerte(X_t) -1/2\right\rVert_\infty \leq 1/2 - \eta $ for $ 0<\eta<1$. • $\sup_{t\in\mathcal{T}}\mathbb{E}\left[\left(Y_{t+h} - \mu_0(d,X_t,h)\right)^2 \vert X_t, D_t=d\right] \leq \epsilon^2_d < \infty$ \end{enumerate}

assumptionFor all $t\in\mathcal{T}$ and $h\in\mathbb{N}_0$ we have that $\mathbb{E}[Y_{t+h} \vert X_t, D_t = d, \{Z^{(0)}_u : u \in \mathcal{T}, u < t\}] = \mathbb{E}[Y_{t+h} \vert X_t, D_t = d]$ and $\mathbb{E}[D_t \vert X_t, \{Z^{(0)}_u : u \in \mathcal{T}, u < t\}] = \mathbb{E}[D_t \vert X_t]$.

theoremLet $S_i^{(h)} = \left\{Z_t^{(h)} : t \in \mathcal{T}_i\right\}$ for $i=1, ..., K\geq 2$ and $h\in\mathbb{N}_0$ be independent stochastic processes generated from the same distribution $\mathcal{P}$ such that $|\mathcal{T}_i|=T/K$ for all $i$. Define the estimator as \begin{align*} \hat{\theta}^{(h)} = \sum_{i=1}^{K}\frac{|\mathcal{T}_i|}{T} \hat{\theta}^{(h)}_{S_i^{(h)}} \quad with\\ \hat{\theta}^{(h)}_{S_i^{(h)}} = \frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} g\left(Z_t^{(h)}, h; \hat\Gamma_{S_{-i}^{(h)}}\right), \end{align*} where the nuisance functions $\hat\Gamma_{S_{-i}^{(h)}}=(\hat\mu_{S_{-i}^{(h)}}, \hat{e}_{S_{-i}^{(h)}})$ are estimated on $S_{-i}^{(h)} = \bigcup_{j = 1, j\neq i}^K S_j^{(h)}$. Then under Assumptions (ref) and (ref) - (ref) it holds that \begin{equation*} \sqrt{T} (\hat\theta^{(h)} - \theta_0^{(h)}) \overset{\mathrm{d}}{\to} \mathcal{N}\left(0, V^{(h)}_0 \right), \end{equation*} with $V^{(h)}_0$ as in Theorem (ref).

procedure[H] \caption{DML estimator for the IRF with cross-fitting on a single stochastic processes} For each forecast horizon $h$, follow the subsequent procedure. \begin{enumerate} • For each $i=1, ..., K$ \begin{enumerate} • Define $\mathcal{T}_{-i} = \{t : t \in \mathcal{T} \wedge \left(t < \inf(\mathcal{T}_i) - k_T \vee t > \sup(\mathcal{T}_i) + k_T\right)\}$ • Fit appropriate machine learners $\hat\Gamma_{S_{-i}^{(h)}} = (\hat\mu_{S_{-i}^{(h)}}(d,x,h), \hat{e}_{S_{-i}^{(h)}}(x))$ on the sample $S_{-i}^{(h)} = \{Z^{(h)}_t : t \in \mathcal{T}_{-i}\}$. • Compute the average of $g\left(z, h; \hat\Gamma_{S_{-i}^{(h)}}\right)$ on $S_i^{(h)}$ as \begin{equation*} \hat{\theta}^{(h)}_{S_i^{(h)}} = \frac{1}{|\mathcal{T}_i|} \sum_{t\in\mathcal{T}_i} g\left(Z_t^{(h)}, h; \hat\Gamma_{S_{-i}^{(h)}}\right). \end{equation*} \end{enumerate} • Compute the IRF estimator at horizon $h$ as \begin{equation*} \hat{\theta}^{(h)} = \sum_{i=1}^{K} \frac{|\mathcal{T}_i|}{T} \hat{\theta}^{(h)}_{S_i^{(h)}}. \end{equation*} \end{enumerate}

assumptionFor $d\in\{0,1\}$, $h\in\mathbb{N}_0$ and some scalar constant $p\geq1$ the following conditions hold. \begin{enumerate} • $k_T = O(T)$. • The nuisance functions $\mu_0(d, x,h)$, $e_0(x)$ and the functions $\mu(d,x,h), e(x)\in\Xi_{T}^{(h)}$ are measurable. • For for $r>p$ and $1/r = 1/r' + 1/r''$, we have $\sup_{t\in\mathcal{T}}\, \lVert\sup_{\mu\in\Xi_{T}^{(h)}} (\mu(d,X_t,h) - \mu_0(d,X_t,h))\lVert_{2r'} < \infty$ and $\sup_{t\in\mathcal{T}} \left\lVerte_0(X_t)-D_t\right\rVert_{2r''} < \infty$. • For $q>p$ and $1/q = 1/q' + 1/q''$, we have $\sup_{t\in\mathcal{T}} \, \lVert\sup_{e\in\Xi_{T}^{(h)}} (e(X_t) - e_0(X_t))\lVert_{2q'} < \infty$ and $\sup_{t\in\mathcal{T}} \left\lVertY_t - \mu_0(D_t,X_t, h)\right\rVert_{2q''} < \infty$. • The stochastic processes $S^{(h)} = \{Z_t^{(h)}:t\in\mathcal{T}\}$ are $\alpha$-mixing with coefficients $\alpha(s)$, satisfying for $T \to \infty$ that $\alpha\left(k_T\right)^{\psi} = o(T^{-1})$, where $\psi = 1/p - 1/\min(r,q)$. \end{enumerate}

enumerate[(i)] • $\alpha$-mixing process: If the stochastic processes $S^{(h)}$ are $\alpha$-mixing of size $-\phi_0$, i.e. $\alpha(s) = O(s^{-\phi})$ for some $\phi>\phi_0$, then Assumption (ref).(ref) is satisfied for $\vartheta>(\phi\psi)^{-1}$ and $0<\psi<1$. Put differently, the slower the decay in the dependence (smaller $\phi$), the larger $\vartheta$ and thus $k_T$ has to be. • Persistent process: For stochastic processes $S^{(h)}$ with very slowly decaying dependence of the form $\alpha(s) = O(s^{2d-1})$ with $0<d<1/2$, we have additionally that $0<\phi<1$ and as a consequence Assumption (ref).(ref) is indeed never satisfied. • Weakly dependent processes: If the stochastic processes $S^{(h)}$ have mixing coefficients $\alpha(s)=O(\rho^s)$ for $0<\rho<1$, then Assumption (ref).(ref) is already satisfied with $\vartheta > 0$. • Independent process: In this case, i.e. when it is assumed that the stochastic processes $S^{(h)}$ are a collection of independent random variables, then $\alpha(s)=0$ for all $s>0$ and Assumption (ref).(ref) is already satisfied for $k_T=0$.

theoremGiven the stochastic processes $S^{(h)} = \{Z_t^{(h)}:t\in\mathcal{T}\}$ for $h\in\mathbb{N}_0$, define $K\geq 2$ sub-sequences $S_i^{(h)} = \left\{Z_t^{(h)} : t \in \mathcal{T}_i\right\}$ such that $\{\mathcal{T}_i : i=1, \dots, K\}$ is a partition of the index set $\mathcal{T}$ where the order of the time indices within each $\mathcal{T}_i$ and across all sub-sequences with length $|\mathcal{T}_i| = T/K$ follows the original order in $\mathcal{T}$. Define the estimator as \begin{align*} \hat{\theta}^{(h)} = \sum_{i=1}^{K}\frac{|\mathcal{T}_i|}{T} \hat{\theta}^{(h)}_{S_i^{(h)}} \quadwith\\ \hat{\theta}^{(h)}_{S_i^{(h)}} = \frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} g\left(Z_t^{(h)}, h; \hat\Gamma_{S_{-i}^{(h)}}\right), \end{align*} where the nuisance functions $\hat\Gamma_{S_{-i}^{(h)}} = (\hat\mu_{S_{-i}^{(h)}}, \hat{e}_{S_{-i}^{(h)}})$ are estimated using the sub-sequences $S_{-i}^{(h)} = \{Z_t^{(h)} : t \in \mathcal{T}_{-i}\}$ for $\mathcal{T}_{-i} = \{t : t \in \mathcal{T} \wedge \left(t < \inf(\mathcal{T}_i) - k_T \vee t > \sup(\mathcal{T}_i) + k_T\right)\}$ respectively. Then under Assumptions (ref) and (ref) - (ref) it holds that \begin{equation*} \sqrt{T} (\hat\theta^{(h)} - \theta_0^{(h)}) \overset{\mathrm{d}}{\to} \mathcal{N}\left(0, V^{(h)}_0 \right), \end{equation*} with $V^{(h)}_0$ as in Theorem (ref).

assumptionThe following conditions hold. \begin{enumerate} • There are some fixed finite constants $C$ and $r> 4$ such that \\$\sup_{t\in\mathcal{T}} \mathbb{E}\left[\left\vert g\left(Z_t^{(h)}, h; \Gamma_0\right)\right\vert^{r}\right] < C$. \item \label{ass:variance_estimator-ii} There exists a measurable function $m(z)$ such that $\sup_{\Gamma \in \Xi_T} |g(z, h; \Gamma)| < m(z)$, where for some finite constant $D$, $\sup_{t\in\mathcal{T}}\mathbb{E}[m(Z_t)^2] < D$. \item \label{ass:variance_estimator-iv} For $q>2$ and some fixed strictly positive and finite constant $C$ we have $\sup_{t\in\mathcal{T}}\left\lVertY_t\right\rVert_q \leq C$. \item For some scalar $0< b_m < b_r \leq 1/2$ it holds that: \begin{enumerate} \item \label{ass:variance_estimator-iii} The bandwidth $m_T$ is a function of the sample size such that $\lim_{T\to\infty} m_T = \infty$ and for $T\to\infty$ it holds that $T^{-b_m} m_T = o(1)$. \item \label{ass:variance_estimator-v} $r_{\mu,T}\leq \delta_TT^{-b_r}$ and $r_{e,T} \leq \delta_TT^{-b_r}$. \end{enumerate} \end{enumerate}

theoremGiven the stochastic processes $S^{(h)} = \{Z_t^{(h)}:t\in\mathcal{T}\}$, define the sub-sequences $S_i^{(h)}$ for $i = 1, ..., K\geq 2$ and $h\in\mathbb{N}_0$ as in Theorem (ref). Furthermore, define $v^{(h)}_t = g\left(Z^{(h)}_t, h; \Gamma_0\right) - \theta^{(h)}_0$ and the corresponding estimated quantities as $\hat{v}^{(h)}_{S_i^{(h)},t} = g\left(Z^{(h)}_t, h; \hat\Gamma_{S_{-i}}\right) - \hat\theta^{(h)}$. $\hat\theta^{(h)}$ and $S_{-i}^{(h)}$ are defined as in Theorem (ref) and the nuisance functions $\hat\Gamma_{S_{-i}^{(h)}} = (\hat\mu_{S_{-i}^{(h)}}, \hat{e}_{S_{-i}^{(h)}})$ are estimated on $S_{-i}^{(h)}$. Let $w(s,m_T) = 1-s/(m_T+1)$, where $m_T$ is a bandwidth parameter and define the additional index sets $\mathcal{T}_{i,s} = \{t \in \mathcal{T}_i \, : \, t-s \geq \inf(\mathcal{T}_i) \}$. Moreover, define the following Newey1987 type variance estimators as \begin{align*} \widehat{V}^{(h)}_{S_i^{(h)}} &= \frac{1}{\vert\mathcal{T}_i\vert}\left( \sum_{t\in\mathcal{T}_i} (\hat{v}^{(h)}_{S_{-i}^{(h)},t})^2 + 2 \sum_{s=1}^{m_T} w(s,m_T) \sum_{t\in\mathcal{T}_{i,s}} \hat{v}^{(h)}_{S_{-i}^{(h)},t} \hat{v}^{(h)}_{S_{-i}^{(h)},t-s} \right). \end{align*} The variance estimator is finally defined as \begin{equation*} \widehat{V}^{(h)} = \sum_{i=1}^{K}\frac{|\mathcal{T}_i|}{T} \widehat{V}^{(h)}_{S_i^{(h)}} \end{equation*} Then for $V^{(h)}_0$ as in Theorem (ref) and under Assumptions (ref) and (ref) - (ref) as $T\to\infty$ it holds that \begin{equation*} \left\vert\widehat{V}^{(h)} - V^{(h)}_0\right\vert \overset{p}{\to} 0 \end{equation*} with measure $\mathcal{P}$.

procedure\caption{Setup of the simulation study} \begin{enumerate} • Draw a realization from the DGP with $T$ observations. • For each evaluated forecast horizon $h = 0, 1, ..., H$: \begin{enumerate}[a)] • Construct $\{S^{(h)}_i : i=1,\dots,K\}$ from the realization. • Find optimal hyperparameters for the estimators for $\mu_{0}(0, X, h)$, $\mu_{0}(1, X, h)$, and $e_0(X)$ by cross-validation using $\{S^{(h)}_i : i=1,\dots,K\}$ as folds and removing $k_T$ observations at the boundary between estimation and inference sample. • Train each of the four learners (DML, RA, DR, LP); and for each learner \begin{enumerate} • compute the IRF estimator $\hat\theta^{(h)}$ according to Procedure (ref) or Procedure (ref) • compute the variance estimator $\widehat{V}^{(h)}$ from Theorem (ref). Following the arguments outlined in Section (ref), we use the approach in NeweyWest1994 to determine the bandwidth $m_T$. \end{enumerate} \end{enumerate} • Repeat steps 1. and 2. $N$ times. \end{enumerate}

appendix\setcounter{equation}{0} \section*{Proofs} \begin{proof}[Proof of Theorem (ref)] We have that \begin{align*} \mathbb{E}\left[Y_{t+h}(d)\right] &= \mathbb{E}\left[\mathbb{E}\left[Y_{t+h}(d)\vert D_t=d, X_t\right]\right]= \mathbb{E}\left[\mathbb{E}\left[Y_{t+h}\vert D_t=d, X_t\right]\right] =\mathbb{E}\left[\mu_0(d,X_t,h)\right], \end{align*} where the second equality follows from Assumption (ref).(ref) and the last equality from Assumption (ref).(ref). It thus follows that $\mathbb{E}\left[Y_{t+h}(1)-Y_{t+h}(0)\right] = \mathbb{E}\left[\mu_0(1,X_t,h)-\mu_0(0,X_t,h)\right]$. \end{proof} \begin{proof}[Proof of Theorem (ref)] Under Assumption (ref) the stochastic process $\mathcal{G}^{(h)}$ satisfies the assumptions for the Central Limit Theorem for $\alpha$-mixing processes herrndorf1984. \end{proof} \begin{proof}[Proof of Theorem (ref)] The proof follows a similar strategy to the one in Wager2022 and Chernozhukov2018 for the i.i.d. case. For the sake of legibility, we ease the notation and drop the reference to the forecast horizon $h$. Let $\tilde\theta$ be the oracle IRF estimator as defined in Theorem (ref). Denote by $\mathcal{E}_T$ the event that $(\hat{\mu}_{S_{-i}}(d,x,h), \hat{e}_{S_{-i}}(x)) \in \Xi_{T}$ for all $i=1, ..., K$. We have that \begin{align*} \sqrt{T} (\hat\theta - \theta_0) &= \sqrt{T} (\hat\theta - \tilde\theta) + \sqrt{T}(\tilde\theta - \theta_0)\\ &= \sqrt{T} \left( \sum_{i=1}^{K}\frac{|\mathcal{T}_i|}{T} \hat{\theta}_{S_i} - \tilde\theta\right) + \sqrt{T}(\tilde\theta - \theta_0)\\ &=\sum_{i=1}^{K}\sqrt{T} \left(\frac{|\mathcal{T}_i|}{T} \hat{\theta}_{S_i} - \frac{|\mathcal{T}_i|}{T} \tilde\theta_{S_i}\right) + \sqrt{T}(\tilde\theta - \theta_0), \end{align*} where $\tilde\theta_{S_i} = \vert\mathcal{T}_i\vert^{-1} \sum_{t\in \mathcal{T}_i}g(Z_t; \Gamma_0)$. We have to show that the summation converges to zero in probability. Note that since $K$ is a finite integer, it suffices to show convergence for one summand. We begin by expanding a summand for some arbitrary $i$ as \begin{align*} \frac{|\mathcal{T}_i|}{T} \hat{\theta}_{S_i} - \frac{|\mathcal{T}_i|}{T} \tilde\theta_{S_i} &= \frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} g(Z_t; \hat\Gamma_{S_{-i}}) - g(Z_t;\Gamma_0) \\ &= \frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} \bigg(\hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) + \frac{D_t}{\hat{e}_{S_{-i}}(X_t)}(Y_t - \hat{\mu}_{S_{-i}}(1,X_t)) \\ &\qquad\qquad\qquad\qquad - \frac{D_t}{{e}_0(X_t)}(Y_t - {\mu}_0(1,X_t))\bigg)\\ &\quad -\frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} \bigg(\hat{\mu}_{S_{-i}}(0,X_t) - \mu_0(0,X_t) + \frac{1-D_t}{1-\hat{e}_{S_{-i}}(X_t)}(Y_t - \hat{\mu}_{S_{-i}}(0,X_t)) \\ &\qquad\qquad\qquad\qquad - \frac{1-D_t}{1-{e}_0(X_t)}(Y_t - {\mu}_0(0,X_t))\bigg). \end{align*} We will prove convergence for the first summation, the second summation can be treated analogously. The first summation can be decomposed as \begin{align*} \frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} & \left(\hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) + \frac{D_t}{\hat{e}_{S_{-i}}(X_t)}(Y_t - \hat{\mu}_{S_{-i}}(1,X_t)) - \frac{D_t}{{e}_0(X_t)}(Y_t - {\mu}_0(1,X_t))\right)\\ =&\underbrace{\frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} \left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right) \left(1-\frac{D_t}{e_0(X_t)} \right)}_{= P_1}\\ &\quad + \underbrace{\frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} D_t \left(Y_t - \mu_0(1,X_t)\right) \left(\frac{1}{\hat{e}_{S_{-i}}(X_t)} -\frac{1}{e_0(X_t)} \right)}_{= P_2}\\ &\quad + \underbrace{\frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} D_t \left(\hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t)\right) \left(\frac{1}{\hat{e}_{S_{-i}}(X_t)} -\frac{1}{e_0(X_t)}\right)}_{= P_3}. \end{align*} We will show that $P_k = o_p(T^{-1/2})$ for $k\in\{1,2,3\}$. Term $P_1$: From the squared $L_2$-norm of $P_1$ we have that \begin{align*} \mathbb{E}\Bigg[ &\Bigg(\frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} \left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right) \left(1-\frac{D_t}{e_0(X_t)} \right) \Bigg)^2 \Bigg]\\ &= \frac{1}{|\mathcal{T}_i|^2}\mathbb{E}\left[\mathbb{E}\left[\left( \sum_{t \in \mathcal{T}_i} \left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right) \left(1-\frac{D_t}{e_0(X_t)} \right) \right)^2 \Bigg| S_{-i} \right] \right]\\ &= \frac{1}{|\mathcal{T}_i|^2}\mathbb{E}\Bigg[ \sum_{t \in \mathcal{T}_i} \sum_{s \in \mathcal{T}_i} \mathbb{E} \Bigg[\left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right) \left(1-\frac{D_t}{e_0(X_t)} \right) \times\\ &\phantom{= \frac{1}{|\mathcal{T}_i|^2}\mathbb{E}\Bigg[ \sum_{t \in \mathcal{T}_i} \sum_{s \in \mathcal{T}_i} \mathbb{E} \Bigg[} \left( \hat{\mu}_{S_{-i}}(1,X_s) - \mu_0(1,X_s) \right) \left(1-\frac{D_s}{e_0(X_s)} \right) \Bigg| S_{-i} \Bigg] \Bigg]\\ &=\frac{1}{|\mathcal{T}_i|^2} \sum_{t \in \mathcal{T}_i} \mathbb{E}\left[\left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right)^2 \left(\frac{1}{e_0(X_t)} -1\right) \right]\\ &\leq \frac{1}{\eta |\mathcal{T}_i|^2 } \sum_{t \in \mathcal{T}_i} \mathbb{E}\left[\left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right)^2 \right] = \frac{o_p(1)}{|\mathcal{T}_i|} = o_p(T^{-1}). \end{align*} The third equality follows from the fact that the sum has mean zero and the inequality at the end follows from Assumption (ref).(ref). The step from the second to the third equality follows from Assumption (ref), which gives for $t<s$ that \begin{align*} \mathbb{E} \Bigg[&\left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right) \left(1-\frac{D_t}{e_0(X_t)} \right) \left( \hat{\mu}_{S_{-i}}(1,X_s) - \mu_0(1,X_s) \right) \left(1-\frac{D_s}{e_0(X_s)} \right) \Bigg| S_{-i} \Bigg]\\ &= \mathbb{E} \Bigg[\left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right) \left(1-\frac{D_t}{e_0(X_t)} \right) \left( \hat{\mu}_{S_{-i}}(1,X_s) - \mu_0(1,X_s) \right)\times\\ &\phantom{= \mathbb{E} \Bigg[}\;\underbrace{\mathbb{E}\left[1-\frac{D_s}{e_0(X_s)} \Bigg| X_s, \{(X_u, Y_u, D_u) : u\in\mathcal{T}_i, u < s)\}, S_{-i} \right]}_{=0} \Bigg| S_{-i} \Bigg]= 0. \end{align*} The same argument can be made for $t>s$. Convergence in the last step finally follows from the fact that $K$ is a fixed and finite integer and thus $\lim_{T\to\infty}|\mathcal{T}_i|=\lim_{T\to\infty} T/K=\infty$ for all $i$ and by noting that conditional on $S_{-i}$, the nuisance function estimator is non-stochastic, and thus conditional on the event $\mathcal{E}_T$, we have that \begin{align*} \sup_{t\in\mathcal{T}_i} \mathbb{E}\left[\left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right)^2 \Bigg\vert S_{-i} \right] & \leq\sup_{t\in\mathcal{T}_i} \sup_{\mu \in \Xi_{T}} \mathbb{E}\left[\left(\mu(1,X_t)-\mu_0(1,X_t)\right)^2 \Bigg\vert S_{-i}\right]\\ & \leq \sup_{t\in\mathcal{T}_i} \sup_{\mu \in \Xi_{T}} \left\lVert\mu(1,X_t)-\mu_0(1,X_t)\right\rVert_2^2 = o_p(1) \end{align*} since by Assumption (ref) $\sup_{t\in\mathcal{T}_i} \sup_{\mu \in \Xi_{T}} \left\lVert\mu(D_t,X_t)-\mu_0(D_t,X_t)\right\rVert_2^2 = (r_{\mu,T})^2 = o_p(1)$. By Lemma 6.1 in Chernozhukov2018 it follows that $ \sup_{t\in\mathcal{T}_i} \mathbb{E}[( \hat{\mu}_{S_{-i}}(1,X_t) \allowbreak - \mu_0(1,X_t) )^2] = o_p(1)$, and we thus conclude that $P_1$ is $o_p(T^{-1/2})$. Term $P_2$: Similarly, from the squared $L_2$-norm of $P_2$ we have that \begin{align*} \mathbb{E}\Bigg[&\Bigg(\frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} D_t \left(Y_t - \mu_0(1,X_t)\right) \left(\frac{1}{\hat{e}_{S_{-i}}(X_t)} -\frac{1}{e_0(X_t)} \right) \Bigg)^2\Bigg]\\ &= \frac{1}{|\mathcal{T}_i|^2} \mathbb{E}\Bigg[\mathbb{E}\Bigg[\Bigg( \sum_{t \in \mathcal{T}_i} D_t \left(Y_t - \mu_0(1,X_t)\right) \left(\frac{1}{\hat{e}_{S_{-i}}(X_t)} -\frac{1}{e_0(X_t)} \right) \Bigg)^2 \Bigg| S_{-i}\Bigg]\Bigg]\\ &= \frac{1}{|\mathcal{T}_i|^2} \mathbb{E}\Bigg[ \sum_{t \in \mathcal{T}_i} \sum_{s \in \mathcal{T}_i} \mathbb{E}\Bigg[ D_t \left(Y_t - \mu_0(1,X_t)\right) \left(\frac{1}{\hat{e}_{S_{-i}}(X_t)} -\frac{1}{e_0(X_t)} \right) \times\\ &\phantom{= \frac{1}{|\mathcal{T}_i|^2} \mathbb{E}\Bigg[ \sum_{t \in \mathcal{T}_i} \sum_{s \in \mathcal{T}_i} \mathbb{E}\Bigg[ } D_s \left(Y_s - \mu_0(1,X_s)\right) \left(\frac{1}{\hat{e}_{S_{-i}}(X_s)} -\frac{1}{e_0(X_s)}\right) \Bigg| S_{-i}\Bigg]\Bigg]\\ &= \frac{1}{|\mathcal{T}_i|^2} \sum_{t \in \mathcal{T}_i} \mathbb{E}\Bigg[ D_t \left(Y_t - \mu_0(1,X_t)\right)^2 \left(\frac{1}{\hat{e}_{S_{-i}}(X_t)} -\frac{1}{e_0(X_t)} \right)^2 \Bigg]\\ &\leq \frac{1}{|\mathcal{T}_i|^2} \frac{\epsilon^2_1}{\eta^2} \sum_{t \in \mathcal{T}_i} \mathbb{E}\left[\left( \hat{e}_{S_{-i}}(X_t) - e_0(X_t) \right)^2 \right] = \frac{o_p(1)}{|\mathcal{T}_i|} = o_p(T^{-1}).\\ \end{align*} The third equality follows from the fact that the sum has mean zero, and the inequality follows from Assumptions (ref).(ref) and (ref).(ref). The crucial step is again to establish that the variance of the sum equals the sum of the variances (from the second to the third equality). This follows from Assumption (ref), which gives that whenever $t<s$, it holds that \begin{align*} \mathbb{E}&\Bigg[ D_t \left(Y_t - \mu_0(1,X_t)\right) \left(\frac{1}{\hat{e}_{S_{-i}}(X_t)} -\frac{1}{e_0(X_t)} \right) \times \\ &\phantom{\mathbb{E}\Bigg[ D_t }\;D_s \left(Y_s - \mu_0(1,X_s)\right) \left(\frac{1}{\hat{e}_{S_{-i}}(X_s)} -\frac{1}{e_0(X_s)}\right) \Bigg| S_{-i}\Bigg]\\ &= \mathbb{E}\Bigg[ D_t \left(Y_t - \mu_0(1,X_t)\right) \left(\frac{1}{\hat{e}_{S_{-i}}(X_t)} -\frac{1}{e_0(X_t)} \right) \times \\ &\phantom{\mathbb{E}\Bigg[ D_t }\;\underbrace{\mathbb{E}\left[D_s(Y_s - \mu_0(1,X_s))\Bigg|X_s, \{(X_u, Y_u, D_u): u \in \mathcal{T}_i, u < s \}, S_{-i} \right]}_{=0} \left(\frac{1}{\hat{e}_{S_{-i}}(X_s)} -\frac{1}{e_0(X_s)}\right) \Bigg| S_{-i}\Bigg]=0, \end{align*} and the same argument can be made for $t>s$. Convergence in the last step finally follows from the fact that $K$ is a fixed and finite integer and thus $\lim_{T\to\infty}|\mathcal{T}_i|=\lim_{T\to\infty} T/K=\infty$ for all $i$ and by noting that conditional on $S_{-i}$, the nuisance function estimator is non-stochastic, and thus conditional on the event $\mathcal{E}_T$, we have that \begin{align*} \sup_{t\in\mathcal{T}_i} \mathbb{E}\left[\left( \hat{e}_{S_{-i}}(X_t) - e_0(X_t) \right)^2 \Bigg\vert S_{-i} \right] & \leq\sup_{t\in\mathcal{T}_i} \sup_{e \in \Xi_{T}} \mathbb{E}\left[\left(e(X_t) - e_0(X_t)\right)^2 \Bigg\vert S_{-i}\right]\\ & \leq \sup_{t\in\mathcal{T}_i} \sup_{e \in \Xi_{T}} \left\lVerte(X_t) - e_0(X_t)\right\rVert_2^2 = (r_{e,T})^2 \end{align*} by definition of the rate $r_{e,T}$ in Assumption (ref). By Lemma 6.1 in Chernozhukov2018 and Assumption (ref) it follows that $ \sup_{t\in\mathcal{T}_i} \mathbb{E}\left[\left( \hat{e}_{S_{-i}}(X_t) - e_0(X_t) \right)^2 \right] = o_p(1)$, and we thus conclude that $P_2$ is $o_p(T^{-1/2})$. Term $P_3$: Finally, from the $L_1$-norm of $P_3$ we get that \begin{align*} \mathbb{E}\Bigg[\Bigg| & \frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} D_t \left(\hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t)\right) \left(\frac{1}{\hat{e}_{S_{-i}}(X_t)} -\frac{1}{e_0(X_t)}\right) \Bigg|\Bigg]\\ &\leq \mathbb{E}\Bigg[ \frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} D_t \left|\hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t)\right| \left|\frac{1}{\hat{e}_{S_{-i}}(X_t)} -\frac{1}{e_0(X_t)}\right| \Bigg]\\ &= \frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} \mathbb{E}\Bigg[ D_t \left|\hat\mu_{S_{-i}}(D_t,X_t) - \mu_0(D_t,X_t)\right| \left|\frac{1}{\hat{e}_{S_{-i}}(X_t)} -\frac{1}{e_0(X_t)}\right| \Bigg]\\ &\leq \frac{1}{\eta}\frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} \mathbb{E}\Bigg[ \left|\hat\mu_{S_{-i}}(D_t,X_t) - \mu_0(D_t,X_t)\right| \left|\hat{e}_{S_{-i}}(X_t) - e_0(X_t)\right| \Bigg] = \frac{o_p(1)}{T^{1/2}}, \end{align*} where the last inequality follows from Assumption (ref).(ref). Convergence in the last equality finally follows from the fact that $K$ is a fixed and finite integer and thus $\lim_{T\to\infty}|\mathcal{T}_i|=\lim_{T\to\infty} T/K = \infty$ for all $i$ and by noting that conditional on $S_{-i}$, the nuisance function estimators are non-stochastic, and thus conditional on the event $\mathcal{E}_T$, we have that \begin{align*} &\sup_{t\in\mathcal{T}_i} \mathbb{E}\Bigg[ \left|\hat\mu_{S_{-i}}(D_t,X_t) - \mu_0(D_t,X_t)\right| \left|\hat{e}_{S_{-i}}(X_t) - e_0(X_t)\right| \Bigg\vert S_{-i} \Bigg]\\ &\;\leq\sup_{t\in\mathcal{T}_i} \mathbb{E}\Bigg[ \left|\hat\mu_{S_{-i}}(D_t,X_t) - \mu_0(D_t,X_t)\right|^2\Bigg\vert S_{-i} \Bigg]^{1/2} \sup_{t\in\mathcal{T}_i} \mathbb{E}\Bigg[\left|\hat{e}_{S_{-i}}(X_t) - e_0(X_t)\right|^2 \Bigg\vert S_{-i} \Bigg]^{1/2}\\ &\; \leq\sup_{t\in\mathcal{T}_i} \sup_{\mu \in \Xi_{T}}\mathbb{E}\Bigg[ \left|\mu(D_t,X_t)-\mu_0(D_t,X_t)\right|^2\Bigg\vert S_{-i} \Bigg]^{1/2} \sup_{t\in\mathcal{T}_i} \sup_{e \in \Xi_{T}}\mathbb{E}\Bigg[\left|e(X_t) - e_0(X_t)\right|^2 \Bigg\vert S_{-i} \Bigg]^{1/2}\\ &\;\leq \sup_{t\in\mathcal{T}_i} \sup_{\mu \in \Xi_{T}} \left\lVert\mu(D_t,X_t)-\mu_0(D_t,X_t)\right\rVert_2 \sup_{t\in\mathcal{T}_i} \sup_{e \in \Xi_{T}} \left\lVerte(X_t) - e_0(X_t)\right\rVert_2 = r_{\mu,T} \cdot r_{e,T} \end{align*} by Cauchy-Schwarz and the definition of the rates $r_{\mu,T}$ and $r_{e,T}$ in Assumption (ref). By Lemma 6.1 in Chernozhukov2018 and Assumption (ref).(ref) it follows that $\sup_{t\in\mathcal{T}_i} \mathbb{E}\left[ \left|\hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t)\right| \left|\hat{e}_{S_{-i}}(X_t) - e_0(X_t)\right| \right] = o_p(T^{-1/2})$. We have shown that $P_1$, $P_2$ and $P_3$ are $o_p(T^{-1/2})$. It follows that $\frac{|\mathcal{T}_i|}{T} \hat{\theta}_{S_i} - \frac{|\mathcal{T}_i|}{T} \tilde\theta_{S_i} = o_p(T^{-1/2})$ for all $i=1, ..., K$ and we can thus conclude that \begin{equation*} \sqrt{T} (\hat\theta - \theta) = \sum_{i=1}^{K}\underbrace{\sqrt{T} \left(\frac{|\mathcal{T}_i|}{T} \hat{\theta}_{S_i} - \frac{|\mathcal{T}_i|}{T} \tilde\theta_{S_i}\right)}_{=o_p(1)} + \underbrace{\sqrt{T}(\tilde\theta - \theta_0)}_{\overset{\mathrm{d}}{\to}\mathcal{N}(0, V_0)}. \end{equation*} \end{proof} \begin{lemma} Let $\Xi$ be a convex subset of some normed vector space, $g: \mathbb{R}^n \times \Xi \to \mathbb{R}$ be a measurable function, $\{Z_t : t \in \mathcal{T}\}$ an $\alpha$-mixing stochastic process with mixing coefficient $\alpha(m)$ and $Z_t$ a real-valued random vector. For $s\geq t$ denote by $\mathcal{F}_{t}^s = \sigma(Z_t,\dots, Z_s)$ the smallest $\sigma$-field such that $Z_t, \dots, Z_{s}$ are measurable. If $\left\lVert\sup_{\Gamma\in\Xi} g(Z_t, \Gamma)\right\rVert_r < \infty$ for some $r\geq p \geq 1$, then for all $t\in\mathcal{T}$ $$\sup_{\Gamma\in\Xi}\mathbb{E}\left[ g(Z_t, \Gamma) | \mathcal{F}^{t-k}_{-\infty} \right] = \sup_{\Gamma\in\Xi}\mathbb{E}\left[ g(Z_t, \Gamma) \right] + O_p(\alpha(k)^{1/p-1/r}).