Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
45,859 characters · 8 sections · 48 citation commands
Constructing an Instrument as a Function of Covariates
Instrumental variable (IV) strategies are widely used to investigate the causal effect of an endogenous variable on an outcome of interest. Since the work of imbens_identification_1994, these estimators are often interpreted as estimating a local average treatment effect (LATE) or, under homogeneity conditions, the population treatment effect of the endogenous variable of interest.
However, estimating this causal effect requires finding valid instruments that satisfy the exclusion restriction, which can be challenging in many applications. One response to this challenge is to construct an instrumental variable that is a nonlinear function of covariates.\footnote{Here we use covariate to mean “included exogenous variable," and reserve “excluded variable” to refer to excluded exogenous variables.} The leading case is to use the “product instrument," the interaction of two covariates. A sample of recent articles published in the American Economic Review that use this approach are munshi_networks_2016, bettinger_virtual_2017, aladangady_housing_2017, and rogall_mobilizing_2021. Other examples include shao_estimating_2014, bharadwaj_fertility_2015, helmers_board_2017, crawford_impact_2020, cage_social_2022, liu_housing_2023, and ghose_leveraging_2024.
In this paper, we study the causal interpretability of these IV estimators that use a nonlinear function of covariates as the instrumental variable, under misspecification. A researcher may be interested in the effect of a variable $D$ on some outcome $Y$, where $D$ may be endogenous to unobserved factors affecting $Y$. The researcher’s model specifies $Y$ as a linear function of $D$ and some covariates $X$, with the causal relationships governed by a parameter $\tilde{\theta}$. They do not have access to an excluded variable $Z$ that is correlated with the endogenous variable $D$ but not the unobserved factors affecting the relationship between $D$ and $Y$ (conditional upon $X$). Instead, a recurring practice is for the researcher to construct an instrumental variable $\tilde{f}(X) = X_{1} \cdot X_{2}$ that is the interaction of covariates, which they argue satisfy the IV assumptions. The researcher then uses IV estimators to evaluate $\tilde{\theta}$.
For example, aladangady_housing_2017 uses a linear model to examine the effect of log house price fluctuations $D$ on log consumer spending $Y$ in the United States while trying to control for potential unobserved confounding caused by preferences or expectations that change over time. They would like to use a measurement of local land availability or zoning regulations $Z$ that is uncorrelated with preference shifts but varies geographically. However, they do not have access to detailed zoning data across regions. Therefore, they use the interaction between long-term real interest rates $X_{1}$ and housing supply elasticity measures $X_{2}$ as an instrument for house price fluctuations and estimate the two-stage least squares (2SLS) model,
for $\tilde{f}(X) = X_{1} \cdot X_{2}$. By including long-term real interest rates $X_{1}$ and housing supply elasticity measures $X_{2}$ in the specification for consumer spending $Y$, they hope to control for the direct effect of interest rates and housing elasticity on spending trends that may be related to housing prices. This relaxes the exogeneity assumption required on the instrument $\tilde{f}(X)$ because even if long-term interest rates or housing elasticity are correlated with unobserved determinants of spending, as long as their interaction is uncorrelated with these unobserved determinants, the model assumptions required for IV are still satisfied.
We show that for IV estimators that use a function of covariates as an instrument, even while imposing homogeneous treatment effects, the researcher's model is uninterpretable as a causal effect under mild forms of misspecification. As the main contribution we show that for any level of bias and for any IV estimand which relies on an instrument that is a transformation of covariates $f(X, Z) = \tilde{f}(X)$, there is a continuous data generating process (DGP) with constant, linear treatment effects under which we can attain that level of bias.
Quantitatively, we investigate the sensitivity of linear IV estimates that use a function of covariates $\tilde{f}(X)$ as an instrument to a reasonable set of nonlinear DGPs. For a collection of influential papers that use linear IV with product instruments, we perform a semi-synthetic exercise by estimating the treatment effect of the endogenous variable $D$ under nonlinear structural functions proposed in the literature. For three out of four pairs of papers we examine, we find that a 95% confidence interval for the treatment effect $\hat{\theta}$ rejects the ground truth in our simulations. These findings emphasize the sensitivity of IV estimands that use product instruments to the true relationship between the outcome $Y$ and covariates $X$ in the true structural function. This paper aims to guide the causal interpretation of these estimands.
