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Logit-Based Alternatives to Two-Stage Least Squares
\address[D. Chetverikov]{Department of Economics, UCLA, Bunche Hall, 8283, 315 Portola Plaza, Los Angeles, CA 90095, USA.} \email{[email removed]} \address[J. Hahn]{Department of Economics, UCLA, Bunche Hall, 8283, 315 Portola Plaza, Los Angeles, CA 90095, USA.} \email{[email removed]} \address[Z. Liao]{Department of Economics, UCLA, Bunche Hall, 8283, 315 Portola Plaza, Los Angeles, CA 90095, USA.} \email{[email removed]} \address[S. Sheng]{Shenzhen Finance Institute, School of Management and Economics, The Chinese University of Hong Kong, Shenzhen, 2001 Longxiang Boulevard, Shenzhen, Guangdong, 518172, China.} \email{[email removed]}
We study the problem of instrumental variable estimation in the setting with a binary treatment and a binary instrument in the presence of controls. Numerous parametric and nonparametric instrumental variable estimators have been proposed in the literature for this setting, and among all of them, perhaps the most important one is the 2SLS estimator. It is simple to compute and has straightforward motivation in the case of constant treatment effects. However, it has been recently demonstrated by BBMT22 that in the case of heterogeneous treatment effects, the 2SLS estimator has multiple issues unless saturated controls\footnote{ When controls are discrete, the vector of controls is said to be saturated if it consists of dummy variables such that for all its realizations, one and only one dummy takes value one.} are being used, which is rarely the case in practice. In this paper, we propose a new instrumental variable estimator that alleviates some of the problems of the 2SLS estimator and is as simple to compute as the 2SLS estimator itself.
Like the 2SLS estimator, our estimator consists of two steps. In the first step, we run a logit regression of the instrument on the set of controls. In the second step, we use a classic instrumental variable estimator for the linear regression of the outcome on the treatment using residuals from the logit regression calculated on the first step as an instrument. We refer to this procedure as the logit-based instrumental variable (IV) estimator .
Under the standard monotonicity (no defiers) and conditional independence conditions, the probability limits of both 2SLS and logit-based IV estimators consist of the sum of complier, always-taker, never-taker, and non-causal terms. Without further conditions, complier, always-taker, and never-taker terms take the form of weighted-average treatment effects but the advantage of the logit-based IV estimator is that the corresponding weights in the complier term are always non-negative, which is not necessarily the case for the 2SLS estimator. This advantage is particularly important under additional conditions guaranteeing that the always-taker, never-taker, and non-causal terms vanish. Under these conditions, the probability limit of the logit-based IV estimator is represented by a convex combination of treatment effects for compliers and the probability limit of the 2SLS estimator is not. Thus, under these conditions, the logit-based IV estimator has a causal interpretation and the 2SLS estimator does not.
In addition, we develop an augmented logit-based IV estimator that has a causal interpretation under conditions that are more plausible than those underlying causal interpretability of the logit-based IV and 2SLS estimators. This estimator is similar to the logit-based IV estimator itself but contains an extra term in the logit regression used in the first step of the logit-based IV estimator. This term in turn originates from the binary regression model of the treatment variable on controls using a subsample of the data corresponding to an ex ante fixed value of the instrument.
Moreover, we construct a Hausman specification test that can be used to check some of the conditions underlying causal interpretability of the logit-based IV estimator. The test is based on the comparison of the logit-based IV and augmented logit-based IV estimators and is easy to perform.
We demonstrate the usefulness of our estimators by applying them to three well-established empirical studies: AE98, ABBKK02, and DH20. We find that our logit-based IV and augmented logit-based IV estimates are nearly identical to the 2SLS estimates reported in AE98 and ABBKK02, but differ substantially from the 2SLS estimates in DH20. Our results indicate that one of the reasons for this discrepancy in the latter case is that the 2SLS estimator may be assigning negative weights to a significant fraction of compliers. Our estimators avoid this issue and are therefore preferable in settings like that in DH20.
Our paper contributes to the large literature discussing causal interpretability of various parametric IV estimators in the case of heterogeneous treatment effects. We therefore provide here only a few key references that are particularly relevant for our work. The literature has been started by IA94, who gave the local average treatment effect interpretation of the instrumental variable estimator in the case of a binary treatment and a binary instrument without controls allowing for heterogeneous treatment effects under the monotonicity (no defiers) assumption. AI95 showed that in a model with saturated controls, the 2SLS estimator that includes all interactions between the instrument and controls in the first step converges in probability to a weighted average of control-specific local average treatment effects. A03, K13, and S20 obtained a similar result for the same and other parametric instrumental variable estimators without saturated controls assuming that the conditional mean function of the instrument given controls is linear. BBMT22 demonstrated that parametric instrumental variable estimators generally lack a causal interpretation if this conditional mean function is not linear.