$$ \end{lemma} \begin{proof}[Proof of Lemma (ref)] We will prove the statement by bounding the $L_p$-norm. First notice that \begin{align*} \left\lVert\sup_{\Gamma\in\Xi}\mathbb{E}\left[ g(Z_t, \Gamma) | \mathcal{F}^{t-k}_{-\infty} \right] - \sup_{\Gamma\in\Xi}\mathbb{E}\left[ g(Z_t, \Gamma) \right]\right\rVert_p \leq \left\lVert\mathbb{E}\left[ \sup_{\Gamma\in\Xi} \left(g(Z_t, \Gamma) - \mathbb{E}\left[g(Z_t, \Gamma)\right]\right) | \mathcal{F}^{t-k}_{-\infty} \right] \right\rVert_p \end{align*} By Theorem 15.2 in Davidson2021 we have \begin{equation*} \begin{aligned} &\left\lVert\mathbb{E}\left[ \sup_{\Gamma\in\Xi} \left(g(Z_t, \Gamma) - \mathbb{E}\left[g(Z_t, \Gamma)\right]\right) | \mathcal{F}^{t-k}_{-\infty} \right] \right\rVert_p \\ &\quad\leq 2(2^{1/2} + 1)\alpha(k)^{1/p-1/r} \left\lVert\sup_{\Gamma\in\Xi} \left(g(Z_t, \Gamma) - \mathbb{E}\left[g(Z_t, \Gamma)\right]\right)\right\rVert_r\\ &\quad\leq 2(2^{1/2} + 1)\alpha(k)^{1/p-1/r}\left( \left\lVert\sup_{\Gamma\in\Xi} g(Z_t, \Gamma) \right\rVert_r + \left\vert\sup_{\Gamma\in\Xi} \mathbb{E}\left[g(Z_t, \Gamma)\right]\right\vert \right)=O(\alpha(k)^{1/p-1/r}) \end{aligned} \end{equation*} where the second inequality follows from the Minkowski inequality. \end{proof} \begin{proof}[Proof of Theorem (ref)] The proof builds on the proof of Theorem (ref) and we continue to omit the horizon $h$ for the sake of legibility. Following Davidson2021, let the smallest $\sigma$-field on which the stochastic process $S_{-i} = \{Z_t : t \in \mathcal{T}_{-i} \}$ is measurable, be denoted as $\mathcal{F}_{-i} = \sigma\left(Z_t : t\in\mathcal{T} \wedge ( t < \inf(\mathcal{T}_i) - k_T \vee t > \sup(\mathcal{T}_i) + k_T) \right)$. Consider again the three summations $P_1$, $P_2$ and $P_3$ from the proof of Theorem (ref). Term $P_1$: From the squared $L_2$-norm of $P_1$ we to obtain \begin{align} \begin{split} \mathbb{E}\Bigg[ &\Bigg(\frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} \left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right) \left(1-\frac{D_t}{e_0(X_t)} \right) \Bigg)^2 \Bigg]\\ &=\frac{1}{|\mathcal{T}_i|^2} \sum_{t \in \mathcal{T}_i} \mathbb{E} \left[\left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right)^2 \left(1-\frac{D_t}{e_0(X_t)} \right)^2 \right]\\ &\phantom{=}+\frac{1}{|\mathcal{T}_i|^2} \sum_{t \in \mathcal{T}_i} \sum_{s \in \mathcal{T}_i, s\neq t} \mathbb{E} \Bigg[\left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right) \left(1-\frac{D_t}{e_0(X_t)} \right) \times\\ &\phantom{=+\frac{1}{|\mathcal{T}_i|^2} \sum_{t \in \mathcal{T}_i} \sum_{s \in \mathcal{T}_i, s\neq t} \mathbb{E} \Bigg[} \left( \hat{\mu}_{S_{-i}}(1,X_s) - \mu_0(1,X_s) \right) \left(1-\frac{D_s}{e_0(X_s)} \right) \Bigg]. \end{split} \end{align} First, note that by applying Hölder's inequality twice we have that for $r>p\geq1$ and $1/r = 1/r' + 1/r''$ \begin{align*} \sup_{t,s \in\mathcal{T}_i} \sup_{\mu\in\Xi_{T}} & \left\lVert \left( \mu(1,X_t) - \mu_0(1,X_t) \right) \left(1-\frac{D_t}{e_0(X_t)} \right) \left( \mu(1,X_s) - \mu_0(1,X_s) \right) \left(1-\frac{D_s}{e_0(X_s)} \right)\right\rVert_r\\ &\leq \sup_{t\in\mathcal{T}_i} \sup_{\mu\in\Xi_{T}} \left\lVert \left( \mu(1,X_t) - \mu_0(1,X_t) \right)^2 \left(1-\frac{D_t}{e_0(X_t)} \right)^2 \right\rVert_r\\ &\leq \sup_{t\in\mathcal{T}_i} \sup_{\mu\in\Xi_{T}} \left\lVert \left( \mu(1,X_t) - \mu_0(1,X_t) \right)^2 \right\rVert_{r'} \left\lVert\left(1-\frac{D_t}{e_0(X_t)} \right)^2 \right\rVert_{r”} \end{align*} which is bounded by Assumption (ref).(ref). Next, for the first summand in (ref), note that conditional on $\mathcal{F}_{-i}$ the estimator is non-stochastic, and thus conditional on the event $\mathcal{E}_T$ we have that \begin{align*} \sup_{t\in\mathcal{T}_i} \mathbb{E} &\left[\left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right)^2 \left(1-\frac{D_t}{e_0(X_t)} \right)^2 \Bigg\vert \mathcal{F}_{-i} \right] \\ &\leq \sup_{t\in\mathcal{T}_i} \sup_{\mu\in\Xi_{T}} \mathbb{E} \left[\left( \mu(1,X_t) - \mu_0(1,X_t) \right)^2 \left(1-\frac{D_t}{e_0(X_t)} \right)^2 \Bigg\vert \mathcal{F}_{-i} \right]\\ &\leq \sup_{t\in\mathcal{T}_i} \sup_{\mu\in\Xi_{T}} \mathbb{E} \left[\left( \mu(1,X_t) - \mu_0(1,X_t) \right)^2 \left(1-\frac{D_t}{e_0(X_t)} \right)^2 \right] + O_p\left(\alpha(k_T)^\psi\right)\\ &= o_p(1) + O_p\left(\alpha(k_T)^\psi\right) \end{align*} by Lemma (ref) in combination with Assumption (ref), and the definition of the rate $r_{\mu,T}$ in Assumption (ref). By Lemma 6.1 in Chernozhukov2018 and Assumptions (ref).(ref) and (ref).(ref) it follows that \begin{align*} \frac{1}{|\mathcal{T}_i|^2} \sum_{t \in \mathcal{T}_i} \mathbb{E} \left[\left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right)^2 \left(1-\frac{D_t}{e_0(X_t)} \right)^2 \right] = \frac{o_p(1)}{|\mathcal{T}_i|} = o_p(T^{-1}). \end{align*} Similarly, for the second summand of the $L_2$-norm of $P_1$, conditional on the event $\mathcal{E}_T$ we have that \begin{align*} \sup_{t,s \in\mathcal{T}_i} \mathbb{E} \Bigg[&\left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right) \left(1-\frac{D_t}{e_0(X_t)} \right) \times \\ &\qquad\qquad \left( \hat{\mu}_{S_{-i}}(1,X_s) - \mu_0(1,X_s) \right) \left(1-\frac{D_s}{e_0(X_s)} \right) \Bigg\vert \mathcal{F}_{-i} \Bigg] \\ &\leq \sup_{t,s \in\mathcal{T}_i} \sup_{e\in\Xi_{T}} \mathbb{E} \bigg[\left( \mu(1,X_t) - \mu_0(1,X_t) \right) \left(1-\frac{D_t}{e_0(X_t)} \right) \times \\ &\qquad\qquad\qquad\qquad \left( \mu(1,X_s) - \mu_0(1,X_s) \right) \left(1-\frac{D_s}{e_0(X_s)} \right) \Bigg\vert \mathcal{F}_{-i} \bigg]\\ &\leq \sup_{t,s \in\mathcal{T}_i} \sup_{e\in\Xi_{T}} \mathbb{E} \bigg[\left( \mu(1,X_t) - \mu_0(1,X_t) \right) \left(1-\frac{D_t}{e_0(X_t)} \right) \times \\ &\qquad\qquad\qquad\qquad\left( \mu(1,X_s) - \mu_0(1,X_s) \right) \left(1-\frac{D_s}{e_0(X_s)} \right) \bigg] + O_p\left(\alpha(k_T)^\psi\right) \end{align*} by Lemma (ref) in combination with Assumption (ref), and the expectation in the last inequality is zero. By Lemma 6.1 in Chernozhukov2018 and Assumptions (ref).(ref) and (ref).(ref) it follows that \begin{align*} &\frac{1}{|\mathcal{T}_i|^2} \sum_{t \in \mathcal{T}_i} \sum_{s \in \mathcal{T}_i, s\neq t} \mathbb{E} \Bigg[\left( \hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t) \right) \left(1-\frac{D_t}{e_0(X_t)} \right) \times \\ &\qquad\qquad\qquad\qquad\qquad \left( \hat{\mu}_{S_{-i}}(1,X_s) - \mu_0(1,X_s) \right) \left(1-\frac{D_s}{e_0(X_s)} \right) \Bigg] = O_p\left(\alpha(k_T)^\psi\right). \end{align*} By Assumption (ref).(ref) it follows that $P_1$ is $o_p(T^{-1/2})$. Following a similar argument, it can be shown that $P_2$ is $o_p(T^{-1/2})$. Convergence for $P_3$ can again be shown by bounding its $L_1$-norm as \begin{align*} \mathbb{E}\Bigg[\Bigg| & \frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} D_t \left(\hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t)\right) \left(\frac{1}{\hat{e}_{S_{-i}}(X_t)} -\frac{1}{e_0(X_t)}\right) \Bigg|\Bigg]\\ &= \frac{1}{|\mathcal{T}_i|}\mathbb{E}\Bigg[\Bigg| \sum_{t \in \mathcal{T}_i} D_t \left(\hat\mu_{S_{-i}}(D_t,X_t) - \mu_0(D_t,X_t)\right) \left(\frac{1}{\hat{e}_{S_{-i}}(X_t)} -\frac{1}{e_0(X_t)}\right) \Bigg|\Bigg]\\ &\leq \frac{1}{\eta}\frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} \mathbb{E}\Bigg[ \left|\hat\mu_{S_{-i}}(D_t,X_t) - \mu_0(D_t,X_t)\right| \left|\hat{e}_{S_{-i}}(X_t) - e_0(X_t)\right| \Bigg]. \end{align*} Noting that conditional on $\mathcal{F}_{-i}$, the nuisance function estimators are non-stochastic, and thus conditional on the event $\mathcal{E}_T$ we have that \begin{align*} \sup_{t\in\mathcal{T}_i} \mathbb{E}\Bigg[ &\left|\hat\mu_{S_{-i}}(D_t,X_t) - \mu_0(D_t,X_t)\right| \left|\hat{e}_{S_{-i}}(X_t) - e_0(X_t)\right| \Bigg\vert \mathcal{F}_{-i} \Bigg]\\ &\leq \sup_{t\in\mathcal{T}_i} \sup_{\mu, e \in \Xi_T} \mathbb{E}\Bigg[ \left|\mu(D_t,X_t) - \mu_0(D_t,X_t)\right| \left|e(X_t) - e_0(X_t)\right| \Bigg\vert \mathcal{F}_{-i} \Bigg]\\ &\leq \sup_{t\in\mathcal{T}_i} \sup_{\mu, e \in \Xi_T} \mathbb{E}\Bigg[ \left|\mu(D_t,X_t) - \mu_0(D_t,X_t)\right| \left|e(X_t) - e_0(X_t)\right| \Bigg] + O_p(\alpha(k_T)^\psi)\\ & \leq \sup_{t\in\mathcal{T}_i} \sup_{\mu \in \Xi_{T}} \left\lVert\mu(D_t,X_t)-\mu_0(D_t,X_t)\right\rVert_2 \sup_{t\in\mathcal{T}_i} \sup_{e \in \Xi_{T}} \left\lVerte(X_t) - e_0(X_t)\right\rVert_2 + O_p(\alpha(k_T)^\psi)\& = r_{\mu,T} \cdot r_{e,T} + O_p(\alpha(k_T)^\psi) \end{align*} by Lemma (ref) in combination with Assumption (ref), Cauchy-Schwarz and the definition of the rates $r_{\mu,T}$ and $r_{e,T}$ in Assumption (ref). By Lemma 6.1 in Chernozhukov2018 and Assumptions (ref).(ref) and (ref).(ref) it follows that $ \sup_{t\in\mathcal{T}_i} \mathbb{E}\left[ \left|\hat{\mu}_{S_{-i}}(1,X_t) - \mu_0(1,X_t)\right| \left|\hat{e}_{S_{-i}}(X_t) - e(X_t)\right| \right] = o_p(T^{-1/2})$. This concludes the proof as we can now apply the same arguments as in the proof of Theorem (ref). \end{proof} \begin{proof}[Proof of Theorem (ref)] Define $v_t(\theta, \Gamma) = g(Z_t; \Gamma) - \theta$, for some nuisance functions $\Gamma = (\mu, e)$, $g(Z_t; \Gamma)$ is the influence function (ref) evaluated using the nuisance functions in $\Gamma$ and we will drop the forecast horizon $h$ everywhere for legibility. Let \begin{equation*} V_T = \mathrm{Var}\left(\frac{1}{\sqrt{T}}\sum_{t\in\mathcal{T}}g(Z_t;\Gamma_0)\right) = \frac{1}{T}\left(\sum_{t=1}^T\mathbb{E}[v_t^2] + 2\sum_{s=1}^{T-1}\sum_{t=s+1}^{T}\mathbb{E}[v_tv_{t-s}]\right) \end{equation*} with $v_t = g(Z_t;\Gamma_0) - \theta_0$ and $\Gamma_0 = (\mu_0, e_0)$, and thus $\lim_{T\to\infty} V_T = V_0$. Next, define \begin{equation*} V_{S_i} = \frac{1}{|\mathcal{T}_i|}\left(\sum_{t\in\mathcal{T}_i}\mathbb{E}[v^2_t] + 2\sum_{s\in\mathcal{T}_i}\sum_{t\in\mathcal{T}_{i,s}}\mathbb{E}[v_tv_{t-s}]\right) \end{equation*} with $\mathcal{T}_{i,s} = \{t \in \mathcal{T}_i \, \vert \, t-s \geq \inf(\mathcal{T}_i) \}$ for $i=1, ..., K$. Then we have that $\lim_{T\to\infty}V_{S_i} = \lim_{T\to\infty} V_T = V_0$, so given $K$ being a finite integer, for $|\widehat{V} - V_T| \overset{p}{\to} 0$ it will be sufficient to show that $|\widehat{V}_{S_i} - V_{S_i}| = o_p(1)$, where \begin{equation*} \widehat{V}_{S_i} = \frac{1}{|\mathcal{T}_i|}\left(\sum_{t\in\mathcal{T}_i}\hat{v}^2_t + 2\sum_{s=1}^{m_T}w(s,m_T)\sum_{t\in\mathcal{T}_{i,s}}\hat{v} _t\hat{v}_{t-s}\right) \end{equation*} with \begin{equation*} \hat{v}_t = g(Z_t;\hat\Gamma_t) - \hat\theta \qquad and \qquad \hat\Gamma_t = \{\hat\Gamma_{S_{-i}} : i \in \{1, ..., K\}, t\in \mathcal{T}_i\}. \end{equation*} Moreover, let \begin{equation*} V^m_{S_i} = \frac{1}{|\mathcal{T}_i|}\left(\sum_{t\in\mathcal{T}_i}v^2_t + 2\sum_{s=1}^{m_T}w(s,m_T)\sum_{t\in\mathcal{T}_{i,s}}v_tv_{t-s}\right). \end{equation*} Applying the triangle inequality, we will follow Newey1987 and prove that the following three terms converge to zero in probability \begin{equation} |\widehat{V}_{S_i} - V_{S_i}| \leq \underbrace{|\widehat{V}_{S_i} - V^m_{S_i}|}_{=P_1} + \underbrace{|V^m_{S_i} -\mathbb{E}[V^m_{S_i}]|}_{=P_2} + \underbrace{|\mathbb{E}[V^m_{S_i}] - V_{S_i}|}_{=P_3}. \end{equation} Since terms $P_2$ and $P_3$ do not contain any estimated quantities, they are $o_p(1)$ following the same arguments as in the proof of Theorem 2 in Newey1987 and Kool1988, provided that Assumptions (ref).(ref)-(ref).(ref) hold. For the term $P_1$, define a function \begin{align*} f(\theta, r) = &\frac{1}{|\mathcal{T}_i|}\sum_{t\in\mathcal{T}_i}v_t(\theta, \Gamma_0 + r(\hat\Gamma_t - \Gamma_0))^2 \\ &+ \frac{2}{|\mathcal{T}_i|}\sum_{s=1}^{m_T}w(s,m_T)\sum_{t\in\mathcal{T}_{i,s}}v_{t}(\theta, \Gamma_0 + r(\hat\Gamma_t - \Gamma_0)) v_{t-s}(\theta, \Gamma_0 + r(\hat\Gamma_{t-s} - \Gamma_0)), \end{align*} so that $\widehat{V}_{S_i} = f(\hat\theta, 1)$ and $V^m_{S_i} = f(\theta_0, 0)$. By the multivariate mean-value theorem, for some $(\tilde\theta, \tilde{r})$ on the line segment from $(\theta_0, 0)$ to $(\hat\theta, 1)$, on the event $\mathcal{E}_T$ we have \begin{equation*} \widehat{V}_{S_i} - V^m_{S_i} = f(\hat\theta, 1) - f(\theta_0, 0) = \frac{\partial f}{\partial\theta}(\tilde\theta, \tilde{r}) (\hat\theta-\theta_0) + \frac{\partial f}{\partial r}(\tilde\theta, \tilde{r}) \end{equation*} and by the triangle inequality \begin{equation} |\widehat{V}_{S_i} - V^m_{S_i}| \leq \underbrace{\left\vert\frac{\partial f}{\partial\theta}(\tilde\theta, \tilde{r}) (\hat\theta-\theta_0)\right\vert}_{=P_{11}} + \underbrace{\left\vert\frac{\partial f}{\partial r}(\tilde\theta, \tilde{r})\right\vert}_{=P_{12}}. \end{equation} We will show that both terms on the right hand side of (ref) are $o_p(1)$.\\ Term $P_{11}$: The partial derivative with respect to $\theta$ in $P_{11}$ is\ \begin{equation*} \begin{aligned} \frac{\partial f}{\partial\theta}(\tilde\theta, \tilde{r}) &= -\frac{2}{|\mathcal{T}_i|}\sum_{t\in\mathcal{T}_i}v_t(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))\\ &\qquad -\frac{2}{|\mathcal{T}_i|}\sum_{s=1}^{m_T}w(s,m_T)\sum_{t\in\mathcal{T}_{i,s}}\left[v_t(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0)) + v_{t-s}(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_{t-s} - \Gamma_0))\right] \end{aligned} \end{equation*} and thus \begin{equation*} \left\vert\frac{\partial f}{\partial\theta}(\tilde\theta, \tilde{r})\right\vert \leq 2\sup_{t\in\mathcal{T}_i}|v_t(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))| + 4m_T\sup_{t\in\mathcal{T}_i}|v_t(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))|. \end{equation*} Then we get for $P_{11}$ that \begin{equation} \begin{aligned} \left\vert\frac{\partial f}{\partial\theta}(\tilde\theta, \tilde{r})\right\vert|\hat\theta-\theta_0| \leq & \ 2\sup_{t\in\mathcal{T}_i}|v_t(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))|\cdot|\hat\theta-\theta_0| \\ &+ 4\frac{m_T}{\sqrt{T}}\sup_{t\in\mathcal{T}_i}|v_t(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))|\cdot\sqrt{T}|\hat\theta-\theta_0|. \end{aligned} \end{equation} By Theorem (ref), on the event $\mathcal{E}_T$, $|\hat\theta-\theta_0| = o_p(1)$ and $\sqrt{T}(\hat\theta-\theta_0) = O_p(1)$. Moreover, by Assumption (ref).