A large literature following imbens_identification_1994 and angrist_identification_1996 studies the interpretation of IV estimators under misspecification. Our work is closest to blandhol_when_2022, which investigates the LATE interpretation of IV estimators that include exogenous covariates $X$. Unlike blandhol_when_2022, we allow for continuous outcomes and continuous treatment effects. blandhol_when_2022 shows that IV estimators with instruments that do not satisfy a condition called “saturated covariates” cannot be interpreted as LATE. As we show in section (ref), the estimators we study are undefined under saturated covariates since there is no variation in the instrument $\tilde{f}(X)$. Appendix (ref) expands on the relationship between our work and blandhol_when_2022 and generalizes some of the results in the latter.
Our work is also closely related to andrews_structural_2023, which studies the causal consistency of nonlinear IV estimators under misspecification. They show that IV estimators with instruments that do not satisfy a condition known as “strong exclusion” cannot approximately describe the effects of the endogenous variable $D$. The IV specifications we study do not have access to excluded variables, and therefore do not satisfy strong exclusion. We differ from andrews_structural_2023 by showing a more significant failure of the causal interpretability of IV estimators using product instruments even when imposing homogeneous, linear treatment effects, meaning the analyst's model satisfies causally correct specification.
Researchers often justify their models using economic reasoning, which can lead them to believe their model is correctly specified. This motivates gao_iv_2023, who introduce a causal estimator that does not depend on excluded covariates but is sensitive to the functional form chosen by the analyst. If the researcher observes the true structural function $Y(D, X, \epsilon)$, or if the model proposed by the researcher $\tilde{Y}(D, X, \tilde{\epsilon}_{i})$ matches the true DGP, then the estimator from gao_iv_2023 and the linear IV specifications we study both remain unbiased. Our work focuses on cases where the researcher does not know the true structural function, so their model may be mildly misspecified.
The remainder of the paper proceeds as follows. Section (ref) introduces the assumptions of our model, which includes the IV conditions assumed by the analyst and the true structural function. Section (ref) establishes our main result and shows that IV estimators that rely on instruments constructed from covariates can be arbitrarily biased. Finally, Section (ref) shows our quantitative results and illustrates the wide range of estimates produced by IV estimators that rely on product instruments across plausible DGPs. We reserve our theoretical proofs for the appendix, with the main text focusing on the key results.
A analyst observes variables $(Y_{i}, D_{i}, X_{i}, Z_{i})$ for units $i \in \{1, \dots, n\}$. All variables are finite-dimensional, and $Y_{i} \in \mathbb{R}$. To illustrate the possibility of misspecification, we introduce a true model that is consistent with the data generating process (DGP) and summarizes the causal relationships between the observed variables, and the analyst's model which may rule out the true DGP.
Under the true model, which is consistent with the true data-generating process, the observed outcome satisfies $Y_{i} = Y(D_{i}, X_{i}, \epsilon_{i})$ for a potential outcome function $d \mapsto Y(d, X_{i}, \epsilon_{i}) \in \mathcal{Y} \subseteq \mathbb{R}$, covariates $X_{i} = (X_{i, 1}, \dots, X_{i, J})^{\top} \in \mathcal{X} \subseteq \mathbb{R}^{\text{dim}(X) \times 1}$, and unobserved heterogeneity $\epsilon_{i}$. The observed endogenous variable $D_{i}$ satisfies $D_{i} = D(X_{i}, Z_{i}, \nu_{i})$ for $D(X_{i}, Z_{i}, \nu_{i}) \in \mathcal{D} \subseteq \mathbb{R}$, a potential endogenous variable function, unobserved heterogeneity $\nu_{i}$, and an excluded exogenous variable $Z_{i} \in \mathcal{Z} \subseteq \mathbb{R}^{\text{dim}(Z) \times 1}$. We impose one additional assumption on the true model in line with instrumental variables (IV) models.