The rest of the paper is organized as follows. In the next section, we introduce the logit-based IV estimator and discuss its causal interpretation. In Section (ref), we discuss the augmented logit-based IV estimator and compare its causal interpretability with that of the logit-based IV estimator itself. In Section (ref), we derive asymptotic normality results for both estimators. In Section (ref), we develop a Hausman test that can be used to check causal interpretability of the logit-based IV estimator. In Section (ref), we present our empirical applications. In the Appendix, we provide all the proofs.
In this section, we propose a logit-based IV estimator and explain its advantages over the 2SLS estimator in the potential outcome model with a binary treatment and a binary instrument in the presence of controls. In particular, we derive a set of conditions under which our logit-based IV estimator has a causal interpretation and the 2SLS estimator does not.
Consider the potential outcome model with a binary treatment $T\in \{0,1\}$ and a binary instrument $Z\in \{0,1\}$:
where $Y(0),Y(1)\in \mathbb{R}$ are potential outcome values and $ T(0),T(1)\in \{0,1\}$ are potential treatment values. In addition, let $X\in \mathcal{X }\subset \mathbb{R}^p$ be a vector of controls, including the constant one. We will assume that the instrument $Z$ is independent of potential values $ (Y(0),Y(1),T(0),T(1))$ conditional on $X$, which is a standard assumption in the program evaluation literature, e.g. see Chapter 4.5.2 in AP09:
In this model, the group variable $G = (T(0),T(1))$ can take four values, $ (0,0)$, $(0,1)$, $(1,0)$, and $(1,1)$, that are typically thought to correspond to sub-populations of never-takers (NT), compliers (CP), defiers (DF), and always-takers (AT), respectively, and it is customary to use letter-based values instead of digit-based values. We will follow this tradition and will write, for example, $G=CP$ instead of $G=(0,1)$.
For each sub-population $g\in \mathcal{G }= \{NT, CP, AT, AT\}$ and each $ x\in \mathcal{X}$, define the $x$-conditional average treatment effect
For any estimator $\widehat \beta$, we will say that it has a {\em causal} interpretation if its probability limit takes the form of a weighted-average treatment effect
where the weights $w_g(x)$ are non-negative for all $(g,x)\in \mathcal{G} \times \mathcal{X}$ and integrate to one: $\sum_{g\in \mathcal{G}}\mathbb{E} [w_g(X)] = 1$. In other words, the estimator has a causal interpretation if its probability limit can be represented as a convex combination of treatment effects. We will say that an estimator has a {\em partially causal} interpretation if takes the form of a weighted-average treatment effect (ref) with non-negative weights that do not necessarily integrate to one.
In addition, we will impose the monotonicity condition as in IA94:
This assumption excludes the sub-population of defiers. In the model without controls $X$, IA94 used this assumption to identify the LATE, the local average treatment effect for compliers:
which is often a quantity of interest. F07 extended this result and showed that the LATE is identified in the model with controls as well, as long as we impose Assumption (ref) in addition to Assumption (ref). F07, as well as following papers, e.g. BCFH17, developed nonparametric and machine learning estimators of the LATE in the model with controls.
In practice, however, empirical researchers often prefer simple parametric alternatives, such as the 2SLS estimator. As argued in BBMT22, this could be problematic. Indeed, not only may the 2SLS estimator not converge in probability to the LATE, it may not have a causal interpretation at all. In particular, it may not take the form of a weighted-average treatment effect and even if it does, the weights may take negative values and may not integrate to one.
To cope with some of these problems, we propose a simple alternative to the 2SLS estimator. Let $(Y_1,T_1,X_1,Z_1),\dots,(Y_n,T_n,X_n,Z_n)$ be a random sample from the distribution of $(Y,T,X,Z)$. Also, let
be the logit function. Our estimator, which we refer to as the logit-based IV estimator, takes the following form.
Note that this estimator differs from the 2SLS estimator by using the logit regression residuals $Z_i - \Lambda(X_i^{\top}\widehat \theta)$ in the second step instead of the linear regression residuals used by the 2SLS estimator. For clarity of presentation, and since this will be helpful for our discussion below, we provide the formal algorithm for the 2SLS estimator as well.