(ref), $\lim_{T\to\infty} m_T / \sqrt{T} = 0$. Thus it remains to show that on the event $\mathcal{E}_T$, $\sup_{t\in\mathcal{T}_i}|v_t(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))|$ remains bounded in probability. By the triangular inequality we have that \begin{equation*} \bigg\vert v_t(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0)) \bigg\vert = \bigg\vert g(Z_t; \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0)) - \tilde\theta \bigg\vert \leq \bigg\vert g(Z_t; \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))\bigg\vert + \vert\tilde\theta \vert. \end{equation*} For the first term after the inequality we have, for some $q>2$ \begin{align} &\left\lVert g(Z_t; \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))\right\rVert_{q} \nonumber \\ &= \bigg\lVert \mu_0(1, X_t) + \tilde{r}(\hat\mu_t(1, X_t) - \mu_0(1,X_t)) \nonumber\\ &\qquad - \mu_0(0,X_t) - \tilde{r}(\hat\mu_t(0,X_t) - \mu_0(0,X_t)) \\ &\qquad + \frac{D_t}{e_0(X_t)+\tilde{r}(\hat{e}_t(X_t) - e_0(X_t))} (Y_t-\mu_0(1,X_t)-\tilde{r}(\hat\mu_t(1,X_t) - \mu_0(1,X_t))) \nonumber\\ &\qquad - \frac{1-D_t}{1-e_0(X_t)-\tilde{r}(\hat{e}_t(X_t) - e_0(X_t))}(Y_t-\mu_0(0,X_t)-\tilde{r}(\hat\mu_t(0,X_t) - \mu_0(0,X_t))) \bigg\rVert_{q} \nonumber\\ &\leq \left\lVert\mu_0(1, X_t)\right\rVert_{q} + \tilde{r}\left\lVert\hat\mu_t(1,X_t) - \mu_0(1, X_t)\right\rVert_{q} + \left\lVert\mu_0(0, X_t)\right\rVert_{q} + \tilde{r}\left\lVert\hat\mu_t(0,X_t) - \mu_0(0, X_t)\right\rVert_{q}\nonumber \\ &\qquad + \bigg\lVert \frac{D_t}{e_0(X_t)+\tilde{r}(\hat{e}_t(X_t) - e_0(X_t))} (Y_t-\mu_0(1,X_t)-\tilde{r}(\hat\mu_t(1,X_t) - \mu_0(1,X_t))) \bigg\rVert_{q}\nonumber\\ &\qquad + \bigg\lVert \frac{1-D_t}{1-e_0(X_t)-\tilde{r}(\hat{e}_t(X_t) - e_0(X_t))}(Y_t-\mu_0(0,X_t)-\tilde{r}(\hat\mu_t(0,X_t) - \mu_0(0,X_t))) \bigg\rVert_{q}\nonumber \end{align} where \begin{equation*} \hat\mu_t = \{\hat\mu_{S_{-i}} : i \in \{1, ..., K\}, t\in \mathcal{T}_i\} \qquad and \qquad \hat{e}_t = \{\hat{e}_{S_{-i}} : i \in \{1, ..., K\}, t\in \mathcal{T}_i\}. \end{equation*} From Assumptions (ref) and (ref).(ref) we have \begin{equation} \left\lVert\hat\mu_t(d,X_t) - \mu_0(d,X_t)\right\rVert_q \leq \frac{C}{\eta^{1/q}} \qquadand\qquad \left\lVert\mu_0(d,X_t)\right\rVert_q \leq \frac{C}{\eta^{1/q}}, \end{equation} for $q>2$, any $\hat\Gamma_t \in \Xi_T$ and $d\in\{0,1\}$, as shown in the proof of Theorem 5.1 in Chernozhukov2018. Using these results, we get the following bound for $|\tilde\theta|$ \begin{equation*} |\tilde\theta| \leq |\tilde\theta - \theta_0| + |\theta_0| \leq |\hat\theta - \theta_0| + |\theta_0| \leq |\hat\theta - \theta_0| + 2\frac{C}{\eta^{1/q}}, \end{equation*} where $|\theta_0| \leq 2C / \eta^{1/q}$ follows from Assumption (ref), as shown in the proof of Theorem 5.1 in Chernozhukov2018. From Theorem (ref), it follows that $|\hat\theta - \theta_0| = o_p(1)$, so $|\tilde\theta| = O_p(1)$. We continue with the term in the second line of the last inequality in (ref). For $q>2$, conditional on $\mathcal{E}_T$, we get \begin{align} \bigg\lVert &\frac{D_t}{e_0(X_t)+\tilde{r}(\hat{e}_t(X_t) - e_0(X_t))} (Y_t-\mu_0(1,X_t)-\tilde{r}(\hat\mu_t(1,X_t) - \mu_0(1,X_t))) \bigg\rVert_{q} \nonumber\\ &\leq \frac{1}{\eta}\bigg\lVert D_t(Y_t-\mu_0(1,X_t)-\tilde{r}(\hat\mu_t(1,X_t) - \mu_0(1,X_t))) \bigg\rVert_{q}\nonumber\\ &\leq \frac{1}{\eta}\bigg\lVert Y_t-\mu_0(1,X_t)-\tilde{r}(\hat\mu_t(1,X_t) - \mu_0(1,X_t)) \bigg\rVert_{q} \\ &\leq \frac{1}{\eta}\left\lVertY_t\right\rVert_q+ \frac{1}{\eta}\left\lVert\mu_0(1,X_t)\right\rVert_q + \frac{\tilde{r}}{\eta}\left\lVert\hat\mu_t(1,X_t) - \mu_0(1,X_t)\right\rVert_{q} \leq \frac{1}{\eta}C + \frac{1}{\eta}\frac{C}{\eta^{1/q}} + \frac{\tilde{r}}{\eta}\frac{C}{\eta^{1/q}},\nonumber \end{align} where the first inequality follows from Assumption (ref). The second inequality follows from $D_t\in\{0,1\}$, and the third from the Minkowski inequality. The fourth follows from (ref). For the term in the third line of the last inequality in (ref), we get for $q>2$ and conditional on $\mathcal{E}_T$ that \begin{equation} \begin{aligned} \bigg\lVert &\frac{1-D_t}{1-e_0(X_t)-\tilde{r}(\hat{e}_t(X_t) - e_0(X_t))}(Y_t-\mu_0(0,X_t)-\tilde{r}(\hat\mu_t(0,X_t) - \mu_0(0,X_t))) \bigg\rVert_{q} \\ &\leq \frac{1}{\eta}\bigg\lVert Y_t-\mu_0(0,X_t)-\tilde{r}(\hat\mu_t(0,X_t) - \mu_0(0,X_t)) \bigg\rVert_{q} \leq \frac{1}{\eta}C + \frac{1}{\eta}\frac{C}{\eta^{1/q}} + \frac{\tilde{r}}{\eta}\frac{C}{\eta^{1/q}},\\ \end{aligned} \end{equation} using the same arguments as for the term in (ref). Since all of the terms in the first line of the last inequality in (ref) are $L_q$-bounded, they are also $O_p(1)$ by Markov's inequality. The same holds for (ref) and (ref), so we can conclude that all of the terms on the right-hand-side of the last inequality in (ref) are $O_p(1)$. As a consequence, $\sup_{t\in\mathcal{T}_i}|v_t(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))| = O_p(1)$ in (ref) and thus $P_{11} = o_p(1)$. Term $P_{12}$: From the partial derivative with respect to $r$ in $P_{12}$ we find \begin{equation*} \begin{aligned} \left|\frac{\partial f}{\partial r}(\tilde\theta, \tilde{r})\right| &= \bigg\vert \frac{2}{|\mathcal{T}_i|}\sum_{t\in\mathcal{T}_i}v_t(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0)) \frac{\partial v_t}{\partial r}(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))\\ &\qquad + \frac{2}{|\mathcal{T}_i|} \sum_{s=1}^{m_T}w(s,m_T) \sum_{t\in\mathcal{T}_{i,s}} v_t(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0)) \frac{\partial v_{t-s}}{\partial r}(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_{t-s} - \Gamma_0))\\ &\qquad + \frac{2}{|\mathcal{T}_i|} \sum_{s=1}^{m_T}w(s,m_T) \sum_{t\in\mathcal{T}_{i,s}} v_{t-s}(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_{t-s} - \Gamma_0)) \frac{\partial v_{t}}{\partial r}(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0)) \bigg\vert \\ & \leq 2\sup_{t\in\mathcal{T}_i}|v_t(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))| \sup_{t\in\mathcal{T}_i}\left\vert \frac{\partial v_t}{\partial r}(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0)) \right\vert \\ &\qquad + 4\frac{m_T}{T^{b_m}}\sup_{t\in\mathcal{T}_i}|v_t(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))| \sup_{t\in\mathcal{T}_i}\left\vert \frac{\partial v_t}{\partial r}(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))\right\vert T^{b_m}. \end{aligned} \end{equation*} From proving $P_{11} = o_p(1)$ we know that $\sup_{t\in\mathcal{T}_i}|v_t(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))| = O_p(1)$. Moreover, $m_T / T^{b_m} = o(1)$ by Assumption (ref).(ref). Thus it remains to show that $\sup_{t\in\mathcal{T}_i}|\frac{\partial v_t}{\partial r}(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))| = o_p(T^{-b_m})$. Since by Assumption (ref) $b_m < b_r$, we will show that the quantity is actually $o_p(T^{-b_r})$. By the triangle inequality, we have \begin{equation} \sup_{t\in\mathcal{T}_i}\left\vert\frac{\partial v_t}{\partial r}(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))\right\vert \leq P_{12,A} + P_{12,B} + P_{12,C} + P_{12,D}, \end{equation} where \begin{equation*} \begin{aligned} P_{12,A} &= \sup_{t\in\mathcal{T}_i}\left\vert \hat\mu_t(1,X_t) - \mu_0(1,X_t) - \left(\hat\mu_t(0,X_t) - \mu_0(0,X_t)\right)\right\vert \\ P_{12,B} &= \sup_{t\in\mathcal{T}_i}\bigg\vert \frac{D_t}{\left(e_0(X_t) + \tilde{r}(\hat{e}_t(X_t) - e_o(X_t))\right)^2} \times\\ &\qquad\quad \left(Y_t - \mu_0(1,X_t) - \tilde{r}(\hat\mu_t(1,X_t) - \mu_0(1,X_t))\right) \left(\hat{e}_t(X_t) - e_0(X_t)\right)\bigg\vert \\ P_{12,C} &= \sup_{t\in\mathcal{T}_i}\left\vert \frac{1-D_t}{1-e_0(X_t) - \tilde{r}(\hat{e}_t(X_t)-e_0(X_t))} \left(\hat\mu_t(0,X_t) - \mu_0(0,X_t)\right)\right\vert \\ P_{12,D} &= \sup_{t\in\mathcal{T}_i}\bigg\vert \frac{1-D_t}{\left( (1-\tilde{r})(1-e_0(X_t)) + \tilde{r}(1-\hat{e}_t(X_t))\right)^2} \times\\ &\qquad\quad \left(Y_t - \mu_0(0,X_t) + \tilde{r}(\hat\mu_t(0,X_t) - \mu_0(0,X_t))\right)\left(\hat{e}_t(X_t) - e_0(X_t)\right)\bigg\vert. \end{aligned} \end{equation*} For $P_{12,A}$, we first establish that on the event $\mathcal{E}_T$ for $d\in\{0,1\}$ we have \begin{equation} \begin{aligned} \sup_{t\in\mathcal{T}_i}&\mathbb{E}\left[ \left\vert\hat\mu_t(d,X_t) - \mu_0(d,X_t)\right\vert^2 \bigg\vert S_{-i}\right] \\ &\leq \sup_{t\in\mathcal{T}_i}\sup_{\mu\in \Xi_T} \mathbb{E}\left[ \left(\mu(d,X_t) - \mu_0(d,X_t)\right)^2 \bigg\vert S_{-i}\right]\\ &\leq \sup_{t\in\mathcal{T}_i}\sup_{\mu\in \Xi_T} \mathbb{E}\left[ \left(\mu(d,X_t) - \mu_0(d,X_t)\right)^2\right] + O_p(\alpha(k_T)^\psi)\\ &\leq r_{\mu,T}^2 + O_p(\alpha(k_T)^\psi) = o_p(T^{-2 b_r}). \end{aligned} \end{equation} The second inequality follows from Lemma (ref), the third by the definition of the rate $r_{\mu,T}$ in Assumption (ref). The final equality follows by Lemma 6.1 in Chernozhukov2018 and Assumptions (ref).(ref), (ref).(ref) and (ref).(ref). We thus conclude that on the event $\mathcal{E}_T$, \begin{equation} \sup_{t\in\mathcal{T}_i} \left\vert\hat\mu_t(d,X_t) - \mu_0(d,X_t)\right\vert = o_p(T^{-b_r}) \end{equation} and as a consequence \begin{equation*} P_{12,A} \leq \sup_{t\in\mathcal{T}_i} \left\vert\hat\mu_t(1,X_t) - \mu_0(1,X_t)\right\vert + \sup_{t\in\mathcal{T}_i} \left\vert\hat\mu_t(0,X_t) - \mu_0(0,X_t)\right\vert = o_p(T^{-b_r}). \end{equation*} For $P_{12,B}$, we first establish that on the event $\mathcal{E}_T$ we have \begin{equation*} \begin{aligned} \sup_{t\in\mathcal{T}_i}\mathbb{E}\left[ \left\vert\hat{e}_t(X_t) - e_0(X_t)\right\vert^2 \big\vert S_{-i}\right] &\leq \sup_{t\in\mathcal{T}_i}\sup_{e \in \Xi_T} \mathbb{E}\left[ (\hat{e}_t(X_t) - e_0(X_t))^2\big\vert S_{-i}\right]\\ &\leq \sup_{t\in\mathcal{T}_i}\sup_{e \in \Xi_T} \mathbb{E}\left[ (\hat{e}_t(X_t) - e_0(X_t))^2\right] + O_p(\alpha(k_T)^\psi)\\ &\leq r_{e,T}^2 + O_p(\alpha(k_T)^\psi) = o_p(T^{-2 b_r})\\ \end{aligned} \end{equation*} by the same arguments as for (ref) and thus on the event $\mathcal{E}_T$, \begin{equation} \sup_{t\in\mathcal{T}_i} \left\vert\hat{e}_t(X_t) - e_0(X_t)\right\vert = o_p(T^{-b_r}). \end{equation} This allows us to write \begin{equation} \begin{aligned} P_{12,B} &\leq \sup_{t\in\mathcal{T}_i}\frac{1}{\eta^2}\left\vert Y_t - (1-\tilde{r})\mu_0(1,X_t)-\tilde{r}\hat\mu_t(1,X_t)\right\vert \left\vert\hat{e}_t(X_t) - e_0(X_t)\right\vert \\ &\leq \sup_{t\in\mathcal{T}_i}\frac{1-\tilde{r}}{\eta^2}\left\vert Y_t - \mu_0(1,X_t)\right\vert \left\vert\hat{e}_t(X_t) - e_0(X_t)\right\vert \\ &\quad + \sup_{t\in\mathcal{T}_i}\frac{\tilde{r}}{\eta^2}\left\vert Y_t - \hat\mu_t(1,X_t)\right\vert \left\vert\hat{e}_t(X_t) - e_0(X_t)\right\vert \\ &\leq \sup_{t\in\mathcal{T}_i}\frac{2}{\eta^2}\left\vert Y_t - \mu_0(1,X_t)\right\vert \left\vert\hat{e}_t(X_t) - e_0(X_t)\right\vert \\ &\quad + \sup_{t\in\mathcal{T}_i}\frac{1}{\eta^2}\left\vert \hat\mu_t(1,X_t) - \mu_0(1,X_t)\right\vert \left\vert\hat{e}_t(X_t) - e_0(X_t)\right\vert, \\ \end{aligned} \end{equation} where the first inequality follows from Assumption (ref) and $D_t\in\{0,1\}$. From Assumption (ref).(ref) and (ref), we have for $q>2$, $d\in\{0,1\}$ and any $\mu \in \Xi_T$ that \begin{equation} \sup_{t\in\mathcal{T}_i}\left\lVertY_t - \mu(d,X_t)\right\rVert_q \leq \sup_{t\in\mathcal{T}_i} \left\lVertY_t\right\rVert_q + \sup_{t\in\mathcal{T}_i}\left\lVert\mu(d,X_t)\right\rVert_q \leq C\left(1+\frac{1}{\eta^{1/q}}\right). \end{equation} Combined with (ref) it follows that for $d\in\{0,1\}$ and on the event $\mathcal{E}_T$ we have \begin{align*} &\sup_{t\in\mathcal{T}_i}\left\lVert|Y_t - \mu(d,X_t)||\hat{e}_t(X_t) - e_0(X_t)|\right\rVert_1 \\ &\quad \leq \sup_{t\in\mathcal{T}_i}\left\lVertY_t - \mu(d,X_t)\right\rVert_2 \left\lVert\hat{e}_t(X_t) - e_0(X_t)\right\rVert_2 \\ &\quad \leq \sup_{t\in\mathcal{T}_i}C\left(1+\frac{1}{\eta^{1/q}}\right) \left\lVert\hat{e}_t(X_t) - e_0(X_t)\right\rVert_2 = o_p(T^{-b_r}) \end{align*} for any $\mu \in \Xi_T$, and as a consequence, combined with (ref), we have $P_{12,B} = o_p(T^{-b_r})$. For $P_{12,C}$, we have by Assumption (ref) and $D_t\in\{0,1\}$ that $P_{12,C} \leq \sup_{t\in\mathcal{T}_i}\frac{1}{\eta}| \hat\mu_t(0,X_t) - \mu_0(0,X_t) |$ and thus $P_{12,C} = o_p(T^{-b_r})$ by (ref). For $P_{12,D}$ finally, we have \begin{equation*} \begin{aligned} P_{12,D} &\leq \sup_{t\in\mathcal{T}_i}\frac{1}{\eta^2}\left\vert (1-\tilde{r})(Y_t-\mu_0(0,X_t)) + \tilde{r}(Y_t - \hat\mu_t(0,X_t)) \right\vert \left\vert\hat{e}_t(X_t) - e_0(X_t)\right\vert \\ &\leq \sup_{t\in\mathcal{T}_i}\frac{2}{\eta^2} \left\vert Y_t-\mu_0(0,X_t)\right\vert \left\vert\hat{e}_t(X_t) - e_0(X_t)\right\vert \\ &\quad + \sup_{t\in\mathcal{T}_i}\frac{1}{\eta^2} \left\vert \hat\mu_t(0,X_t)-\mu_0(0,X_t)\right\vert \left\vert\hat{e}_t(X_t) - e_0(X_t)\right\vert, \end{aligned} \end{equation*} where the first inequality again follows from Assumption (ref) and $D_t\in\{0,1\}$, and the second from the same arguments as in (ref). Using (ref), (ref) and (ref) lets us conclude that $P_{12,D} = o_p(T^{-b_r})$. This shows that all terms on the right hand side of (ref) are $o_p(T^{-b_r})$ and thus $\sup_{t\in\mathcal{T}_i}|\frac{\partial v_t}{\partial r}(\tilde\theta, \Gamma_0 + \tilde{r}(\hat\Gamma_t - \Gamma_0))| = o_p(T^{-b_r})$. As a consequence, $P_{11} = o_p(1)$ and $P_{12} = o_p(1)$ in (ref). We conclude that $|\widehat{V}_{S_i} - V^m_{S_i}| = o_p(1)$ and thus $|\widehat{V}_{S_i} - V_{S_i}| = o_p(1)$ so that $|\widehat{V} - V_T| \overset{p}{\to} 0$. \end{proof}