Under Assumption (ref), we can write $Y_{i} = Y(D_{i}, X_{i}, \epsilon_{i})$.
Definition (ref) establishes the linear IV estimand to estimate the causal effect of the treatment $D$ on the outcome $Y(D_{i}, X_{i}, \epsilon_{i})$ under a linear model. This is the estimand computed by linear method of moments and two-stage least squares estimators for an instrumental variable $f(X, Z)$ when there is a single endogenous treatment $D$.
The analyst’s model is a special case of the true model. Under the analyst’s model, the causal effects of the endogenous variable and included variables on the outcome are governed by a parameter $\tilde{\theta} \in \mathbb{R}$. Here, we focus on the case where the analyst proposes and estimates a linear model with homogeneous treatment effects,
The analyst would like to estimate the causal effect of $D$ on the outcome $Y$ in Equation (ref), but is unwilling to assume unconfoundedness ($D \perp \!\!\! \perp \tilde{\epsilon} \mid X$). Therefore, to use IV estimation, the analyst constructs an instrument $f(X, Z)$ under two assumptions sufficient to estimate $\tilde{\theta}$.
Assumption (ref) says that the instrument $f(X, Z)$ and included exogenous covariates $X$ must be uncorrelated ($\perp$) with the unobserved heterogeneity $\tilde{\epsilon}$ in the outcome under their proposed model.
Assumption (ref) is necessary for $\theta_{IV}$ in Definition (ref) to be defined, so the denominator is not equal to zero. Otherwise, the treatment effect $\tilde{\theta}$ is unidentified using linear IV estimation.
In this paper, we focus on the case where the analyst does not have access to an excluded exogenous variable $Z$ to create an instrumental variable $f(X, Z)$ that is a nontrivial function of $Z$. Therefore, the analyst chooses $f(X, Z)$ to be a nonlinear function of only covariates $f(X, Z) = \tilde{f}(X) \in \mathbb{R}$ to use IV estimation.
Lemma (ref) makes it clear why the analyst's instrumental variable $\tilde{f}(X)$ must be nonlinear in $X$ to use the IV estimator from Definition (ref). If $\tilde{f}(X)$ is linear in each element of $X$, then the linear IV estimand $\theta_{IV}$ becomes unidentified because the denominator $\mathrm{Cov}\mathopen{}\mathclose\bgroup\originalleft(D,f_{\perp}\aftergroup\egroup\originalright) = \mathrm{Cov}\mathopen{}\mathclose\bgroup\originalleft(D, 0\aftergroup\egroup\originalright) = 0$.
Lemma (ref) demonstrates that an instrument based solely on included exogenous covariates automatically satisfies Assumption (ref) (exogeneity) for IV estimation when the analyst assumes a stronger exogeneity condition for these covariates. Some researchers using instruments of the form \(f(X, Z) = \tilde{f}(X)\) argue that such instruments “require weaker identifying assumptions than using [\(X_1\) or \(X_2\)] alone” bettinger_virtual_2017. By including \(X_i\) in the second-stage regression, they “control for the direct effect of [\(X_i\)] that may be related to [\(Y_i\)]” aladangady_housing_2017. If the covariates \(X\) included in the second-stage equation \(\tilde{Y}(D_i, X_i, \tilde{\epsilon}_i)\) satisfy conditional exogeneity (\(\mathbb{E}[\tilde{\epsilon} \mid X] = 0\)), then a nonlinear function of \(X\) can be used as an instrument without needing to independently verify its exogeneity under Assumption (ref).
The following proposition shows that this estimand corresponds to the causal effect $\tilde{\theta}$ of $D_{i}$ on $Y(D_{i}, X_{i}, \epsilon_{i})$ under the analyst's model if the true data generating process follows the model assumed by the analyst.
Claim (ref) shows that if the analyst's model is correctly specified and their constructed instrument $f(X, Z) = \tilde{f}(X)$ satisfies Assumption (ref) (relevance), IV estimators $\theta_{IV}$\footnote{Note that $\theta_{IV}$ is implicitly indexed by $\tilde{f}(X)$} are consistent for the treatment effect. gao_iv_2023 show that even for nonlinear models, if the analyst's model matches the true DGP, the causal effect $\theta$ of $D$ on $Y$ can be recovered locally. We next move to analyze what happens when the analyst's model is not correctly specified.