Algorithm (ref) may not be the usual way to define the 2SLS estimator but it is easy to verify that it does define the 2SLS estimator by applying the Frisch-Waugh-Lovell theorem.\footnote{Notably, our logit-based IV estimator is not related to the \textquotedblleft forbidden regression\textquotedblright \ discussed in Chapter 4.6.1 of AP09, which refers to the use of the fitted value $\hat{T}_{i}$\ in the second stage OLS regression of $Y_{i}$ on $\hat T_{i}$ and $X_{i}$ obtained from a non-linear regression of $T_{i}$ on $Z_{i}$ and $X_{i}$. Instead, our approach replaces the linear predictor $X_{i}^{\top }\widehat{ \gamma }$\ of $Z_{i}$\ by a nonlinear predictor $\Lambda (X_{i}^{\top } \widehat{\theta })$, so we are \textquotedblleft partialling out\textquotedblright \ $X_{i}$\ using a nonlinear model. Importantly, under the assumption of constant treatment effects, so that $Y(1) - Y(0) = \varrho$, our logit-based IV estimator is consistent for $\varrho$ under the same conditions as those required for consistency of the 2SLS estimator, as opposed to the forbidden regression, which requires extra conditions. In particular, it is easy to check that the logit-based IV estimator is consistent for $\varrho$ under Assumption (ref) as long as the function $x\mapsto \mathbb E[Y(0)|X=x]$ is linear.}
Without further assumptions, neither logit-based IV nor 2SLS estimators may have a causal interpretation. To see why this is so, we first need to introduce some additional notation. For all $x\in \mathcal{X}$, let
denote the $x$-conditional fractions of compliers, always-takers, and never-takers in the population, respectively. Also, let
and
be the probability limits of the estimators $\widehat \theta$ and $\widehat \gamma$ appearing in Algorithms (ref) and (ref). Moreover, for all $x\in \mathcal{X}$, let
The following theorem derives the probability limits of both $\widehat \beta_{\Lambda}$ and $\widehat \beta_{2SLS}$.
The proof of this theorem, as well as of all other results in the main text, is rather simple and is provided in the Appendix. In fact, the expression for the probability limit of the 2SLS estimator $\widehat \beta_{2SLS}$ in this theorem is closely related to that in Proposition 1 of BBMT22.
Theorem (ref) shows that without further assumptions, the probability limits of both logit-based IV and 2SLS estimators take the form of a sum of complier, always-taker, never-taker, and non-causal terms, appearing in expressions (ref), (ref), (ref), and (ref), respectively. The last term is referred to here as a non-causal term because it does not take the form of a functional of the treatment effect $Y(1)-Y(0)$ . In particular, this term depends non-trivially on the level of potential outcomes $Y(0)$ and $Y(1)$. The representation for the probability limits given in Theorem (ref) is not unique as different values of the parameter $s$ give different representations. For example, it is always possible to get rid of the never-taker term by substituting $s=1$ and it is always possible to get rid of the always-taker term by substituting $s = 0$. However, without further assumptions, it is not possible to get rid of both always-taker and never-taker terms at the same time.
Theorem (ref) identifies at least two problems with both logit-based IV and 2SLS estimators. First, the presence of the non-causal term means that neither the logit-based IV nor the 2SLS estimator in general converges in probability to a weighted-average treatment effect. Second, the always-taker term in (ref) shows that the $x$-conditional average treatment effect for always-takers $ \Delta_{AT}(x)$ has the weight $$ \frac{\omega_{AT}(x)(\mathbb{E}[Z|X=x] - h_a(x))}{\mathbb{E}[T(Z - h_a(x))]}, $$ which may be negative for both logit-based IV and 2SLS estimators, and the same applies to the never-taker term in (ref). In addition, although the weights in the causal terms (ref), (ref), and (ref) do integrate to one, as we show below (Corollary (ref)), eliminating the non-causal term in (ref) may also cause parts of the causal terms to drop out. In such cases, the resulting probability limit takes the form of a weighted-average treatment effect, but the weights no longer integrate to one. These are all the problems discussed in BBMT22 in the case of the 2SLS estimator.
Theorem (ref) also identifies the key advantage of the logit-based IV estimator in comparison with the 2SLS estimator: for the former, all the weights in the complier term are non-negative and this is not necessarily the case for the latter. To see this advantage of the logit-based IV estimator more clearly, we now impose two additional assumptions that ensure that the always-taker, never-taker, and non-causal terms drop out and the weights still integrate to one.