appendix\begin{center} Supplementary material for “Semiparametric inference for impulse response functions using double/debiased machine learning” \end{center} \section{Hyperparameter tuning} \subsection{Tuning in the simulation study} Random forests: We fix the number of trees to 500. For each simulation replication, the maximal depth of each tree ($d$), the minimal number of observations in the leafs of the trees ($\ell$) and the maximal fraction of features considered for each node split ($m_{try}$) are determined by cross-validation using the $K$ sub-processes as folds. We perform a simple grid search over $\{\{d,\ell, m_{try}\}~\lvert~d\in\{5, 10,20,50\} \wedge \ell \in \{1,5,10\} \wedge m_{try} \in \{0.3, 1.0\}\}$ and select the hyperparameter-combination yielding the best predictive cross-validation performance. Gradient boosted trees: We perform a two-stage tuning procedure. In the first stage, we fix the learning rate of the tree booster to 0.1, set the number of boosting rounds to some very high number (10'000) and abort the estimation process if the predictive validation error has not decreased since 50 rounds of boosting. The maximum tree depth for the base learners ($d$), the minimum sum of instance weight needed in a child ($w$), the subsampling ratio for observations used to construct each tree ($s^o$), and the subsampling ratio for features when constructing each tree ($s^f$) are determined by cross-validation using the $K$ sub-processes as folds. We perform a simple grid search over $\{\{d,w,s^o, s^f\}~\lvert~d\in\{1, ...,10\} \wedge w \in \{1, ..., 10\} \wedge s^o\in\{0.25,0.5,0.75, 1\} \wedge s^f\in\{0.25,0.5,0.75, 1\}\}$ and select the hyperparameter-combination yielding the best predictive cross-validation performance. In the second stage, using the optimal tree-hyperparameters from stage 1, we select the learning rate most frequently yielding the best predictive cross-validation performance. We perform a search over the candidate set $\{0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.1, 0.25, 0.5\}$. Finally, the optimal number of boosting rounds is determined as the average early-stopped boosting round for the selected learning rate. For computational reasons, we repeat this procedure on $R=50$ simulated examples and fix the hyperparameters for all simulation replications to the hyperparameter combination most frequently yielding the best cross-validation performance.\\\\ \subsection{Tuning in the empirical application} Random forests: We fix the number of trees to 500. The maximal depth of each tree ($d$), the minimal number of observations in the leafs of the trees ($\ell$) and the size of the random subsets of features to consider when splitting a node ($\bar{m}$) are determined by cross-validation using the $K$ sub-processes as folds. We perform a simple grid search over the same candidate sets as for the simulation study (cf. Appendix (ref)) and select the hyperparameter-combination yielding the best predictive cross-validation performance. Gradient boosted trees: We perform a two-stage tuning procedure. In the first stage, we fix the learning rate of the tree booster to 0.1, the number of boosting rounds to 500 and find optimal tree parameters by cross-validation. For this, we perform a simple grid search over $\{\{d,w,s^o,s^f\}~\lvert~d\in\{1, ...,10\} \wedge w \in \{1, ..., 10\} \wedge s^o\in\{0.25,0.5,0.75, 1\} \wedge s^f\in\{0.25,0.5,0.75, 1\}\}$, for the same parameters as in Appendix (ref), and select the hyperparameter-combination yielding the best predictive cross-validation performance. In the second stage, using the optimal tree-hyperparameters from stage 1, we finally select the learning rate and number of boosting rounds yielding the best predictive cross-validation performance. We perform a search over all combinations of $\{0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.1, 0.25, 0.5\}$ for the learning rate and $\{10, 110, 210, 310, 410, 510\}$ for the number of boosting rounds. All estimators are tuned using 10-fold blocked cross-validation on the full sample, where at the boundary of the folds 24 observations are dropped to eliminate dependence between the folds. \section{Simulation with an empirically calibrated data generating process} We calibrate the data generating process defined in Section (ref) of the main text using monthly U.S. data from 1982 to 2012 obtained from the empirical study in Angrist2018. In more detail, the outcome process follows \begin{equation*} Y_t = c + b(X_t) + \left(D_t -0.5\right)\tau(X_t) + \gamma Y_{t-1} + \epsilon_t. \end{equation*} We calibrate the parameters $c$ and $\gamma$, the innovation process $\epsilon_t$ and the process governing a set of confounder variables $X_t$. The vector $X_t$ contains the changes in the federal funds rate and in the ten-year Treasury yield, percentage changes in the S&P 1000 index, the M1 money stock, civilian employment, and industrial production, as well as percentage-point changes in the unemployment rate. The outcome variable $Y_t$ is the monthly percentage changes in core personal consumption expenditures. The specification of $b(X_t)$, $\tau(X_t)$ and $e(X_t)$ are kept identical to those used in the simulation study in Section (ref) of the main text. The innovation process is assumed to follow a GARCH($p,q$) specification. Following Lazarus2018,Adamek2024, the confounder process $X_t$ is modelled using a dynamic factor model as \begin{equation*} \begin{aligned} X_t &= \Lambda F_t + U_t\\ F_t &= \sum_{j=1}^{p_F} \Phi_j F_{t-j} + V_t, \qquad V_t \sim \mathcal{N}(0,I)\\ U_{i,t} &= \sum_{j=1}^{p_U} \phi_j U_{i,t-j} + \eta_t, \qquad \eta_{i,t} \sim \mathcal{N}(0,\sigma^2_{i,\eta}) \quad for i=1,\dots, \dim(X_t). \end{aligned} \end{equation*} Model orders are selected using the Bayesian Information Criterion (BIC). Specifically, we determine the GARCH($p,q$) orders for $\epsilon_t$, the number of latent factors $F_t$, the lag length $p_F$ of the factor VAR, and the AR order $p_U$ of the idiosyncratic components by minimizing the BIC. For the sample at hand, this procedure selects an ARCH(1) process for $\epsilon_t$, one latent factor following a VAR(1) process, and univariate AR(1) processes for the components of $U_t$. Simulation samples are generated by drawing independently from the distributions of $\epsilon_t$, $V_t$ and $\eta_t$. \section{Additional tables and figures} \begin{table}[htbp] \caption{Simulation results for $K$ independent nonlinear baseline DGPs ($n=12$ and $\sigma_\epsilon=1.0$) and random forest nuisance function estimates} \resizebox{\textwidth}{!}{\begin{threeparttable}\begin{tabular}{rrrrrrrrrrrrrrrrrr} \toprule \multicolumn{18}{c}{$h=0$, $\theta_0^{(h)}=0.3321$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & -0.020 & 0.749 & 0.749 & 0.969 & 0.959 & 0.379 & 0.438 & 0.579 & 0.530 & 0.275 & 0.400 & 0.486 & 0.732 & 0.099 & 0.361 & 0.375 & 0.868 \\ 250 & 0.025 & 0.406 & 0.407 & 0.954 & 0.953 & 0.276 & 0.275 & 0.390 & 0.512 & 0.198 & 0.256 & 0.323 & 0.752 & 0.110 & 0.245 & 0.269 & 0.872 \\ 500 & 0.027 & 0.212 & 0.214 & 0.944 & 0.944 & 0.215 & 0.199 & 0.293 & 0.469 & 0.139 & 0.185 & 0.231 & 0.776 & 0.139 & 0.181 & 0.228 & 0.821 \\ 1'000 & 0.015 & 0.131 & 0.132 & 0.962 & 0.959 & 0.174 & 0.130 & 0.217 & 0.431 & 0.095 & 0.121 & 0.154 & 0.809 & 0.135 & 0.121 & 0.181 & 0.747 \\ 8'000 & 0.009 & 0.044 & 0.045 & 0.957 & 0.955 & 0.177 & 0.052 & 0.184 & 0.015 & 0.042 & 0.044 & 0.061 & 0.818 & 0.138 & 0.043 & 0.145 & 0.094 \\ \midrule \multicolumn{18}{c}{$h=1$, $\theta_0^{(h)}=0.1992$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.017 & 0.762 & 0.762 & 0.958 & 0.953 & 0.412 & 0.439 & 0.602 & 0.467 & 0.296 & 0.402 & 0.499 & 0.729 & 0.072 & 0.369 & 0.376 & 0.904 \\ 250 & 0.057 & 0.398 & 0.402 & 0.971 & 0.951 & 0.306 & 0.284 & 0.418 & 0.441 & 0.215 & 0.263 & 0.340 & 0.741 & 0.099 & 0.269 & 0.287 & 0.912 \\ 500 & 0.053 & 0.220 & 0.226 & 0.949 & 0.941 & 0.251 & 0.197 & 0.319 & 0.330 & 0.163 & 0.182 & 0.244 & 0.708 & 0.135 & 0.190 & 0.233 & 0.872 \\ 1'000 & 0.037 & 0.142 & 0.147 & 0.951 & 0.951 & 0.211 & 0.130 & 0.248 & 0.236 & 0.119 & 0.123 & 0.171 & 0.731 & 0.128 & 0.134 & 0.186 & 0.829 \\ 8'000 & 0.014 & 0.045 & 0.047 & 0.934 & 0.934 & 0.219 & 0.052 & 0.225 & 0.001 & 0.052 & 0.045 & 0.069 & 0.748 & 0.142 & 0.046 & 0.149 & 0.134 \\ \midrule \multicolumn{18}{c}{$h=2$, $\theta_0^{(h)}=0.1195$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.058 & 0.821 & 0.823 & 0.951 & 0.942 & 0.394 & 0.466 & 0.610 & 0.458 & 0.288 & 0.433 & 0.520 & 0.713 & 0.043 & 0.412 & 0.414 & 0.930 \\ 250 & 0.067 & 0.460 & 0.464 & 0.967 & 0.938 & 0.289 & 0.299 & 0.416 & 0.434 & 0.205 & 0.280 & 0.347 & 0.797 & 0.081 & 0.313 & 0.324 & 0.926 \\ 500 & 0.067 & 0.248 & 0.257 & 0.942 & 0.939 & 0.243 & 0.206 & 0.319 & 0.307 & 0.162 & 0.197 & 0.255 & 0.725 & 0.109 & 0.219 & 0.245 & 0.911 \\ 1'000 & 0.040 & 0.161 & 0.166 & 0.945 & 0.945 & 0.196 & 0.145 & 0.244 & 0.238 & 0.111 & 0.141 & 0.179 & 0.764 & 0.112 & 0.154 & 0.191 & 0.886 \\ 8'000 & 0.014 & 0.052 & 0.054 & 0.933 & 0.932 & 0.213 & 0.057 & 0.221 & 0.002 & 0.051 & 0.050 & 0.071 & 0.787 & 0.126 & 0.054 & 0.137 & 0.367 \\ \midrule \multicolumn{18}{c}{$h=3$, $\theta_0^{(h)}=0.0717$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.057 & 0.849 & 0.851 & 0.943 & 0.939 & 0.347 & 0.487 & 0.598 & 0.475 & 0.257 & 0.459 & 0.526 & 0.756 & 0.026 & 0.461 & 0.461 & 0.930 \\ 250 & 0.072 & 0.483 & 0.489 & 0.954 & 0.936 & 0.257 & 0.326 & 0.416 & 0.416 & 0.188 & 0.310 & 0.363 & 0.816 & 0.060 & 0.350 & 0.355 & 0.929 \\ 500 & 0.066 & 0.273 & 0.281 & 0.943 & 0.941 & 0.216 & 0.224 & 0.311 & 0.337 & 0.147 & 0.216 & 0.261 & 0.759 & 0.090 & 0.243 & 0.259 & 0.937 \\ 1'000 & 0.045 & 0.181 & 0.187 & 0.935 & 0.932 & 0.181 & 0.159 & 0.241 & 0.252 & 0.107 & 0.156 & 0.189 & 0.786 & 0.094 & 0.177 & 0.201 & 0.908 \\ 8'000 & 0.013 & 0.059 & 0.061 & 0.931 & 0.931 & 0.195 & 0.063 & 0.205 & 0.005 & 0.047 & 0.057 & 0.074 & 0.818 & 0.107 & 0.061 & 0.123 & 0.579 \\ \midrule \multicolumn{18}{c}{$h=4$, $\theta_0^{(h)}=0.0430$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.071 & 0.896 & 0.899 & 0.941 & 0.934 & 0.299 & 0.513 & 0.594 & 0.487 & 0.222 & 0.486 & 0.534 & 0.771 & 0.027 & 0.481 & 0.482 & 0.935 \\ 250 & 0.084 & 0.532 & 0.539 & 0.936 & 0.931 & 0.233 & 0.352 & 0.422 & 0.455 & 0.173 & 0.336 & 0.378 & 0.758 & 0.061 & 0.391 & 0.396 & 0.918 \\ 500 & 0.057 & 0.308 & 0.313 & 0.937 & 0.930 & 0.188 & 0.241 & 0.305 & 0.404 & 0.129 & 0.235 & 0.269 & 0.790 & 0.073 & 0.263 & 0.273 & 0.932 \\ 1'000 & 0.040 & 0.200 & 0.203 & 0.936 & 0.932 & 0.157 & 0.173 & 0.234 & 0.278 & 0.093 & 0.170 & 0.193 & 0.807 & 0.071 & 0.193 & 0.206 & 0.937 \\ 8'000 & 0.014 & 0.066 & 0.068 & 0.931 & 0.930 & 0.176 & 0.067 & 0.189 & 0.009 & 0.044 & 0.064 & 0.078 & 0.846 & 0.088 & 0.067 & 0.111 & 0.740 \\ \midrule \multicolumn{18}{c}{$h=5$, $\theta_0^{(h)}=0.0258$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.069 & 0.951 & 0.954 & 0.933 & 0.929 & 0.254 & 0.542 & 0.598 & 0.519 & 0.192 & 0.516 & 0.551 & 0.775 & 0.026 & 0.523 & 0.524 & 0.922 \\ 250 & 0.081 & 0.634 & 0.639 & 0.931 & 0.929 & 0.199 & 0.364 & 0.415 & 0.493 & 0.151 & 0.352 & 0.383 & 0.791 & 0.064 & 0.420 & 0.425 & 0.921 \\ 500 & 0.056 & 0.337 & 0.342 & 0.936 & 0.931 & 0.165 & 0.264 & 0.312 & 0.414 & 0.117 & 0.257 & 0.283 & 0.805 & 0.060 & 0.285 & 0.292 & 0.941 \\ 1'000 & 0.039 & 0.217 & 0.221 & 0.941 & 0.939 & 0.141 & 0.186 & 0.233 & 0.340 & 0.086 & 0.183 & 0.202 & 0.834 & 0.051 & 0.208 & 0.214 & 0.938 \\ 8'000 & 0.013 & 0.070 & 0.072 & 0.931 & 0.931 & 0.155 & 0.072 & 0.171 & 0.027 & 0.039 & 0.069 & 0.079 & 0.875 & 0.069 & 0.073 & 0.100 & 0.844 \\ \bottomrule\end{tabular} \begin{tablenotes}[flushleft] • Note: The table depicts simulation results across $N=1'000$ draws obtained for the scenario with $K$ independent stochastic processes, with $K=10$. Except for the LP estimator, nuisance functions are estimated with random forest. For the DML estimator we set $k_T=T/10$ to obtain estimation samples of the same size as in the setting with one stochastic process. For sample size $T=125$, probabilities are winsorized at 1%. The parameters of the data generating process are $n=12$, $\sigma_\epsilon=1.0$, $\gamma=0.6$, $p=2$, $q=1$, $\sigma_u=1.0$, $\alpha_A=0.3$, $\alpha_M=0.3$, $\rho_A=0.35$, $\rho_M=0.7$, $\beta_1=0.3$, $\beta_2=0.5$. $C_a$($\cdot$) and $C_b$($\cdot$) in the tables denote the coverage at the given confidence level using asymptotic and fixed-bandwidth critical values, respectively. \end{tablenotes} \end{threeparttable}} \end{table} \begin{table}[htbp] \caption{Simulation results for a nonlinear DGP ($n=20$, $\sigma_\epsilon=1.0$) and random forest nuisance function estimates} \resizebox{\textwidth}{!