When the analyst estimates a linear model like Equation (ref), any nonlinear effect of the covariates $X$ on the outcome $Y$ is captured in the unobserved heterogeneity $\tilde{\epsilon}$. This means that IV estimators that rely on moment conditions interacting $\tilde{\epsilon}$ and the covariates $X$ will be sensitive to nonlinearity in the true structural function $Y(D_{i}, X_{i}, \epsilon_{i})$.
In Theorem (ref), we illustrate this sensitivity using a toy case where the true DGP imposes linear, homogeneous treatment effects. We show that if the analyst uses an instrumental variable $f(X, Z) = \tilde{f}(X)$ that is only a function of covariates, the bias of the IV estimand $\theta_{IV}$ can be arbitrarily bad.
The proof of Theorem (ref) is constructive -- we propose an adversarial $h(\cdot)$ which has a nonlinear term proportional to $\rho$. As we raise $\rho$, increasing the nonlinearity of the function $h(\cdot)$, the bias of the IV estimand $\theta_{IV}$ for the true treatment effect $\theta$ increases. This demonstrates the close relationship between the nonlinearity of the true structural function $Y(D_{i}, X_{i}, \epsilon_{i})$ in $X_{i}$ and the bias of the linear IV estimand.
For the product instrument $\tilde{f}(X) = X_{1} \cdot X_{2}$ used in the papers cited in the introduction, Corollary (ref) characterizes the bias of the IV estimand for a single linear DGP. It shows that the bias is proportional to the coefficient on the interaction term $X_{1} \cdot X_{2}$. As the structural function includes increasingly nonlinear interactions, the bias in the IV estimand $\theta_{IV}$ grows accordingly. This result highlights that sizable violations of the exclusion restriction—leading to substantial bias—can arise when the interaction of excluded covariates strongly affects the outcome. For instance, if $X_{1}$ is a binary variable such as gender, then a large coefficient on $X_{1} \cdot X_{2}$ in the structural function $Y(D_i, X_i, \epsilon_i)$ suggests that the relationship between $X_{2}$ and the outcome differs markedly between men and women. To visualize this connection, Figure (ref) plots IV estimates $\hat{\beta}_{IV}$ against data generated from the model in Corollary (ref), illustrating how this misspecification undermines the interpretability of the IV estimand.
Examining panel (a) of Figure (ref) illustrates the sensitivity of these IV estimates to misspecification in the true relationship between $X$ and $Y$. As Corollary (ref) implies, we see that by increasing the coefficient on the interaction term $X_{1} \cdot X_{2}$ in the true structural equation $Y(D_{i}, X_{i}, \epsilon_{i})$, we can inflate the bias of the IV estimate $\hat{\theta}$ while holding the true treatment effect $\theta$ constant. In contrast, the consistency of the IV estimates in panel (b) underscores the robustness of the IV estimator against misspecification in the true relationship between $X$ and $Y$ for partially linear DGPs when using a product instrument.
Remark (ref) gives an alternative interpretation of the bias for IV estimators. Remark (ref) underlines the generality of Theorem (ref): for IV estimands that rely on instruments of covariates $\tilde{f}(X)$, if the analyst's model does not capture how the covariates $X$ enter the true structural function $Y(D_{i}, X_{i}, \epsilon_{i})$, then the IV estimand can be very biased.
We have shown that the practice of using instruments constructed from nonlinear functions of covariates $f(X, Z) = \tilde{f}(X)$ can misbehave in interesting ways theoretically. In Section (ref), we demonstrate that this behavior is important in empirical work.
Researchers often have strong economic reasoning to justify their proposed model and may believe that their model matches the true DGP. In these cases, Claim (ref) (IV validity) shows that the IV estimand $\theta_{IV}$ gives the average causal effect of $D$ on $Y(D_{i}, X_{i}, \epsilon_{i})$.
However, it is also often the case in the literature that non-nested models are used to estimate the same treatment effect. In these instances, every model cannot match the true DGP, and therefore some of these models must be misspecified.