As we will see below, this assumption ensures that the always-taker, never-taker, and non-causal terms drop out from the general formulas in Theorem (ref). Although non-standard, this assumption seems rather attractive. Indeed, it specifies a linear regression model for the conditional mean of $(Y(1) - Y(0))T(0) + Y(0)$ given $X$, and linear regression models have a long tradition in econometrics. In empirical work, even if regression functions are not believed to be exactly linear, they are believed to be approximately linear. In this sense, Assumption (ref) is in line with a traditional regression analysis in economics. Moreover, as long as the conditional mean function $x\mapsto \mathbb{E}[(Y(1) - Y(0))T(0) + Y(0)|X=x]$ is continuous, it can be well approximated by a linear combination of, say, polynomial transformations of $x$. In such a case, Assumption (ref) can be made more plausible if we replace $X$ by a set of polynomial, or other technical, transformations of $X$. In practice, this amounts to replacing all the $X_i$ by, say, the $q(X_i)$, where $q(\cdot) = (q_1(\cdot),\dots,q_k(\cdot))^{\top}$ is a vector of corresponding transformations. Moreover, under Assumption (ref),
which implies that Assumption (ref) is testable; se Remark (ref) below for details.
This assumption ensures that the weights in Theorem (ref) integrate to one despite the fact that some terms in the theorem drop out because of Assumption (ref). It specifies a linear regression model for the conditional mean of $T(0)$ given $X$. Given that $T(0)$ is a binary random variable, the conditional mean $\mathbb{E}[T(0)|X]$ takes values in the $(0,1)$ interval, and so this assumption may be less plausible than Assumption (ref). However, we will use this assumption mainly to make the comparison between the logit-based IV and 2SLS estimators particularly transparent. Without this assumption, our logit-based IV estimator still has a partially causal interpretation as we will explain in Corollary (ref) below. Also, under Assumption (ref),
which implies that Assumption (ref) is testable as well. In addition, like Assumption (ref), it can be made more plausible if we replace $X$ by a set of appropriate transformations of $ X$. Finally, we will discuss in the next section how one can modify the logit-based IV estimator to accommodate more plausible versions of Assumption (ref).
By combining Theorem (ref) with Assumptions (ref) and (ref), we obtain the following corollary.
This corollary provides a clean comparison between logit-based IV and 2SLS estimators. It shows that under our assumptions, both estimators converge in probability to weighted-average treatment effects for compliers, and the weights do integrate to one:
However, in the case of the 2SLS estimator, some of the weights may take negative values, which happens whenever $h_{2SLS}(X) = X^{\top}\gamma_0$ exceeds one with positive probability. At the same time, the weights of the logit-based IV estimator are always non-negative, as the logit function $\Lambda(\cdot)$ takes values in the $ (0,1)$ interval. Thus, under our assumptions, the logit-based IV estimator has a causal interpretation and the 2SLS estimator does not. This explains the main advantage of the logit-based IV estimator relative to the traditionally used 2SLS estimator.
We now discuss two extensions of Corollary (ref). First, without imposing Assumption (ref), we obtain somewhat more convoluted expressions for the probability limits of the logit-based IV and 2SLS estimators, which are nonetheless useful to make comparisons between these two estimators.
This corollary shows that without imposing Assumption (ref), both logit-based IV and 2SLS estimators still converge in probability to weighted-average treatment effects for compliers but the weights now do not integrate to one. On the other hand, as long as the fraction of always-takers is not too large relative to the fraction of compliers, so that the denominators in (ref) remain non-negative for all $x\in \mathcal{X}$, the main advantage of the logit-based IV estimator remains valid: its corresponding weights are still non-negative and the estimator has a partially causal interpretation, whereas the weights of the 2SLS estimator may be negative and the estimator does not have a partially causal interpretation.
It is also worth noting that, for both the logit-based IV and the 2SLS estimators, the weights may sum to more or less than one. Indeed, inspection of the formula in (ref) shows that, for both $a=\Lambda$ and $a=2SLS$, we would have $\mathbb{E}[w_a(X)] = 1$ if it were the case that $$ \mathbb{E}[\omega_{AT}(X)(\mathbb{E}[Z|X] - h_a(X))] = 0. $$ However, in both cases, $$ \mathbb{E}[\mathbb{E}[Z|X] - h_a(X)] = \mathbb E[Z - h_a(X)] = 0, $$ and so the term $\mathbb{E}[\omega_{AT}(X)(\mathbb{E}[Z|X] - h_a(X))]$ may be either positive or negative. It follows that the failure of the weights to integrate to one can generate both attenuation and amplification biases.