}{\begin{threeparttable}\begin{tabular}{rrrrrrrrrrrrrrrrrr} \toprule \multicolumn{18}{c}{$h=0$, $\theta_0^{(h)}=0.3333$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.180 & 0.791 & 0.811 & 0.937 & 0.920 & 0.382 & 0.454 & 0.593 & 0.524 & 0.293 & 0.418 & 0.511 & 0.712 & 0.101 & 0.370 & 0.383 & 0.823 \\ 250 & 0.082 & 0.385 & 0.393 & 0.959 & 0.945 & 0.279 & 0.289 & 0.402 & 0.519 & 0.210 & 0.269 & 0.342 & 0.719 & 0.117 & 0.248 & 0.274 & 0.857 \\ 500 & 0.037 & 0.213 & 0.216 & 0.953 & 0.940 & 0.199 & 0.194 & 0.278 & 0.529 & 0.136 & 0.183 & 0.228 & 0.780 & 0.122 & 0.175 & 0.213 & 0.850 \\ 1'000 & 0.031 & 0.133 & 0.136 & 0.952 & 0.949 & 0.165 & 0.127 & 0.208 & 0.465 & 0.098 & 0.120 & 0.155 & 0.807 & 0.126 & 0.118 & 0.173 & 0.769 \\ 8'000 & 0.010 & 0.044 & 0.045 & 0.934 & 0.934 & 0.156 & 0.048 & 0.163 & 0.018 & 0.038 & 0.044 & 0.058 & 0.835 & 0.133 & 0.043 & 0.140 & 0.123 \\ \midrule \multicolumn{18}{c}{$h=1$, $\theta_0^{(h)}=0.2000$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.207 & 0.827 & 0.852 & 0.935 & 0.920 & 0.418 & 0.449 & 0.613 & 0.488 & 0.317 & 0.411 & 0.519 & 0.698 & 0.058 & 0.381 & 0.386 & 0.884 \\ 250 & 0.103 & 0.393 & 0.406 & 0.965 & 0.952 & 0.301 & 0.283 & 0.413 & 0.447 & 0.220 & 0.261 & 0.341 & 0.724 & 0.088 & 0.268 & 0.282 & 0.900 \\ 500 & 0.061 & 0.224 & 0.232 & 0.948 & 0.945 & 0.235 & 0.194 & 0.304 & 0.381 & 0.158 & 0.183 & 0.242 & 0.712 & 0.114 & 0.186 & 0.218 & 0.898 \\ 1'000 & 0.053 & 0.144 & 0.154 & 0.931 & 0.929 & 0.203 & 0.134 & 0.243 & 0.283 & 0.121 & 0.127 & 0.175 & 0.734 & 0.126 & 0.132 & 0.182 & 0.827 \\ 8'000 & 0.020 & 0.045 & 0.049 & 0.925 & 0.924 & 0.202 & 0.047 & 0.207 & 0.001 & 0.052 & 0.043 & 0.068 & 0.738 & 0.139 & 0.047 & 0.147 & 0.153 \\ \midrule \multicolumn{18}{c}{$h=2$, $\theta_0^{(h)}=0.1200$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.222 & 0.878 & 0.905 & 0.906 & 0.898 & 0.378 & 0.477 & 0.609 & 0.488 & 0.283 & 0.439 & 0.522 & 0.728 & 0.019 & 0.424 & 0.425 & 0.903 \\ 250 & 0.127 & 0.433 & 0.451 & 0.937 & 0.926 & 0.294 & 0.293 & 0.415 & 0.434 & 0.218 & 0.272 & 0.349 & 0.747 & 0.075 & 0.296 & 0.305 & 0.936 \\ 500 & 0.075 & 0.249 & 0.260 & 0.941 & 0.937 & 0.224 & 0.209 & 0.307 & 0.357 & 0.153 & 0.199 & 0.251 & 0.716 & 0.095 & 0.214 & 0.234 & 0.927 \\ 1'000 & 0.049 & 0.164 & 0.171 & 0.928 & 0.927 & 0.185 & 0.145 & 0.235 & 0.260 & 0.106 & 0.140 & 0.175 & 0.764 & 0.098 & 0.149 & 0.179 & 0.901 \\ 8'000 & 0.022 & 0.051 & 0.056 & 0.919 & 0.919 & 0.198 & 0.052 & 0.205 & 0.001 & 0.052 & 0.049 & 0.072 & 0.758 & 0.127 & 0.054 & 0.138 & 0.362 \\ \midrule \multicolumn{18}{c}{$h=3$, $\theta_0^{(h)}=0.0720$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.228 & 0.987 & 1.013 & 0.924 & 0.901 & 0.353 & 0.486 & 0.601 & 0.534 & 0.271 & 0.450 & 0.526 & 0.747 & 0.022 & 0.454 & 0.454 & 0.914 \\ 250 & 0.137 & 0.487 & 0.506 & 0.932 & 0.924 & 0.266 & 0.328 & 0.422 & 0.467 & 0.201 & 0.311 & 0.370 & 0.746 & 0.053 & 0.340 & 0.344 & 0.925 \\ 500 & 0.083 & 0.289 & 0.301 & 0.941 & 0.936 & 0.216 & 0.228 & 0.314 & 0.376 & 0.152 & 0.218 & 0.266 & 0.735 & 0.085 & 0.244 & 0.259 & 0.925 \\ 1'000 & 0.058 & 0.176 & 0.186 & 0.940 & 0.939 & 0.180 & 0.152 & 0.236 & 0.287 & 0.109 & 0.149 & 0.185 & 0.784 & 0.089 & 0.165 & 0.187 & 0.926 \\ 8'000 & 0.024 & 0.058 & 0.062 & 0.917 & 0.917 & 0.185 & 0.056 & 0.194 & 0.001 & 0.051 & 0.055 & 0.075 & 0.807 & 0.111 & 0.061 & 0.127 & 0.563 \\ \midrule \multicolumn{18}{c}{$h=4$, $\theta_0^{(h)}=0.0432$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.225 & 1.014 & 1.038 & 0.912 & 0.902 & 0.315 & 0.520 & 0.608 & 0.555 & 0.248 & 0.491 & 0.550 & 0.752 & 0.022 & 0.500 & 0.500 & 0.894 \\ 250 & 0.130 & 0.543 & 0.558 & 0.927 & 0.906 & 0.246 & 0.350 & 0.427 & 0.499 & 0.186 & 0.334 & 0.383 & 0.753 & 0.038 & 0.365 & 0.367 & 0.922 \\ 500 & 0.079 & 0.321 & 0.331 & 0.933 & 0.933 & 0.194 & 0.241 & 0.310 & 0.431 & 0.139 & 0.233 & 0.271 & 0.771 & 0.062 & 0.261 & 0.268 & 0.935 \\ 1'000 & 0.058 & 0.194 & 0.202 & 0.948 & 0.944 & 0.166 & 0.166 & 0.234 & 0.309 & 0.101 & 0.162 & 0.191 & 0.811 & 0.072 & 0.181 & 0.195 & 0.937 \\ 8'000 & 0.022 & 0.065 & 0.068 & 0.930 & 0.929 & 0.168 & 0.062 & 0.179 & 0.014 & 0.047 & 0.061 & 0.077 & 0.842 & 0.092 & 0.067 & 0.113 & 0.736 \\ \midrule \multicolumn{18}{c}{$h=5$, $\theta_0^{(h)}=0.0259$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.208 & 1.055 & 1.075 & 0.914 & 0.903 & 0.250 & 0.526 & 0.582 & 0.583 & 0.196 & 0.496 & 0.533 & 0.801 & -0.002 & 0.515 & 0.515 & 0.911 \\ 250 & 0.136 & 0.593 & 0.608 & 0.919 & 0.902 & 0.217 & 0.361 & 0.421 & 0.548 & 0.170 & 0.343 & 0.383 & 0.778 & 0.027 & 0.381 & 0.382 & 0.942 \\ 500 & 0.070 & 0.339 & 0.346 & 0.925 & 0.925 & 0.167 & 0.253 & 0.303 & 0.482 & 0.119 & 0.245 & 0.272 & 0.802 & 0.042 & 0.277 & 0.280 & 0.937 \\ 1'000 & 0.055 & 0.213 & 0.220 & 0.934 & 0.932 & 0.151 & 0.178 & 0.234 & 0.356 & 0.094 & 0.175 & 0.198 & 0.838 & 0.055 & 0.199 & 0.206 & 0.935 \\ 8'000 & 0.021 & 0.070 & 0.073 & 0.934 & 0.933 & 0.152 & 0.067 & 0.166 & 0.025 & 0.043 & 0.067 & 0.079 & 0.859 & 0.075 & 0.071 & 0.103 & 0.822 \\ \bottomrule\end{tabular} \begin{tablenotes}[flushleft] • Note: The table depicts simulation results across $N=1'000$ draws obtained for the scenario with one stochastic process. Except for the LP estimator, nuisance functions are estimated with random forest. For the DML estimator we use 10-fold cross-fitting and set $k_T=T/10$. For sample size $T=125$, probabilities are winsorized at 1%. The parameters of the data generating process are $n=20$, $\sigma_\epsilon=1.0$, $\gamma=0.6$, $p=2$, $q=1$, $\sigma_u=1.0$, $\alpha_A=0.3$, $\alpha_M=0.3$, $\rho_A=0.35$, $\rho_M=0.7$, $\beta_1=0.3$, $\beta_2=0.5$. $C_a$($\cdot$) and $C_b$($\cdot$) in the tables denote the coverage at the given confidence level using asymptotic and fixed-bandwidth critical values, respectively. \end{tablenotes} \end{threeparttable}} \end{table} \begin{table}[htbp] \caption{Simulation results for a nonlinear DGP ($n=12$, $\sigma_\epsilon=3.0$) and random forest nuisance function estimates} \resizebox{\textwidth}{!}{\begin{threeparttable}\begin{tabular}{rrrrrrrrrrrrrrrrrr} \toprule \multicolumn{18}{c}{$h=0$, $\theta_0^{(h)}=0.3321$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.055 & 1.287 & 1.288 & 0.941 & 0.930 & 0.364 & 0.789 & 0.869 & 0.497 & 0.263 & 0.750 & 0.794 & 0.758 & 0.082 & 0.713 & 0.717 & 0.907 \\ 250 & 0.060 & 0.779 & 0.781 & 0.953 & 0.943 & 0.273 & 0.534 & 0.600 & 0.528 & 0.203 & 0.516 & 0.555 & 0.800 & 0.142 & 0.488 & 0.508 & 0.917 \\ 500 & 0.015 & 0.429 & 0.429 & 0.951 & 0.945 & 0.188 & 0.368 & 0.413 & 0.471 & 0.120 & 0.357 & 0.376 & 0.824 & 0.142 & 0.342 & 0.370 & 0.917 \\ 1'000 & 0.019 & 0.285 & 0.285 & 0.946 & 0.945 & 0.161 & 0.264 & 0.309 & 0.429 & 0.088 & 0.260 & 0.274 & 0.838 & 0.146 & 0.245 & 0.285 & 0.871 \\ 8'000 & 0.003 & 0.091 & 0.091 & 0.951 & 0.950 & 0.161 & 0.089 & 0.184 & 0.106 & 0.033 & 0.088 & 0.094 & 0.918 & 0.143 & 0.083 & 0.165 & 0.599 \\ \midrule \multicolumn{18}{c}{$h=1$, $\theta_0^{(h)}=0.1992$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.101 & 1.430 & 1.433 & 0.918 & 0.918 & 0.370 & 0.846 & 0.924 & 0.523 & 0.271 & 0.815 & 0.859 & 0.759 & 0.060 & 0.782 & 0.784 & 0.922 \\ 250 & 0.047 & 1.051 & 1.052 & 0.942 & 0.936 & 0.293 & 0.580 & 0.650 & 0.459 & 0.210 & 0.565 & 0.603 & 0.769 & 0.122 & 0.555 & 0.568 & 0.931 \\ 500 & 0.043 & 0.474 & 0.476 & 0.946 & 0.945 & 0.214 & 0.388 & 0.443 & 0.429 & 0.134 & 0.381 & 0.404 & 0.832 & 0.128 & 0.392 & 0.412 & 0.920 \\ 1'000 & 0.049 & 0.310 & 0.314 & 0.949 & 0.947 & 0.198 & 0.279 & 0.342 & 0.345 & 0.111 & 0.275 & 0.297 & 0.842 & 0.153 & 0.271 & 0.311 & 0.914 \\ 8'000 & 0.010 & 0.102 & 0.102 & 0.947 & 0.946 & 0.207 & 0.097 & 0.228 & 0.047 & 0.045 & 0.098 & 0.108 & 0.913 & 0.153 & 0.096 & 0.181 & 0.642 \\ \midrule \multicolumn{18}{c}{$h=2$, $\theta_0^{(h)}=0.1195$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.159 & 1.382 & 1.391 & 0.932 & 0.921 & 0.352 & 0.847 & 0.917 & 0.512 & 0.258 & 0.816 & 0.856 & 0.784 & 0.033 & 0.814 & 0.815 & 0.935 \\ 250 & 0.044 & 1.106 & 1.107 & 0.932 & 0.925 & 0.258 & 0.600 & 0.653 & 0.497 & 0.182 & 0.592 & 0.619 & 0.777 & 0.080 & 0.599 & 0.604 & 0.925 \\ 500 & 0.059 & 0.503 & 0.507 & 0.941 & 0.938 & 0.202 & 0.414 & 0.460 & 0.417 & 0.130 & 0.408 & 0.428 & 0.814 & 0.111 & 0.420 & 0.434 & 0.933 \\ 1'000 & 0.058 & 0.340 & 0.345 & 0.931 & 0.931 & 0.194 & 0.302 & 0.360 & 0.336 & 0.114 & 0.300 & 0.321 & 0.814 & 0.143 & 0.302 & 0.334 & 0.913 \\ 8'000 & 0.010 & 0.108 & 0.109 & 0.962 & 0.962 & 0.199 & 0.102 & 0.224 & 0.040 & 0.043 & 0.104 & 0.112 & 0.908 & 0.138 & 0.104 & 0.173 & 0.721 \\ \midrule \multicolumn{18}{c}{$h=3$, $\theta_0^{(h)}=0.0717$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.116 & 1.482 & 1.486 & 0.936 & 0.923 & 0.284 & 0.852 & 0.898 & 0.556 & 0.201 & 0.821 & 0.845 & 0.795 & -0.010 & 0.825 & 0.825 & 0.936 \\ 250 & 0.074 & 0.860 & 0.864 & 0.930 & 0.928 & 0.233 & 0.602 & 0.646 & 0.481 & 0.160 & 0.586 & 0.608 & 0.795 & 0.058 & 0.600 & 0.603 & 0.934 \\ 500 & 0.062 & 0.536 & 0.540 & 0.931 & 0.923 & 0.191 & 0.432 & 0.472 & 0.409 & 0.127 & 0.429 & 0.447 & 0.800 & 0.102 & 0.448 & 0.459 & 0.929 \\ 1'000 & 0.060 & 0.353 & 0.358 & 0.924 & 0.922 & 0.183 & 0.310 & 0.360 & 0.331 & 0.112 & 0.311 & 0.331 & 0.815 & 0.128 & 0.319 & 0.344 & 0.911 \\ 8'000 & 0.010 & 0.114 & 0.114 & 0.950 & 0.950 & 0.180 & 0.107 & 0.210 & 0.046 & 0.040 & 0.110 & 0.117 & 0.905 & 0.119 & 0.110 & 0.162 & 0.808 \\ \midrule \multicolumn{18}{c}{$h=4$, $\theta_0^{(h)}=0.0430$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.145 & 1.489 & 1.496 & 0.935 & 0.915 & 0.252 & 0.832 & 0.869 & 0.549 & 0.187 & 0.811 & 0.832 & 0.831 & 0.012 & 0.833 & 0.833 & 0.937 \\ 250 & 0.041 & 1.358 & 1.358 & 0.916 & 0.916 & 0.208 & 0.639 & 0.672 & 0.485 & 0.150 & 0.623 & 0.641 & 0.787 & 0.058 & 0.631 & 0.634 & 0.935 \\ 500 & 0.039 & 0.576 & 0.578 & 0.938 & 0.926 & 0.151 & 0.448 & 0.473 & 0.421 & 0.097 & 0.443 & 0.453 & 0.821 & 0.063 & 0.456 & 0.461 & 0.929 \\ 1'000 & 0.055 & 0.363 & 0.367 & 0.938 & 0.936 & 0.160 & 0.316 & 0.354 & 0.345 & 0.098 & 0.315 & 0.330 & 0.835 & 0.104 & 0.324 & 0.341 & 0.920 \\ 8'000 & 0.010 & 0.117 & 0.118 & 0.949 & 0.949 & 0.159 & 0.110 & 0.194 & 0.056 & 0.037 & 0.113 & 0.119 & 0.911 & 0.100 & 0.110 & 0.148 & 0.859 \\ \midrule \multicolumn{18}{c}{$h=5$, $\theta_0^{(h)}=0.0258$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.167 & 1.619 & 1.627 & 0.915 & 0.898 & 0.253 & 0.897 & 0.932 & 0.558 & 0.199 & 0.874 & 0.897 & 0.813 & 0.053 & 0.913 & 0.914 & 0.922 \\ 250 & 0.050 & 1.062 & 1.063 & 0.920 & 0.912 & 0.176 & 0.650 & 0.674 & 0.469 & 0.125 & 0.638 & 0.650 & 0.775 & 0.046 & 0.643 & 0.645 & 0.931 \\ 500 & 0.025 & 0.587 & 0.588 & 0.950 & 0.941 & 0.122 & 0.446 & 0.463 & 0.450 & 0.078 & 0.444 & 0.451 & 0.829 & 0.045 & 0.458 & 0.461 & 0.938 \\ 1'000 & 0.042 & 0.363 & 0.366 & 0.947 & 0.941 & 0.129 & 0.313 & 0.339 & 0.357 & 0.076 & 0.313 & 0.322 & 0.855 & 0.074 & 0.323 & 0.331 & 0.950 \\ 8'000 & 0.003 & 0.121 & 0.121 & 0.942 & 0.941 & 0.134 & 0.112 & 0.175 & 0.082 & 0.026 & 0.115 & 0.118 & 0.919 & 0.077 & 0.113 & 0.136 & 0.912 \\ \bottomrule\end{tabular} \begin{tablenotes}[flushleft] • Note: The table depicts simulation results across $N=1'000$ draws obtained for the scenario with one stochastic process. Except for the LP estimator, nuisance functions are estimated with random forest. For the DML estimator we use 10-fold cross-fitting and set $k_T=T/10$. For sample size $T=125$, probabilities are winsorized at 1%. The parameters of the data generating process are $n=12$, $\sigma_\epsilon=3.0$, $\gamma=0.6$, $p=2$, $q=1$, $\sigma_u=1.0$, $\alpha_A=0.3$, $\alpha_M=0.3$, $\rho_A=0.35$, $\rho_M=0.7$, $\beta_1=0.3$, $\beta_2=0.5$. $C_a$($\cdot$) and $C_b$($\cdot$) in the tables denote the coverage at the given confidence level using asymptotic and fixed-bandwidth critical values, respectively. \end{tablenotes} \end{threeparttable}} \end{table} \begin{table}[htbp] \caption{Simulation results for an empirically calibrated DGP (see Section (ref)) and random forest nuisance function estimates} \resizebox{\textwidth}{!