For example, recall from Section (ref) that aladangady_housing_2017 uses IV estimation to investigate the effect of log changes in house price $D$ on log consumer expenditure $Y$ in the United States using a model of the form, $$Y_{i} = \tilde{Y}(D_{i}, X_{i}, \tilde{\epsilon}_{i}) = \tilde{\alpha}_{0} + \tilde{\alpha}_{1} D_{i} + \tilde{\pi}^{\top} X_{i} + \tilde{\epsilon}_{i},$$ for $X\in \mathbb{R}^{2}$. Real interest rates $X_{1}$ and housing supply elasticity measures $X_{2}$ as a function of future expectations $X_{3}$ are included as covariates.
Similarly, disney_house_2010 estimates the effect of log house price growth $D$ on log consumption $Y$ in the United Kingdom, but includes interest rates $X_{1}$ and future expectations $X_{3}$ in their life-cycle model for expenditure. They use ordinary least-squares to estimate the model, $$Y_{i} = \tilde{Y}(D_{i}, X_{i}, \tilde{\epsilon}_{i}) = \tilde{\alpha}_{0} + \tilde{\alpha}_{1} D_{i} + \tilde{\pi}^{\top} \log(X_{i}) + \tilde{\epsilon}_{i},$$ for $X \in \mathbb{R}^{7}$,\footnote{The authors also include gross household income, passive savings, estimated unanticipated house growth, and household characteristics as covariates.} where we write $\log(X)$ to be $\log(X_{i}) = (\log(X_{1}), \log(X_{3}), \dots)$.
Both authors justify their models with economic intuition. Still, they do not agree on the functional form of consumer expenditure, or the way the covariates $X$ enter the model $\tilde{Y}(D_{i}, X_{i}, \epsilon_{i})$. Theorem (ref) (model bias) shows that if the author's model does not match the true structural function, IV estimators that use an instrument constructed from covariates can have a high level of bias. Therefore, it is important to understand the sensitivity of these estimates to how the covariates $X$ affect the outcome $Y$ in cases when there are several reasonable choices to model the DGP.
Table (ref) gives additional examples of competing models used in the literature. For each of these examples, we compare: on one hand, (i) a linear IV model that relies on the product instrument $\tilde{f}(X) = X_{1} \cdot X_{2}$; and on the other hand (ii) a richer nonlinear model proposed in that literature to estimate the same treatment effect. For each pair of papers, the linear IV model is written on the top row and the non-nested alternative model is given on the bottom row.
Table (ref) shows that cases often arise when there is no consensus about the functional form of the true structural function $Y(D_{i}, X_{i}, \epsilon_{i})$. Therefore, it is likely that the true DGP is nonlinear in the covariates $X$.
To assess the sensitivity of linear IV models that use product instruments, Table (ref) performs a semi-synthetic exercise for each pair of papers in Table (ref). Specifically, for each pair of papers, we interpret the richer, nonlinear specification as the true structural equation and calibrate the DGP to match that model. We restrict one degree of freedom for the parameters $(\alpha^{\intercal}, \pi^{\intercal})$ by forcing $\theta = \mathbb{E}\mathopen{}\mathclose\bgroup\originalleft\{\pdv{}{D_{i}} Y(D_{i}, X_{i}, \epsilon_{i})\aftergroup\egroup\originalright\} = 1$. We then use 2SLS to estimate $\hat{\theta}$ in the linear model IV model from Equation (ref).
For example, using the structural model offered by disney_house_2010 we construct synthetic draws of $(Y_{i}, D_{i}, X_{i})$ such that $\theta = \alpha_{1} = 1$. Then, we fit the linear IV model used by aladangady_housing_2017 and compare how the IV estimate $\hat{\theta}$ compares to $\theta$ in the synthetic structural model. Full details are given in Appendix (ref).
Table (ref) shows estimates of $\hat{\theta}$ from Equation (ref) for the estimand $\theta_{IV}$ in Definition (ref) using linear IV estimation.