Second, we can relax Assumptions (ref) and (ref) without losing causal interpretability of the logit-based IV estimator. Indeed, consider the following assumption.
This assumption relaxes Assumptions (ref) and (ref) as it reduces to Assumptions (ref) and (ref) if we plugin $s=1$. It requires that there exists a linear combination of the conditional mean functions $x\mapsto \mathbb{E}[(Y(1)-Y(0))T(1)+Y(0)|X=x]$ and $x\mapsto \mathbb{E} [(Y(1)-Y(0))T(0)+Y(0)|X=x]$ that is linear and that the linear combination of the conditional mean functions $x\mapsto \mathbb{E}[T(1)|X=x]$ and $ x\mapsto \mathbb{E}[T(0)|X=x]$ with the same weights is linear as well. All the comments we made about Assumptions (ref) and (ref) apply to this assumption as well. In particular, one can show that equations (ref) and (ref) can be equivalently rewritten as
and
respectively. Thus, Assumption (ref) is testable.
This corollary shows that under Assumptions (ref) and (ref), Assumption (ref) is sufficient for causal interpretability of the logit-based IV estimator. The weights, however, take a more complicated form than those in Corollary (ref). In comparison with Corollary (ref), this corollary also shows that the weights for the 2SLS estimator may be negative not only when $\mathbb P(X^\top\gamma_0 >1)>0$ but also when $\mathbb P(X^\top\gamma_0 < 0) > 0$ (the latter case is relevant when $s$ in Assumption (ref) is close to zero). In practice, we therefore recommend calculating the fractions of observations $i$ with $X_i^\top\widehat\gamma > 1$ and with $X_i^\top\widehat\gamma < 0$ and disregarding the 2SLS estimator in favor of the logit-based IV estimator whenever at least one fraction is non-trivial.
Finally, we demonstrate that the logit-based IV estimator has a causal interpretation even if Assumption (ref) is not satisfied as long as it consistently estimates the conditional mean function $x\mapsto \mathbb{E}[Z|X=x]$ on the first step. Indeed, consider the following assumption.
Note that in general, the assumption that the conditional mean function $ x\mapsto \mathbb{E}[Z|X=x]$ takes the logit form means that there exists some $\theta \in \mathbb{R}^p$ such that $\mathbb{E}[Z|X] = \Lambda(X^{\top}\theta)$ with probability one. However, given the definition of $\theta_0$ in (ref), this $\theta$ should be equal to $\theta_0$, and so we simply assume that $\mathbb{E}[Z|X] = \Lambda(X^{\top}\theta_0)$.
Together with Corollary (ref), this corollary shows that our logit-based IV estimator has a double robustness property, meaning that it has a causal interpretation if at least one of two conditions holds: either Assumption (ref) or Assumption (ref) is satisfied. Interestingly, however, the weights $ w_{\Lambda,s}(\cdot)$ and $w_0(\cdot)$ appearing in Corollaries (ref) and (ref) may be different, which means that even though we have a causal interpretation in both cases, the interpretation of the parameter we are estimating depends on which condition is being satisfied.
In this section, we replace Assumption (ref) by a more plausible assumption and show how one can modify the logit-based IV estimator in order to obtain an estimator that still has a causal interpretation. Our modified estimator is similar to the original logit-based IV estimator but includes an extra covariate in the logit regression of $Z$ on $X$.
Let $\Phi (\cdot )$ be a function mapping $\mathbb{R}$ to $[0,1]$ and consider the following alternative to Assumption (ref) .
If we set $\Phi (t)=\min (\max (0,t),1)$ for all $t\in \mathbb{R}$, then Assumption (ref) relaxes Assumption (ref). Indeed, in this case, two assumptions are the same if the support of $X^{\top }\psi _{0}$ is contained in the $[0,1]$ interval but Assumption (ref) does not actually require the support of $X^{\top }\psi _{0}$ to be contained in the $[0,1]$ interval. We are, however, primarily interested in the cases where the function $\Phi (\cdot )$ is nonlinear and smooth, e.g. $\Phi (\cdot )$ takes the logit or the probit form. In these cases, Assumption (ref) seems more plausible than Assumption (ref), as the function $x\mapsto \Phi (x^{\top }\psi _{0})$ is smooth and automatically satisfies the constraint that the conditional mean of $T(0)$ given $X$ takes values in the $(0,1)$ interval, without restricting the support of $X$. Like Assumptions (ref) and (ref), Assumption (ref) is testable and can be made more plausible if we replace $X$ by a set of appropriate transformations of $X$.
