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On the Role of the Zero Conditional Mean Assumption for Causal Inference in Linear Models
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The zero conditional mean assumption imposing that the regression error is mean independent of the covariates and has mean zero is a building block of linear regression models. Under this assumption, many econometrics textbooks give the model coefficients a causal interpretation angrist2017undergraduate. For example, the econometrics textbook most used in economics, wooldridge2019introductory, discusses the example of yield as outcome and fertilizer as regressor as follows “if fertilizer amounts are chosen independently of other features of the plots, then [zero conditional mean error] will hold: the average land quality will not depend on the amount of fertilizer. However, if more fertilizer is put on the higher-quality plots of land, then the expected value of $u$ [the error] changes with the level of fertilizer, and [zero conditional mean error] fails.”.\footnote{According to \href{https://opensyllabus.org/result/field?id=Economics}{Opensyllabus.org} accessed on March 18th 2022.} As we will show, the first sentence in this quote is true, but the second not necessarily. Notice that a similar approach is used in other textbooks, e.g., cameron2005microeconometrics, hayashi2000econometrics, and stock2019introduction.
\textcolor{black}{Although the literature has recognized the advantages of the potential outcome framework over the traditional linear modeling approach IW09, we clarify the source of ambiguity that may stem from using a non causal model. In particular,} this note highlights that there are cases where the zero conditional mean assumption holds, but the parameter estimated via OLS suffers from omitted variable bias and has no causal interpretation. To illustrate this, note that any regression with a fully saturated model would always be causal if the zero conditional mean assumption would be sufficient for the causal interpretation of the parameters. Consider the following two examples:
In our opinion, the ambiguity occurs because the difference between unobserved variables in a causal model and the statistical error in the statistical model is often not made explicit. Many books use the same error term to discuss both statistical properties of OLS as an estimator and omitted variable bias. This might induce in the reader the false idea that zero conditional mean error implies no confounding. However, the reverse is true, that is, no confounding implies zero conditional mean error. While the lessons learned in both estimation and identification are still valid, we think that the conceptual ambiguity is worth clarifying and to be kept in mind when teaching econometrics.
The goal of this note is to exactly show where and why the ambiguity arises and to propose alternative and less ambiguous ways of presenting the assumptions needed for a causal interpretation of the linear model coefficients. We also show how linear instrumental variable (IV) and panel data models are affected by the same issue. In the Online Appendix we also report several numerical examples to further illustrate our results.
Consider the simple regression
and assume zero conditional mean of the error term, i.e., $E[\varepsilon_i|D_i]=0$. From a purely statistical point of view this is equivalent to assuming that the conditional expectation of $Y$ given $D$ is linear, i.e. $E[Y_i|D_i]=\gamma +\lambda D_i$. We now show that this is not sufficient for a causal interpretation of $\lambda$, which can be arbitrarily different from the causal effect of $D$ on $Y$. To this end, let $Y_i(d)$ be the potential outcome of individual $i$ if $D_i=d$ and assume that $$ Y_i(d)=\alpha+\tau d+ \beta U_i. $$ This implies that the causal effect of a one-unit increase in $D$ on $Y$ is constant and equal to $\tau$, i.e. $Y_i(d+1) - Y_i(d)=\tau$.
For example, $Y_i(d)$ could represent the wage individual $i$ would receive if she had completed $d$ years of education and $U_i$ is unobserved ability. Assume that $E[U_i|D_i]=\mu+\delta D_i$ such that we can write $U_i=E[U_i|D_i]+\nu_i=\mu+\delta D_i+\nu_i$ with $E[\nu_i|D_i]=0$. Notice that this assumption is always satisfied if $D$ is binary or if $U$ and $D$ are drawn from a bivariate normal distribution. Under the stable unit treatment value assumption (SUTVA), i.e. $Y_i=Y_i(d)$ if $D_i=d$, we have:
Therefore, the conditional expectation of the observed outcome given $D$ is linear, $E[Y_i|D_i]=\gamma +\lambda D_i$, and the zero conditional mean assumption, $E[\varepsilon_i|D_i]=\beta E[\nu_i|D_i]=0$, is satisfied. However, although $\lambda=\tau+\beta \delta$ represents how a change in $D$ changes $E[Y|D]$, it can be arbitrarily different from the causal effect of $D$ on $Y$, which is given by $\tau$. Notice that $\beta \delta$ is the usual omitted variable bias that is well known and discussed in any econometric textbook. Most books imply that if $D$ is uncorrelated with $\varepsilon$ there is no omitted variable bias. This is contradicted by our derivation showing that $E[\varepsilon_i|D_i]=0$, and as a consequence $Cov[\varepsilon_i,D_i]=0$, can hold in the presence of omitted variable bias.
We conjecture that this tension is due to the notion that any omitted variable would be included in the error term and that this would violate the zero conditional mean assumption (see the quote in the introduction). However, it is often not clear which error term is referred to. There is the statistical error term $\varepsilon$ and the omitted variable in the causal model $U$, but they are not necessarily the same, i.e. $$ \varepsilon_i\equiv Y_i-E[Y_i|D_i] = \beta \nu_i \neq \beta U_i $$
in our example. This ambiguity disappears only in the absence of the omitted variable $U$, i.e., the statistical error is the only unobserved part of the potential outcome. In that case, assuming zero conditional mean error suffices for causal interpretation.
Notice that $E[\varepsilon_i|D_i]=0$ does not imply that the conditional mean independence assumption on the potential outcome holds, i.e $E[Y(d)|D_i=d]=E[Y(d)|D_i=d'], d\neq d'$. Indeed, $E[Y(d)|D_i=d]-E[Y(d)|D_i=d']=\beta \delta(d-d')$. Therefore this assumption on the potential outcomes is only satisfied if there is no omitted variable bias, so that $\beta \delta =0$. However, it would be satisfied if we put a zero conditional mean assumption on the unobservable part of the potential outcome, i.e. $E[\beta U_i|D_i] = 0$. This shows that assuming zero conditional mean of the unobserved part in the potential outcome is sufficient for a causal interpretation.