}{\begin{threeparttable}\begin{tabular}{rrrrrrrrrrrrrrrrrr} \toprule \multicolumn{18}{c}{$h=0$, $\theta_0^{(h)}=0.2021$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.058 & 0.343 & 0.348 & 0.966 & 0.948 & 0.244 & 0.273 & 0.366 & 0.279 & 0.203 & 0.263 & 0.332 & 0.681 & 0.146 & 0.260 & 0.298 & 0.883 \\ 250 & 0.042 & 0.207 & 0.211 & 0.935 & 0.931 & 0.210 & 0.190 & 0.283 & 0.238 & 0.168 & 0.184 & 0.249 & 0.629 & 0.153 & 0.189 & 0.243 & 0.845 \\ 500 & 0.038 & 0.130 & 0.136 & 0.949 & 0.945 & 0.177 & 0.129 & 0.219 & 0.242 & 0.130 & 0.125 & 0.180 & 0.641 & 0.154 & 0.127 & 0.200 & 0.752 \\ 1'000 & 0.034 & 0.096 & 0.101 & 0.919 & 0.919 & 0.159 & 0.090 & 0.182 & 0.148 & 0.105 & 0.088 & 0.137 & 0.625 & 0.152 & 0.091 & 0.177 & 0.580 \\ 8'000 & -0.002 & 0.031 & 0.032 & 0.949 & 0.949 & 0.153 & 0.033 & 0.156 & 0.000 & 0.035 & 0.031 & 0.047 & 0.760 & 0.153 & 0.033 & 0.157 & 0.001 \\ \midrule \multicolumn{18}{c}{$h=1$, $\theta_0^{(h)}=0.0521$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.040 & 0.347 & 0.349 & 0.965 & 0.939 & 0.108 & 0.271 & 0.291 & 0.330 & 0.088 & 0.267 & 0.282 & 0.791 & 0.029 & 0.274 & 0.276 & 0.941 \\ 250 & 0.025 & 0.245 & 0.247 & 0.962 & 0.944 & 0.100 & 0.192 & 0.216 & 0.258 & 0.079 & 0.191 & 0.206 & 0.789 & 0.041 & 0.199 & 0.203 & 0.936 \\ 500 & 0.027 & 0.151 & 0.153 & 0.933 & 0.929 & 0.084 & 0.137 & 0.161 & 0.265 & 0.062 & 0.137 & 0.151 & 0.792 & 0.043 & 0.141 & 0.147 & 0.918 \\ 1'000 & 0.019 & 0.100 & 0.102 & 0.947 & 0.946 & 0.073 & 0.095 & 0.119 & 0.194 & 0.047 & 0.095 & 0.106 & 0.808 & 0.042 & 0.097 & 0.106 & 0.927 \\ 8'000 & -0.000 & 0.035 & 0.035 & 0.954 & 0.953 & 0.072 & 0.034 & 0.079 & 0.019 & 0.015 & 0.034 & 0.037 & 0.894 & 0.040 & 0.034 & 0.053 & 0.780 \\ \midrule \multicolumn{18}{c}{$h=2$, $\theta_0^{(h)}=0.0134$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.037 & 0.362 & 0.364 & 0.969 & 0.947 & 0.067 & 0.263 & 0.271 & 0.309 & 0.054 & 0.261 & 0.266 & 0.860 & 0.007 & 0.272 & 0.272 & 0.946 \\ 250 & 0.008 & 0.228 & 0.228 & 0.955 & 0.944 & 0.045 & 0.191 & 0.196 & 0.288 & 0.034 & 0.190 & 0.193 & 0.814 & 0.003 & 0.196 & 0.196 & 0.941 \\ 500 & 0.010 & 0.150 & 0.150 & 0.945 & 0.943 & 0.038 & 0.136 & 0.141 & 0.245 & 0.026 & 0.137 & 0.139 & 0.830 & 0.005 & 0.138 & 0.138 & 0.951 \\ 1'000 & 0.006 & 0.108 & 0.108 & 0.934 & 0.934 & 0.034 & 0.099 & 0.105 & 0.184 & 0.020 & 0.101 & 0.103 & 0.828 & 0.007 & 0.101 & 0.101 & 0.938 \\ 8'000 & 0.000 & 0.036 & 0.036 & 0.954 & 0.953 & 0.036 & 0.034 & 0.050 & 0.050 & 0.008 & 0.035 & 0.036 & 0.916 & 0.010 & 0.035 & 0.037 & 0.936 \\ \midrule \multicolumn{18}{c}{$h=3$, $\theta_0^{(h)}=0.0035$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.032 & 0.362 & 0.363 & 0.950 & 0.936 & 0.042 & 0.282 & 0.285 & 0.331 & 0.034 & 0.277 & 0.279 & 0.811 & 0.001 & 0.285 & 0.285 & 0.932 \\ 250 & 0.009 & 0.238 & 0.239 & 0.935 & 0.929 & 0.027 & 0.196 & 0.198 & 0.284 & 0.020 & 0.197 & 0.198 & 0.790 & -0.006 & 0.199 & 0.199 & 0.940 \\ 500 & 0.005 & 0.151 & 0.151 & 0.947 & 0.945 & 0.023 & 0.134 & 0.136 & 0.231 & 0.015 & 0.135 & 0.136 & 0.840 & -0.003 & 0.135 & 0.135 & 0.957 \\ 1'000 & 0.003 & 0.106 & 0.106 & 0.939 & 0.937 & 0.022 & 0.096 & 0.098 & 0.233 & 0.013 & 0.098 & 0.099 & 0.852 & 0.000 & 0.098 & 0.098 & 0.947 \\ 8'000 & 0.000 & 0.037 & 0.037 & 0.945 & 0.945 & 0.022 & 0.034 & 0.040 & 0.054 & 0.005 & 0.036 & 0.036 & 0.925 & 0.002 & 0.035 & 0.035 & 0.944 \\ \midrule \multicolumn{18}{c}{$h=4$, $\theta_0^{(h)}=0.0009$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.023 & 0.376 & 0.376 & 0.945 & 0.933 & 0.034 & 0.290 & 0.292 & 0.315 & 0.026 & 0.286 & 0.287 & 0.777 & -0.001 & 0.296 & 0.296 & 0.925 \\ 250 & 0.006 & 0.228 & 0.228 & 0.944 & 0.938 & 0.025 & 0.190 & 0.192 & 0.316 & 0.020 & 0.191 & 0.192 & 0.810 & 0.003 & 0.195 & 0.195 & 0.940 \\ 500 & 0.006 & 0.155 & 0.155 & 0.940 & 0.936 & 0.017 & 0.141 & 0.142 & 0.252 & 0.011 & 0.142 & 0.142 & 0.815 & -0.005 & 0.145 & 0.145 & 0.938 \\ 1'000 & 0.006 & 0.103 & 0.103 & 0.956 & 0.955 & 0.020 & 0.093 & 0.095 & 0.185 & 0.012 & 0.095 & 0.096 & 0.871 & 0.001 & 0.095 & 0.095 & 0.951 \\ 8'000 & -0.000 & 0.036 & 0.036 & 0.953 & 0.953 & 0.016 & 0.033 & 0.037 & 0.065 & 0.003 & 0.035 & 0.035 & 0.936 & 0.000 & 0.035 & 0.035 & 0.956 \\ \midrule \multicolumn{18}{c}{$h=5$, $\theta_0^{(h)}=0.0002$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.024 & 0.370 & 0.371 & 0.944 & 0.930 & 0.022 & 0.283 & 0.284 & 0.305 & 0.017 & 0.281 & 0.282 & 0.909 & -0.001 & 0.285 & 0.285 & 0.935 \\ 250 & 0.012 & 0.237 & 0.238 & 0.936 & 0.934 & 0.016 & 0.194 & 0.194 & 0.304 & 0.013 & 0.193 & 0.193 & 0.796 & -0.004 & 0.197 & 0.197 & 0.946 \\ 500 & 0.006 & 0.152 & 0.152 & 0.960 & 0.954 & 0.017 & 0.135 & 0.136 & 0.226 & 0.012 & 0.136 & 0.136 & 0.851 & -0.003 & 0.137 & 0.137 & 0.957 \\ 1'000 & 0.006 & 0.104 & 0.105 & 0.953 & 0.950 & 0.017 & 0.096 & 0.097 & 0.171 & 0.011 & 0.097 & 0.097 & 0.869 & 0.000 & 0.096 & 0.096 & 0.953 \\ 8'000 & 0.000 & 0.036 & 0.036 & 0.950 & 0.950 & 0.013 & 0.033 & 0.036 & 0.068 & 0.003 & 0.035 & 0.035 & 0.931 & 0.000 & 0.034 & 0.034 & 0.960 \\ \bottomrule\end{tabular} \begin{tablenotes}[flushleft] • Note: The table depicts simulation results across $N=1'000$ draws obtained for the scenario with one stochastic process. Except for the LP estimator, nuisance functions are estimated with random forest. For the DML estimator we use 10-fold cross-fitting and set $k_T=T/10$. For sample size $T=125$, probabilities are winsorized at 1%. The parameters of the data generating process are empirically calibrated using monthly U.S. data from 1982 to 2012 obtained from the empirical study in Angrist2018. $C_a$($\cdot$) and $C_b$($\cdot$) in the tables denote the coverage at the given confidence level using asymptotic and fixed-bandwidth critical values, respectively. \end{tablenotes} \end{threeparttable}} \end{table} \begin{table}[htbp] \caption{Simulation results for a baseline nonlinear DGP ($n=12$, $\sigma_\epsilon=1.0$) and gradient boosting nuisance function estimates} \resizebox{\textwidth}{!}{\begin{threeparttable}\begin{tabular}{rrrrrrrrrrrrrrrrrr} \toprule \multicolumn{18}{c}{$h=0$, $\theta_0^{(h)}=0.3321$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.166 & 1.285 & 1.295 & 0.937 & 0.937 & 0.071 & 0.525 & 0.530 & 0.800 & -0.019 & 0.514 & 0.515 & 0.900 & 0.090 & 0.357 & 0.368 & 0.871 \\ 250 & 0.090 & 0.631 & 0.638 & 0.922 & 0.922 & 0.111 & 0.391 & 0.406 & 0.732 & 0.081 & 0.313 & 0.324 & 0.883 & 0.128 & 0.251 & 0.282 & 0.847 \\ 500 & 0.043 & 0.269 & 0.273 & 0.961 & 0.955 & 0.280 & 0.242 & 0.370 & 0.398 & 0.158 & 0.199 & 0.254 & 0.805 & 0.135 & 0.173 & 0.220 & 0.829 \\ 1'000 & 0.028 & 0.158 & 0.160 & 0.942 & 0.939 & 0.073 & 0.189 & 0.203 & 0.662 & 0.038 & 0.141 & 0.146 & 0.919 & 0.138 & 0.126 & 0.187 & 0.730 \\ 8'000 & 0.008 & 0.040 & 0.041 & 0.960 & 0.960 & 0.115 & 0.042 & 0.122 & 0.089 & 0.014 & 0.040 & 0.043 & 0.952 & 0.131 & 0.046 & 0.139 & 0.141 \\ \midrule \multicolumn{18}{c}{$h=1$, $\theta_0^{(h)}=0.1992$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.166 & 1.322 & 1.332 & 0.952 & 0.938 & 0.090 & 0.547 & 0.554 & 0.751 & 0.009 & 0.514 & 0.514 & 0.898 & 0.060 & 0.362 & 0.367 & 0.923 \\ 250 & 0.105 & 0.641 & 0.650 & 0.955 & 0.954 & 0.135 & 0.398 & 0.421 & 0.701 & 0.090 & 0.315 & 0.327 & 0.877 & 0.111 & 0.265 & 0.287 & 0.914 \\ 500 & 0.067 & 0.266 & 0.275 & 0.959 & 0.957 & 0.320 & 0.236 & 0.397 & 0.286 & 0.175 & 0.196 & 0.263 & 0.777 & 0.123 & 0.193 & 0.229 & 0.887 \\ 1'000 & 0.044 & 0.154 & 0.160 & 0.953 & 0.952 & 0.097 & 0.187 & 0.211 & 0.553 & 0.052 & 0.137 & 0.146 & 0.916 & 0.139 & 0.133 & 0.193 & 0.792 \\ 8'000 & 0.016 & 0.042 & 0.045 & 0.937 & 0.936 & 0.134 & 0.044 & 0.141 & 0.013 & 0.021 & 0.042 & 0.047 & 0.928 & 0.144 & 0.053 & 0.153 & 0.183 \\ \midrule \multicolumn{18}{c}{$h=2$, $\theta_0^{(h)}=0.1195$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.153 & 1.439 & 1.447 & 0.959 & 0.935 & 0.104 & 0.561 & 0.570 & 0.732 & -0.006 & 0.541 & 0.541 & 0.913 & 0.031 & 0.410 & 0.411 & 0.931 \\ 250 & 0.107 & 0.723 & 0.731 & 0.944 & 0.937 & 0.149 & 0.426 & 0.451 & 0.662 & 0.082 & 0.329 & 0.339 & 0.887 & 0.078 & 0.306 & 0.316 & 0.927 \\ 500 & 0.072 & 0.300 & 0.308 & 0.952 & 0.952 & 0.310 & 0.266 & 0.408 & 0.346 & 0.171 & 0.219 & 0.278 & 0.800 & 0.104 & 0.221 & 0.244 & 0.911 \\ 1'000 & 0.051 & 0.176 & 0.183 & 0.930 & 0.926 & 0.104 & 0.210 & 0.234 & 0.478 & 0.058 & 0.156 & 0.167 & 0.898 & 0.128 & 0.160 & 0.204 & 0.854 \\ 8'000 & 0.015 & 0.049 & 0.051 & 0.943 & 0.943 & 0.131 & 0.051 & 0.141 & 0.014 & 0.019 & 0.048 & 0.052 & 0.935 & 0.128 & 0.055 & 0.140 & 0.352 \\ \midrule \multicolumn{18}{c}{$h=3$, $\theta_0^{(h)}=0.0717$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.132 & 1.513 & 1.519 & 0.958 & 0.944 & 0.124 & 0.622 & 0.634 & 0.694 & 0.007 & 0.591 & 0.591 & 0.894 & 0.012 & 0.448 & 0.448 & 0.937 \\ 250 & 0.097 & 0.797 & 0.803 & 0.933 & 0.930 & 0.143 & 0.476 & 0.497 & 0.611 & 0.074 & 0.364 & 0.371 & 0.901 & 0.058 & 0.340 & 0.345 & 0.934 \\ 500 & 0.078 & 0.327 & 0.337 & 0.953 & 0.946 & 0.289 & 0.287 & 0.407 & 0.391 & 0.163 & 0.238 & 0.289 & 0.840 & 0.087 & 0.249 & 0.264 & 0.915 \\ 1'000 & 0.054 & 0.196 & 0.204 & 0.949 & 0.946 & 0.113 & 0.228 & 0.255 & 0.442 & 0.058 & 0.168 & 0.178 & 0.915 & 0.112 & 0.178 & 0.210 & 0.902 \\ 8'000 & 0.014 & 0.055 & 0.057 & 0.946 & 0.946 & 0.128 & 0.058 & 0.141 & 0.021 & 0.017 & 0.055 & 0.057 & 0.937 & 0.117 & 0.061 & 0.131 & 0.479 \\ \midrule \multicolumn{18}{c}{$h=4$, $\theta_0^{(h)}=0.0430$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.150 & 1.681 & 1.688 & 0.960 & 0.937 & 0.125 & 0.678 & 0.689 & 0.695 & 0.041 & 0.696 & 0.697 & 0.902 & 0.021 & 0.496 & 0.497 & 0.935 \\ 250 & 0.069 & 0.867 & 0.870 & 0.927 & 0.924 & 0.142 & 0.510 & 0.529 & 0.591 & 0.077 & 0.405 & 0.412 & 0.873 & 0.049 & 0.377 & 0.380 & 0.922 \\ 500 & 0.068 & 0.363 & 0.369 & 0.952 & 0.940 & 0.245 & 0.299 & 0.387 & 0.447 & 0.148 & 0.261 & 0.300 & 0.852 & 0.062 & 0.270 & 0.277 & 0.929 \\ 1'000 & 0.051 & 0.219 & 0.225 & 0.942 & 0.942 & 0.110 & 0.247 & 0.271 & 0.408 & 0.054 & 0.181 & 0.189 & 0.934 & 0.092 & 0.192 & 0.213 & 0.921 \\ 8'000 & 0.014 & 0.061 & 0.063 & 0.948 & 0.948 & 0.126 & 0.063 & 0.141 & 0.026 & 0.017 & 0.060 & 0.063 & 0.939 & 0.100 & 0.067 & 0.121 & 0.662 \\ \midrule \multicolumn{18}{c}{$h=5$, $\theta_0^{(h)}=0.0258$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.099 & 1.794 & 1.796 & 0.949 & 0.927 & 0.148 & 0.705 & 0.720 & 0.669 & 0.082 & 0.729 & 0.734 & 0.898 & 0.032 & 0.551 & 0.552 & 0.921 \\ 250 & 0.077 & 0.937 & 0.940 & 0.931 & 0.926 & 0.119 & 0.532 & 0.545 & 0.584 & 0.068 & 0.421 & 0.427 & 0.884 & 0.036 & 0.394 & 0.395 & 0.918 \\ 500 & 0.063 & 0.384 & 0.389 & 0.951 & 0.943 & 0.216 & 0.308 & 0.376 & 0.491 & 0.131 & 0.271 & 0.301 & 0.862 & 0.047 & 0.283 & 0.287 & 0.943 \\ 1'000 & 0.039 & 0.233 & 0.236 & 0.946 & 0.938 & 0.097 & 0.256 & 0.274 & 0.445 & 0.047 & 0.193 & 0.198 & 0.927 & 0.071 & 0.202 & 0.214 & 0.933 \\ 8'000 & 0.010 & 0.068 & 0.068 & 0.951 & 0.951 & 0.116 & 0.071 & 0.136 & 0.040 & 0.013 & 0.067 & 0.068 & 0.941 & 0.082 & 0.078 & 0.113 & 0.789 \\ \bottomrule\end{tabular} \begin{tablenotes}[flushleft] • Note: The table depicts simulation results across $N=1'000$ draws obtained for the scenario with one stochastic process. Except for the LP estimator, nuisance functions are estimated with gradient boosting. For the DML estimator we use 10-fold cross-fitting and set $k_T=T/10$. For sample size $T=125$, probabilities are winsorized at 1%. The parameters of the data generating process are $n=12$, $\sigma_\epsilon=1.0$, $\gamma=0.6$, $p=2$, $q=1$, $\sigma_u=1.0$, $\alpha_A=0.3$, $\alpha_M=0.3$, $\rho_A=0.35$, $\rho_M=0.7$, $\beta_1=0.3$, $\beta_2=0.5$. $C_a$($\cdot$) and $C_b$($\cdot$) in the tables denote the coverage at the given confidence level using asymptotic and fixed-bandwidth critical values, respectively. \end{tablenotes} \end{threeparttable}} \end{table} \begin{table}[htbp] \caption{Simulation results for a linear DGP ($n=12$, $\sigma_\epsilon=1.0$) and random forest nuisance function estimates} \resizebox{\textwidth}{!