In Table (ref), we observe that for all four pairs of papers, a 95% confidence interval for the estimated treatment effect $\hat{\theta}$ does not include the ground truth of $\theta = 1$, i.e. the average partial treatment effect (APE) in the true structural function $Y(D_{i}, X_{i}, \epsilon_{i})$. For instance, IV estimates tend be negative when using the models from aladangady_housing_2017, where the DGP is linear with log-transformed covariates $X$. Meanwhile, for the models used by helmers_board_2017 and liu_housing_2023, where the true structural function follows an exponential and logit design, IV estimates based on instruments that are functions of covariates tend to be close to zero, underestimating the APE. This illustrates how the bias of the IV estimand can be very positive or negative when using product instruments, depending on the underlying DGP.
These findings underscore the sensitivity of these IV estimates $\hat{\theta}$ to the functional form chosen to model the relationship between covariates $X$ and the outcome $Y$ and illustrate their bias under sensible choices for the DGP. Even for reasonable choices of $Y(D_{i}, X_{i}, \epsilon_{i})$, IV estimates of the treatment effect $\hat{\theta}$ with constructed instruments can differ substantially from the true APE and imply different causal conclusions.
Another natural question is how sensitive causal estimates are to the analyst’s modeling choices when applied to real economic data. To examine this, Table (ref) explores how estimates of $\theta$ vary when fitting different functional forms to data from published empirical studies. Specifically, we consider the subset of papers published in the American Economic Review (AER) since 2014 that employ instrumental variable (IV) strategies relying on product instruments of the form $\tilde{f}(X) = X_{1} \cdot X_{2}$.\footnote{We only use papers for which the data is publicly available.} For each of these studies,\footnote{Paper (1) is rogall_mobilizing_2021 and paper (2) is munshi_networks_2016.} we re-estimate the parameter $\tilde{\theta}$ using the linear IV model,
for various nonlinear functions $h(\cdot)$ and investigate the range of estimates produced.
Within each paper, we find that the estimated causal effect $\tilde{\theta}$ is sensitive to the choice of functional form for $h(X_{i})$. In paper (1), for instance, a 95% confidence interval based on the reported linear specification does not include the estimate obtained using a log-transformed specification, where $h(X_{i}) = (\log(1 + X_{i,1}), \dots, \log(1 + X_{i,q}))^{\top}$. Similarly, in paper (2a), none of the estimates from nonlinear specifications lie within the 95% confidence interval derived from the original linear model.
These results demonstrate that the analyst’s estimate of $\tilde{\theta}$ can vary substantially depending on the functional form used to model the relationship between covariates $X$ and the outcome $Y$. Even among plausible specifications of $h(X_{i})$, the magnitude and statistical significance of the estimated treatment effect can differ markedly.
Together, this evidence highlights the lack of robustness of IV estimators based on product instruments to nonlinearities in how covariates $X$ enter the structural function $Y(D_{i}, X_{i}, \eta_{i})$. Substantive conclusions drawn from such models may be sensitive to modeling assumptions.
Overall, our results underscore the importance of functional form choices in IV models where instruments are functions of covariates, $f(X, Z) = \tilde{f}(X)$. The selected functional form can significantly influence how the treatment effect is interpreted.
In this paper, we have provided theoretical and empirical evidence showing that product instruments, or more generally, instruments that are transformations of covariates, can be very biased. Our framework shows that even when imposing homogeneous linear effects between an endogenous variable and outcome, these IV estimates cannot be interpreted as a causal effect under mild forms of misspecification. Theoretically, we show that there exists a simple DGP with constant linear effects that can lead to unbounded bias of the IV estimand. Empirically, we illustrate the sensitivity of IV estimates that use instruments constructed from nonlinear functions of covariates.
When a researcher does not have access to an excluded, exogenous variable, we recommend caution when discussing the causal interpretability of IV estimators. Alternatively, if the researcher believes their instrumental variable $f(X, Z) = \tilde{f}(X)$ satisfies the IV assumptions in Section (ref), it is important that they strongly defend their choice of model $\tilde{Y}(\cdot)$, i.e. the relationship between the outcome $Y$ and covariates $X$, to defend the validity of their estimate.