To avoid such confusion, it is important to emphasize that exogeneity cannot be used in a mere statistical model but needs to refer to a causal model instead. In Section (ref) we propose some alternative formulation of the assumption needed that do not necessarily require introducing the potential outcome framework.
Adding control variables does not change our results as we show in the \hyperref[app:cov]{Appendix} as well as in the simulations we run in the Online Appendix.
The consequences for linear IV models of our results are twofold. First, the way standard IV is presented in most books and scientific articles is as a solution to the so called “endogeneity” issue, i.e., correlation between (some of) the regressor(s) - $D$ in our example of equation ((ref)), and the regression error, $\varepsilon$ in equation ((ref)). As our example demonstrates, even if $D$ is “exogenous”, i.e., uncorrelated with $\varepsilon$, we cannot exclude the presence of omitted variables that make $\lambda$ a pseudo-parameter that does not have a causal interpretation. Second, consider again the regression model
even if $D$ is endogenous i.e., $Cov[D,\varepsilon]\neq 0$, estimating $\lambda$ by 2SLS using an instrument $Z$ that is uncorrelated with $\varepsilon$ does not guarantee a causal interpretation. To see this, assume that $Y_i(d)=\alpha+\tau d+ \beta U_i$, $D_i=\pi Z_i + \eta_i$, and $U_i=\mu+\delta Z_i + \xi_i$ with $E[\eta_i|Z_i]=E[\xi_i|Z_i]=0$ such that $E[D_i|Z_i]=\pi Z_i$, and $E[U_i|Z_i]=\mu+\delta Z_i$. Following a similar derivation as in Section (ref), we can write the observed outcome as
This implies that
Therefore, $E[\varepsilon_i|Z_i]=0$ is satisfied, but $\lambda$, the coefficient of a 2SLS regression of $Y$ on $D$ using $Z$ as an instrument, does not have a causal interpretation unless $\beta \delta =0$. hahn2011conditional provide an example of a data generating process (DGP) involving a non-linear (and non-separable) outcome equation, whereby 2SLS identifies a pseudo-parameter that has no causal interpretation. Our derivation shows that the DGP does not need to feature a non-linearity or non-additivity for the 2SLS estimand to lack a causal interpretation despite $E[\varepsilon_i|Z_i]=0$. See the Online Appendix for a numerical illustration of these results.
In panel data models it is often implied that in the regression
assuming $E[\varepsilon_{it}| D_{it},\gamma_i]=0$ is sufficient to give $\lambda$ a causal interpretation.
Without loss of generality, assume that we only have two periods ($t=1,2$) and that $$ Y_{it}(d)=\alpha_i+\tau d+ \beta U_{it}. $$ Letting $\Delta$ be the first difference operator such that $\Delta W_i= W_{i2}-W_{i1}$, we have that first difference in the potential outcome for a given value of $\Delta D_i$, $d'$, is given by $$ \Delta Y_i(d')=\tau d'+ \beta \Delta U_i. $$ Assume that $E[\Delta U_i|\Delta D_i]=\mu+\delta \Delta D_i$. Thus, following a similar derivation as in Section (ref), we have that
This shows that $E[\varepsilon_{it}| D_{it},\gamma_i]=0$ is satisfied but $\lambda$ does not have a causal interpretation unless $\beta \delta =0$.
Many econometrics textbooks, especially at the undergraduate level directly define $\varepsilon_i\equiv Y_i-E[Y_i|D_i]$. Other books like wooldridge2010econometric define the regression error as the unobserved part of the outcome, $\beta U_i$ in our example, but implicitly rely on the assumption that $\varepsilon_i=\beta U_i$, which we have proven not to be always satisfied. This shows that it is not enough to assume that the linear regression model is structural to solve the potential ambiguity. Consider again the potential outcome of Section (ref): $Y_i(d)=\alpha+\tau d+ \beta U_i$. As we showed, the observed outcome can be written in two alternative ways: either as $Y_i=\alpha+\tau D_i+ \beta U_i$, or as $Y_i=\gamma+\lambda D_i+\varepsilon_i$. Both equations represent the same structural model. Thus, when imposing a conditional zero mean assumption one needs to be very careful in explaining what is meant by the unobservable part. In fact, both $\beta U_i$ and $\varepsilon_i$ are possible structural errors depending on how we write the model.
Therefore, if one wants to avoid introducing a causal framework, such as the potential outcomes, it has to be clear that the structural error is meant to include all unobserved variables including omitted variables that might affect both $D$ and $Y$. Our derivation also shows the importance of invoking SUTVA. If this assumption fails, even if we carefully distinguish between statistical and structural errors, we would not be able to interpret the model coefficients as causal. We suggest introducing this assumption early on in the definition of the problem.
Another possibility is to make clear that a marginal effect is not necessarily causal and that for causal inference one needs to have a causal model, and move the discussion around the identification of causal effects to where causal models are actually introduced.
We have shown that having exogenous (mean independent) regressors in a linear regression model it is not sufficient to avoid omitted variable bias. Although OLS will consistently estimate the true regression coefficient, the latter does not in general have a causal interpretation. This has strong implications for the way we teach econometrics and extends also to linear IV and panel data models, where it is even more common to rely on a lack of correlation or mean independence between the instrument/regressors and the error term of the observed outcome equation, to make causal claims. Ultimately, our results demonstrate that causal inference requires causal rather than merely stochastic assumptions.
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