}{\begin{threeparttable}\begin{tabular}{rrrrrrrrrrrrrrrrrr} \toprule \multicolumn{18}{c}{$h=0$, $\theta_0^{(h)}=0.3321$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.121 & 1.375 & 1.381 & 0.927 & 0.918 & 0.851 & 0.550 & 1.013 & 0.270 & 0.669 & 0.498 & 0.834 & 0.503 & -0.006 & 0.219 & 0.219 & 0.911 \\ 250 & 0.060 & 1.043 & 1.044 & 0.944 & 0.935 & 0.675 & 0.325 & 0.749 & 0.095 & 0.525 & 0.292 & 0.601 & 0.334 & 0.004 & 0.146 & 0.146 & 0.956 \\ 500 & 0.069 & 0.348 & 0.355 & 0.959 & 0.941 & 0.537 & 0.203 & 0.574 & 0.031 & 0.388 & 0.185 & 0.430 & 0.240 & -0.001 & 0.106 & 0.106 & 0.947 \\ 1'000 & 0.062 & 0.164 & 0.176 & 0.944 & 0.942 & 0.445 & 0.126 & 0.463 & 0.000 & 0.280 & 0.116 & 0.303 & 0.186 & -0.000 & 0.074 & 0.074 & 0.956 \\ 8'000 & 0.019 & 0.040 & 0.044 & 0.942 & 0.942 & 0.381 & 0.066 & 0.386 & 0.000 & 0.108 & 0.038 & 0.114 & 0.152 & -0.001 & 0.025 & 0.025 & 0.940 \\ \midrule \multicolumn{18}{c}{$h=1$, $\theta_0^{(h)}=0.1992$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.173 & 1.539 & 1.549 & 0.916 & 0.900 & 0.893 & 0.590 & 1.070 & 0.287 & 0.708 & 0.542 & 0.892 & 0.485 & -0.030 & 0.317 & 0.319 & 0.918 \\ 250 & 0.067 & 1.273 & 1.275 & 0.943 & 0.921 & 0.732 & 0.364 & 0.817 & 0.119 & 0.567 & 0.332 & 0.657 & 0.354 & -0.005 & 0.224 & 0.224 & 0.934 \\ 500 & 0.105 & 0.372 & 0.386 & 0.945 & 0.936 & 0.597 & 0.223 & 0.638 & 0.023 & 0.433 & 0.207 & 0.480 & 0.262 & -0.008 & 0.151 & 0.151 & 0.947 \\ 1'000 & 0.094 & 0.195 & 0.216 & 0.913 & 0.910 & 0.509 & 0.148 & 0.530 & 0.002 & 0.324 & 0.139 & 0.352 & 0.200 & 0.002 & 0.109 & 0.110 & 0.945 \\ 8'000 & 0.028 & 0.050 & 0.057 & 0.922 & 0.922 & 0.447 & 0.073 & 0.453 & 0.000 & 0.128 & 0.046 & 0.136 & 0.163 & 0.001 & 0.038 & 0.038 & 0.950 \\ \midrule \multicolumn{18}{c}{$h=2$, $\theta_0^{(h)}=0.1195$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.217 & 1.495 & 1.511 & 0.917 & 0.892 & 0.818 & 0.659 & 1.050 & 0.372 & 0.649 & 0.615 & 0.894 & 0.610 & -0.042 & 0.454 & 0.456 & 0.918 \\ 250 & 0.082 & 1.058 & 1.061 & 0.917 & 0.905 & 0.656 & 0.418 & 0.778 & 0.238 & 0.510 & 0.393 & 0.644 & 0.546 & -0.014 & 0.326 & 0.326 & 0.931 \\ 500 & 0.114 & 0.412 & 0.427 & 0.944 & 0.932 & 0.547 & 0.270 & 0.610 & 0.102 & 0.399 & 0.257 & 0.475 & 0.483 & -0.009 & 0.218 & 0.218 & 0.954 \\ 1'000 & 0.102 & 0.237 & 0.258 & 0.898 & 0.895 & 0.470 & 0.192 & 0.508 & 0.033 & 0.303 & 0.185 & 0.355 & 0.429 & 0.004 & 0.165 & 0.165 & 0.936 \\ 8'000 & 0.027 & 0.066 & 0.072 & 0.926 & 0.924 & 0.420 & 0.084 & 0.428 & 0.000 & 0.119 & 0.061 & 0.134 & 0.457 & 0.000 & 0.055 & 0.055 & 0.957 \\ \midrule \multicolumn{18}{c}{$h=3$, $\theta_0^{(h)}=0.0717$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.221 & 1.641 & 1.656 & 0.920 & 0.895 & 0.696 & 0.709 & 0.993 & 0.431 & 0.554 & 0.669 & 0.869 & 0.724 & -0.051 & 0.562 & 0.564 & 0.917 \\ 250 & 0.055 & 1.522 & 1.523 & 0.912 & 0.899 & 0.569 & 0.488 & 0.750 & 0.346 & 0.438 & 0.463 & 0.637 & 0.660 & -0.019 & 0.418 & 0.418 & 0.913 \\ 500 & 0.113 & 0.462 & 0.475 & 0.951 & 0.937 & 0.478 & 0.310 & 0.570 & 0.220 & 0.352 & 0.301 & 0.463 & 0.642 & -0.006 & 0.281 & 0.281 & 0.945 \\ 1'000 & 0.106 & 0.275 & 0.295 & 0.913 & 0.902 & 0.413 & 0.227 & 0.471 & 0.100 & 0.270 & 0.223 & 0.350 & 0.590 & 0.007 & 0.209 & 0.209 & 0.935 \\ 8'000 & 0.027 & 0.082 & 0.086 & 0.938 & 0.937 & 0.376 & 0.092 & 0.387 & 0.000 & 0.107 & 0.077 & 0.132 & 0.638 & 0.000 & 0.071 & 0.071 & 0.952 \\ \midrule \multicolumn{18}{c}{$h=4$, $\theta_0^{(h)}=0.0430$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.283 & 1.587 & 1.612 & 0.892 & 0.892 & 0.594 & 0.744 & 0.952 & 0.530 & 0.478 & 0.714 & 0.860 & 0.807 & -0.023 & 0.660 & 0.661 & 0.915 \\ 250 & 0.092 & 1.054 & 1.058 & 0.907 & 0.897 & 0.476 & 0.545 & 0.724 & 0.409 & 0.369 & 0.524 & 0.640 & 0.715 & -0.016 & 0.493 & 0.493 & 0.917 \\ 500 & 0.103 & 0.489 & 0.499 & 0.961 & 0.935 & 0.396 & 0.345 & 0.526 & 0.335 & 0.291 & 0.337 & 0.446 & 0.733 & -0.016 & 0.326 & 0.326 & 0.957 \\ 1'000 & 0.102 & 0.318 & 0.334 & 0.914 & 0.909 & 0.357 & 0.261 & 0.442 & 0.194 & 0.237 & 0.257 & 0.350 & 0.693 & 0.010 & 0.247 & 0.247 & 0.937 \\ 8'000 & 0.023 & 0.097 & 0.100 & 0.942 & 0.941 & 0.329 & 0.100 & 0.343 & 0.002 & 0.092 & 0.091 & 0.130 & 0.756 & -0.001 & 0.084 & 0.084 & 0.948 \\ \midrule \multicolumn{18}{c}{$h=5$, $\theta_0^{(h)}=0.0258$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.280 & 1.696 & 1.719 & 0.902 & 0.886 & 0.500 & 0.809 & 0.951 & 0.560 & 0.411 & 0.777 & 0.879 & 0.794 & 0.001 & 0.738 & 0.738 & 0.919 \\ 250 & 0.091 & 1.178 & 1.182 & 0.923 & 0.916 & 0.407 & 0.579 & 0.708 & 0.489 & 0.313 & 0.558 & 0.640 & 0.765 & -0.009 & 0.532 & 0.532 & 0.927 \\ 500 & 0.091 & 0.531 & 0.539 & 0.958 & 0.938 & 0.327 & 0.377 & 0.499 & 0.402 & 0.241 & 0.370 & 0.442 & 0.779 & -0.020 & 0.364 & 0.364 & 0.953 \\ 1'000 & 0.094 & 0.348 & 0.360 & 0.913 & 0.912 & 0.303 & 0.283 & 0.414 & 0.270 & 0.202 & 0.280 & 0.346 & 0.757 & 0.008 & 0.272 & 0.272 & 0.946 \\ 8'000 & 0.019 & 0.109 & 0.111 & 0.934 & 0.934 & 0.282 & 0.107 & 0.302 & 0.009 & 0.077 & 0.102 & 0.128 & 0.821 & -0.003 & 0.096 & 0.096 & 0.946 \\ \bottomrule\end{tabular} \begin{tablenotes}[flushleft] • \textsc{Note}: The table depicts simulation results across $N=1'000$ draws obtained for the scenario with one stochastic process. The outcome variable is generated from a linear DGP, i.e. $b(X_t) = 0.5 \sum_{i=1}^5X_{i,t}$ and $\tau(X_t) = \theta_0^{(0)}$. Except for the LP estimator, nuisance functions are estimated with random forest. For the DML estimator we use 10-fold cross-fitting and set $k_T=T/10$. For sample size $T=125$, probabilities are winsorized at 1%. The parameters of the data generating process are $n=12$, $\sigma_\epsilon=1.0$, $\gamma=0.6$, $p=2$, $q=1$, $\sigma_u=1.0$, $\alpha_A=0.3$, $\alpha_M=0.3$, $\rho_A=0.35$, $\rho_M=0.7$, $\beta_1=0.3$, $\beta_2=0.5$. $C_a$($\cdot$) and $C_b$($\cdot$) in the tables denote the coverage at the given confidence level using asymptotic and fixed-bandwidth critical values, respectively. \end{tablenotes} \end{threeparttable}} \end{table} \begin{table}[htbp] \caption{Simulation results for a linear DGP with interactions ($n=12$, $\sigma_\epsilon=1.0$) and random forest nuisance function estimates} \resizebox{\textwidth}{!}{\begin{threeparttable}\begin{tabular}{rrrrrrrrrrrrrrrrrr} \toprule \multicolumn{18}{c}{$h=0$, $\theta_0^{(h)}=0.3321$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.115 & 1.926 & 1.930 & 0.810 & 0.786 & 0.900 & 1.076 & 1.403 & 0.404 & 0.714 & 1.020 & 1.245 & 0.513 & 0.556 & 0.858 & 1.023 & 0.579 \\ 250 & 0.079 & 1.248 & 1.251 & 0.814 & 0.810 & 0.700 & 0.726 & 1.009 & 0.468 & 0.551 & 0.702 & 0.892 & 0.580 & 0.595 & 0.589 & 0.837 & 0.527 \\ 500 & 0.085 & 0.618 & 0.624 & 0.839 & 0.830 & 0.571 & 0.511 & 0.767 & 0.473 & 0.419 & 0.495 & 0.649 & 0.633 & 0.624 & 0.404 & 0.743 & 0.364 \\ 1'000 & 0.084 & 0.395 & 0.403 & 0.856 & 0.849 & 0.479 & 0.367 & 0.603 & 0.487 & 0.312 & 0.359 & 0.476 & 0.684 & 0.632 & 0.297 & 0.698 & 0.206 \\ 8'000 & 0.025 & 0.066 & 0.070 & 0.947 & 0.945 & 0.416 & 0.086 & 0.425 & 0.000 & 0.124 & 0.064 & 0.140 & 0.460 & 0.329 & 0.053 & 0.333 & 0.000 \\ \midrule \multicolumn{18}{c}{$h=1$, $\theta_0^{(h)}=0.1992$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.173 & 1.778 & 1.787 & 0.863 & 0.843 & 0.821 & 0.825 & 1.164 & 0.350 & 0.627 & 0.759 & 0.985 & 0.543 & 0.308 & 0.592 & 0.667 & 0.771 \\ 250 & 0.113 & 1.003 & 1.009 & 0.881 & 0.878 & 0.662 & 0.562 & 0.868 & 0.330 & 0.498 & 0.525 & 0.723 & 0.519 & 0.375 & 0.425 & 0.567 & 0.705 \\ 500 & 0.112 & 0.502 & 0.514 & 0.859 & 0.859 & 0.542 & 0.390 & 0.668 & 0.300 & 0.381 & 0.368 & 0.530 & 0.539 & 0.413 & 0.303 & 0.512 & 0.567 \\ 1'000 & 0.103 & 0.300 & 0.318 & 0.868 & 0.863 & 0.464 & 0.276 & 0.540 & 0.267 & 0.289 & 0.264 & 0.392 & 0.571 & 0.433 & 0.222 & 0.487 & 0.341 \\ 8'000 & 0.029 & 0.062 & 0.069 & 0.919 & 0.919 & 0.449 & 0.082 & 0.456 & 0.000 & 0.127 & 0.058 & 0.139 & 0.355 & 0.231 & 0.051 & 0.237 & 0.006 \\ \midrule \multicolumn{18}{c}{$h=2$, $\theta_0^{(h)}=0.1195$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.202 & 1.573 & 1.586 & 0.901 & 0.894 & 0.707 & 0.730 & 1.016 & 0.383 & 0.532 & 0.668 & 0.854 & 0.627 & 0.151 & 0.564 & 0.584 & 0.905 \\ 250 & 0.042 & 2.297 & 2.297 & 0.920 & 0.912 & 0.555 & 0.497 & 0.745 & 0.312 & 0.412 & 0.465 & 0.621 & 0.607 & 0.225 & 0.415 & 0.472 & 0.895 \\ 500 & 0.111 & 0.467 & 0.480 & 0.917 & 0.909 & 0.461 & 0.341 & 0.574 & 0.280 & 0.320 & 0.323 & 0.455 & 0.595 & 0.266 & 0.301 & 0.402 & 0.818 \\ 1'000 & 0.106 & 0.281 & 0.301 & 0.887 & 0.880 & 0.406 & 0.248 & 0.476 & 0.198 & 0.254 & 0.237 & 0.347 & 0.566 & 0.302 & 0.220 & 0.374 & 0.645 \\ 8'000 & 0.025 & 0.070 & 0.074 & 0.935 & 0.934 & 0.411 & 0.086 & 0.419 & 0.000 & 0.111 & 0.065 & 0.129 & 0.540 & 0.156 & 0.061 & 0.167 & 0.301 \\ \midrule \multicolumn{18}{c}{$h=3$, $\theta_0^{(h)}=0.0717$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.208 & 1.625 & 1.638 & 0.915 & 0.901 & 0.594 & 0.720 & 0.933 & 0.454 & 0.453 & 0.669 & 0.808 & 0.709 & 0.072 & 0.612 & 0.617 & 0.929 \\ 250 & 0.105 & 0.913 & 0.919 & 0.913 & 0.912 & 0.462 & 0.498 & 0.679 & 0.394 & 0.339 & 0.470 & 0.579 & 0.670 & 0.128 & 0.459 & 0.476 & 0.911 \\ 500 & 0.102 & 0.470 & 0.481 & 0.940 & 0.927 & 0.388 & 0.334 & 0.512 & 0.296 & 0.269 & 0.322 & 0.420 & 0.686 & 0.171 & 0.331 & 0.372 & 0.903 \\ 1'000 & 0.102 & 0.285 & 0.303 & 0.911 & 0.908 & 0.350 & 0.243 & 0.426 & 0.191 & 0.222 & 0.236 & 0.324 & 0.658 & 0.214 & 0.237 & 0.319 & 0.827 \\ 8'000 & 0.023 & 0.082 & 0.085 & 0.934 & 0.933 & 0.363 & 0.092 & 0.375 & 0.001 & 0.097 & 0.078 & 0.124 & 0.706 & 0.104 & 0.074 & 0.128 & 0.709 \\ \midrule \multicolumn{18}{c}{$h=4$, $\theta_0^{(h)}=0.0430$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.242 & 1.594 & 1.612 & 0.919 & 0.899 & 0.504 & 0.742 & 0.897 & 0.486 & 0.389 & 0.701 & 0.802 & 0.722 & 0.045 & 0.696 & 0.697 & 0.924 \\ 250 & 0.108 & 0.967 & 0.973 & 0.905 & 0.900 & 0.399 & 0.529 & 0.662 & 0.418 & 0.300 & 0.504 & 0.587 & 0.712 & 0.090 & 0.512 & 0.520 & 0.920 \\ 500 & 0.086 & 0.552 & 0.559 & 0.938 & 0.921 & 0.319 & 0.353 & 0.476 & 0.365 & 0.222 & 0.343 & 0.409 & 0.755 & 0.104 & 0.363 & 0.378 & 0.934 \\ 1'000 & 0.090 & 0.299 & 0.313 & 0.938 & 0.935 & 0.294 & 0.251 & 0.386 & 0.251 & 0.187 & 0.244 & 0.308 & 0.751 & 0.146 & 0.255 & 0.294 & 0.911 \\ 8'000 & 0.021 & 0.094 & 0.096 & 0.938 & 0.938 & 0.318 & 0.097 & 0.333 & 0.004 & 0.085 & 0.089 & 0.123 & 0.786 & 0.071 & 0.084 & 0.110 & 0.866 \\ \midrule \multicolumn{18}{c}{$h=5$, $\theta_0^{(h)}=0.0258$} \\ & \multicolumn{5}{c}{DML} & \multicolumn{4}{c}{RA} & \multicolumn{4}{c}{DR} & \multicolumn{4}{c}{LP} \\ T & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & $C_{a}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) & Bias & std$(\hat{\theta}_h)$ & RMSE & $C_{b}$(95%) \\ \cmidrule(lr){1-1}\cmidrule(lr){2-6}\cmidrule(lr){7-10}\cmidrule(lr){11-14}\cmidrule(lr){15-18} 125 & 0.242 & 1.669 & 1.686 & 0.899 & 0.899 & 0.428 & 0.786 & 0.894 & 0.516 & 0.342 & 0.745 & 0.820 & 0.757 & 0.043 & 0.770 & 0.771 & 0.912 \\ 250 & 0.106 & 0.977 & 0.983 & 0.930 & 0.920 & 0.328 & 0.552 & 0.642 & 0.482 & 0.244 & 0.528 & 0.582 & 0.749 & 0.047 & 0.539 & 0.541 & 0.934 \\ 500 & 0.084 & 0.533 & 0.540 & 0.943 & 0.923 & 0.269 & 0.370 & 0.457 & 0.403 & 0.190 & 0.362 & 0.409 & 0.775 & 0.067 & 0.386 & 0.392 & 0.941 \\ 1'000 & 0.084 & 0.317 & 0.328 & 0.942 & 0.937 & 0.251 & 0.265 & 0.365 & 0.303 & 0.162 & 0.260 & 0.306 & 0.797 & 0.102 & 0.274 & 0.292 & 0.939 \\ 8'000 & 0.016 & 0.104 & 0.105 & 0.942 & 0.942 & 0.273 & 0.103 & 0.292 & 0.009 & 0.071 & 0.099 & 0.121 & 0.829 & 0.045 & 0.095 & 0.106 & 0.924 \\ \bottomrule\end{tabular} \begin{tablenotes}[flushleft] • \textsc{Note}: The table depicts simulation results across $N=1'000$ draws obtained for the scenario with one stochastic process. The outcome variable is generated from a linear DGP, i.e. $b(X_t) = 0.5 \sum_{i=1}^5X_{i,t}$ and $\tau(X_t) = \theta_0^{(0)} + \sum_{i=1}^3X_{i,t} - \sum_{i=4}^5X_{i,t}$. Except for the LP estimator, nuisance functions are estimated with random forest. For the DML estimator we use 10-fold cross-fitting and set $k_T=T/10$. For sample size $T=125$, probabilities are winsorized at 1%. The parameters of the data generating process are $n=12$, $\sigma_\epsilon=1.0$, $\gamma=0.6$, $p=2$, $q=1$, $\sigma_u=1.0$, $\alpha_A=0.3$, $\alpha_M=0.3$, $\rho_A=0.35$, $\rho_M=0.7$, $\beta_1=0.3$, $\beta_2=0.5$. $C_a$($\cdot$) and $C_b$($\cdot$) in the tables denote the coverage at the given confidence level using asymptotic and fixed-bandwidth critical values, respectively. \end{tablenotes} \end{threeparttable}} \end{table} \begin{figure}[htbp] \caption{Estimated cumulative effects of target rate changes on the bond yield curve} \begin{minipage}{\textwidth} \textsc{Note}: The figure shows the estimated cumulative effects of target rate changes on the bond yield curve for the time period July 1989 to December 2008. The left (right) column shows the effect of decreasing (increasing) the target rate by 25 basis points. The estimated effects on the yield curve are depicted for 0 (top row), 12 (middle row), and 24 (bottom row) months after the target rate change. The nuisance functions are estimated by random forest using 10-fold cross-fitting removing $k_T=24$ observations from the estimation sample at the boundary to the inference sample. The shaded areas represent 68% and 95% confidence intervals with fixed-bandwidth critical values Kiefer2005. The variances are estimated using bandwidth determined by the procedure of NeweyWest1994. \end{minipage} \end{figure}