As it turns out, we can modify our logit-based IV estimator in a way so that it has a causal interpretation even if we replace Assumption (ref) by Assumption (ref). The modification, which yields the augmented logit-based IV estimator, is explained in the algorithm below.
This estimator requires that we know the link function $\Phi$ in the single-index structure $\Phi(X^{\top}\psi_0)$ used to model the conditional mean of $T(0)$ given $X$. However, from a practical point of view, different link functions, such as logit or probit, often lead to similar results, and the researcher can always check whether it is indeed the case by trying several link functions. Also, as pointed out above, the researcher can test whether a particular link function is consistent with the data. Finally, the researcher can estimate the link function $\Phi$, as described in Chapter 2 of H09, for example.
To derive the probability limit of this estimator, define
Note that $\bar \psi_0$ equals $\psi_0$ if Assumption (ref) is satisfied, but may be different from $\psi_0$ otherwise. Also, define
Moreover, define
for all $x\in \mathcal{X}$. The following theorem provides the probability limit of the estimator $\widehat \beta_{A\Lambda}$ imposing only Assumptions (ref) and (ref).
This theorem shows that the probability limit of the augmented logit-based IV estimator has the same structure as those of the logit-based IV and 2SLS estimators. In particular, it shows that imposing only a minimal set of conditions used in the program evaluation literature is not sufficient to give the augmented logit-based IV estimator a causal interpretation, which is similar to conclusions in the previous section and in BBMT22 for the logit-based IV and 2SLS estimators, respectively. To obtain a causal interpretation, we need to impose additional conditions. The following corollary provides the probability limit of the estimator $\widehat{\beta } _{A\Lambda }$ under the condition that either Assumption (ref) or Assumption (ref) is satisfied, along with Assumptions (ref), (ref), and (ref) used in the previous section.
This corollary demonstrates that the augmented logit-based IV estimator has a causal interpretation in a wider set of cases than the logit-based IV estimator itself. In particular, both have a causal interpretation if Assumption (ref) is satisfied, along with other conditions, but the former has a causal interpretation even if Assumption (ref) is not satisfied, as long as the correct link function $\Phi$ is being used. Interestingly, however, the weights $ w_{A\Lambda}(\cdot)$ appearing in this theorem are generally different from the weights $w_{\Lambda}(\cdot)$ appearing in Corollary (ref), which means that even when both estimators have a causal interpretation, they generally estimate different quantities.
In principle, Corollary (ref) could be extended by relaxing Assumptions (ref) and (ref) in the same way that Corollary (ref) extends Corollary (ref). However, implementing such an extension would substantially complicate Step 1 of Algorithm (ref). In particular, it would require estimating $\overline\psi_0$ in a model of the form
for some {\em unknown} $s\in [0,1]$. Developing this extension would add considerable technical complexity without yielding additional conceptual insights, and we therefore leave it for future work. Alternatively, we could construct a variant of Algorithm (ref) that uses the subsample with $Z=1$ in the first step rather than $Z=0$. The analysis for this variant proceeds analogously to Theorem (ref) and Corollary (ref) upon replacing $T(0)$ by $T(1)$ in Assumptions (ref), (ref), and (ref). Since this modification is conceptually identical and does not change the main conclusions, we do not present it separately.
Moreover, we could also consider a version of the augmented logit-based IV estimator in Algorithm (ref) that treats the data with $Z=0$ and $Z=1$ symmetrically. For such a version, on the first step of Algorithm (ref), we run the maximum likelihood estimator separately for the data with $Z=0$ and $Z=1$ to obtain $\widehat\psi_1$ and $\widehat\psi_2$. Then on the second and third steps, instead of including just one $\widehat C_i = \Phi(X_i^\top \widehat\psi)$, we include both $\widehat C_{i,1} = \Phi(X_i^\top\widehat\psi_1)$ and $\widehat C_{i,2} = \Phi(X_i^\top\widehat\psi_2)$, with the corresponding coefficients $\widehat\kappa_1$ and $\widehat\kappa_2$. Such an estimator has a causal interpretation if Assumptions (ref), (ref), and (ref) hold either as stated, with $T(0)$, or as modified above, with $T(1)$ replacing $T(0)$.
To conclude this section, we note that as in the previous section, Assumptions (ref), (ref), and (ref) are not needed for a causal interpretation of the augment logit-based IV estimator if the conditional mean function $ x\mapsto \mathbb{E}[Z|X=x]$ takes the logit form, i.e. Assumption (ref) is satisfied. Indeed, we have the following result.
Together with Corollary (ref), this corollary shows that the augmented logit-based IV estimator has a triple robustness property, meaning that it has a causal interpretation if at least one of three conditions holds: either Assumptions (ref) and (ref) are satisfied, Assumptions (ref) and (ref) are satisfied, or Assumption (ref) is satisfied. In the latter case, the logit-based IV and the augmented logit-based IV estimators have the same probability limits as the weights $w_0(\cdot)$ in (ref) coincide with the weights $w_0(\cdot)$ in (ref), i.e. $ \widehat \beta_{\Lambda}$ and $\widehat \beta_{A\Lambda}$ consistently estimate the same quantity.
In this section, we describe the asymptotic distribution of the logit-based IV estimator $\widehat{\beta }_{\Lambda }$ and of the augmented logit-based IV estimator $\widehat{\beta }_{A\Lambda }$.\ We do so without imposing Assumptions (ref), (ref), (ref), and (ref) and without assuming that the conditional mean function $x\mapsto \mathbb{E}[Z|X=x]$ takes the logit form (Assumption (ref)). We thus allow for general misspecification, with the probability limits $\beta _{\Lambda }$ and $\beta _{A\Lambda }$ of the estimators being given by formulas in Theorems (ref) and (ref), respectively.
To describe the asymptotic distribution of $\widehat{\beta }_{\Lambda }$, let
which is a vector of coefficients in the weighted projection of $Y-T\beta _{\Lambda }$ on $X$. Also, let
for all $i=1,\dots ,n$ and let $\ell ^{\Lambda }$ be defined analogously with $(Y,T,X,Z)$ replacing $(Y_{i},T_{i},X_{i},Z_{i})$. The following theorem derives the asymptotic distribution of $\widehat{\beta }_{\Lambda }$.
To describe the asymptotic distribution of $\widehat{\beta }_{A\Lambda }$, let $C=\Phi (X^{\top }\overline{\psi }_{0})$ and $C_{i}=\Phi (X_{i}^{\top } \overline{\psi }_{0})$ for all $i=1,\dots ,n$. Also, let $W=(X^{\top },C)^{\top }$ and $ W_{i}=(X_{i}^{\top },C_{i})^{\top }$ for all $i=1,\dots ,n$. In addition, let $A_{0}=\mathbb{E}[\Lambda ^{\prime }(X^{\top }\overline{ \theta }_{0}+C\overline{\kappa }_{0})WW^{\top }]$ and
Moreover, let $$ S_i = \mathds{1}\{Z_i = 0\}\left(\frac{T_i\Phi'(X_i^\top\overline\psi_0)}{\Phi(X_i^\top\overline\psi_0)} - \frac{(1-T_i)\Phi'(X_i^\top\overline\psi_0)}{1 - \Phi(X_i^\top\overline\psi_0)}\right)X_i $$ for all $i = 1,\dots,n$. Further, let $e$ be the vector in $\mathbb{R}^{p+1}$ such that its last component is one and all other components are zero. Further, let
which is the vector of coefficients in the weighted projection of $Y-T\beta _{A\Lambda }$ on $W$,
Finally, let
for all $i=1,\dots ,n$ and let $\ell _{1}^{A\Lambda }$ and $\ell _{2}^{A\Lambda }$ be defined analogously with $(Y,T,X,Z)$ replacing $ (Y_{i},T_{i},X_{i},Z_{i})$. The following theorem derives the asymptotic distribution of $\widehat{\beta }_{A\Lambda }$.
In this theorem, the terms $\ell_{i,1}^{A\Lambda}$ are analogous to the terms $\ell_{i}^{\Lambda}$ in Theorem (ref) and the terms $\ell_{i,2}^{A\Lambda}$ capture the extra noise appearing in Step 1 of Algorithm (ref).
The asymptotic variances $\sigma_{\Lambda}^2$ and $\sigma_{A\Lambda}^2$ appearing in these theorems can clearly be estimated by a plugin method, and it is standard to provide conditions under which such estimators will be consistent. We omit more detailed discussion for the sake of paper brevity.
In Sections (ref) and (ref), we showed that one of the cases where the logit-based IV and augmented logit-based IV estimators $\widehat \beta_{\Lambda}$ and $ \widehat \beta_{A\Lambda}$ have a causal interpretation is the case where the conditional mean function $x\mapsto \mathbb{E}[Z|X=x]$ takes the logit form, i.e. Assumption (ref) is satisfied. In this section, we develop a Hausman test to check whether Assumption (ref) is indeed satisfied, following the original work in H78. Throughout this section, we will implicitly maintain Assumptions (ref) and (ref).
To describe the Hausman test, observe that under Assumption (ref), it follows from Corollaries (ref) and (ref) that $\beta_{\Lambda} = \beta_{A\Lambda}$. Therefore, to test whether Assumption (ref) is satisfied, it makes sense to check whether the estimators $\widehat \beta_{\Lambda}$ and $\widehat \beta_{A\Lambda}$ are sufficiently close to each other. In turn, the asymptotic distribution of the difference $\widehat \beta_{\Lambda} - \widehat \beta_{A\Lambda}$ can be obtained from the asymptotic expansions in Theorems (ref) and (ref). Indeed, under Assumption (ref), we have $\overline \theta_0 = \theta_0$ and $\overline \kappa_0 = 0$, and so, by Theorems (ref) and (ref),
where
The Hausman test therefore rejects the null hypothesis that Assumption (ref) is satisfied if the test statistic
exceeds the critical value $z_{1- \alpha/2}$, where $\widehat \sigma_{H}$ is the plugin estimator of $\sigma_{H}$, $\alpha$ is the nominal level of the test and $z_{1- \alpha/2}$ is the number such that a standard normal random variable exceeds this number with probability $\alpha/2$.
Being parametric, the test we have just described may not have power against some alternatives. However, as long as Assumption (ref) is not satisfied, we will generically have $(\overline \theta_0,\overline \kappa_0)\neq (\theta_0,0)$, in which case the function $ h_{A\Lambda}(\cdot)$ is different from the function $h_{\Lambda}(\cdot)$ and, as follows from Theorems (ref) and (ref), $\beta_{\Lambda}$ is different from $\beta_{A\Lambda}$. Thus, the Hausman test will have power against most alternatives.
In addition, we can consider a split-sample version of the Hausman test. To describe it, randomly split the whole sample $\mathcal{I }= \{1,\dots,n\}$ into two subsamples $\mathcal{I}_1$ and $\mathcal{I}_2$ of comparable size and calculate the logit-based IV estimator using the subsample $\mathcal{I}_1$ and the augmented logit-based IV estimator using the subsample $\mathcal{I}_2$. Call these estimators $\widehat \beta_{\Lambda,1}$ and $\widehat \beta_{A\Lambda,2}$, respectively. Then, under Assumption (ref), these two estimators have the same probability limits and, by Theorems (ref) and (ref),
where
and $\alpha_0$ is the limit of the ratio $n/|\mathcal I_1|$. The split-sample Hausman test therefore rejects the null hypothesis that Assumption (ref) is satisfied if the test statistic
exceeds the critical value $z_{1- \alpha/2}$, where $\widehat \sigma_{H,2}$ is the plugin estimator of $\sigma_{H,2}$ and the rest is the same as before.
Being split-sample, this version of the Hausman test may be somewhat less powerful than the one described above. However, it is more robust in terms of size control if it incidentally happens that $\sigma_H^2$ is close to zero, in which case the distribution of the test statistic in (ref) may not be well approximated by the standard normal distribution.
In this section, we compare our logit-based IV estimators with the 2SLS estimator in three empirical applications: AE98, ABBKK02, and DH20. All three applications involve a binary treatment and a binary instrument, making them well-suited for our logit-based IV estimators.
AE98 study the effect of childbearing on women's labor supply. We focus on their 2SLS specification in Table 7, Column 2, based on a sample of 394,840 women aged 21--35 with two or more children in the 1980 Census. The outcome $Y$ represents several labor market measures: an indicator for having worked for pay in the previous year, weeks worked in the previous year, average hours worked per week, labor income, or the log of family income. The treatment $T$ is a binary indicator for having more than two children. The instrument $Z$ is a binary indicator for whether the first two children are of the same sex. The vector of controls $X$ includes age, age at first birth, plus indicators for the first child being a boy, the second child being a boy, and the mother being Black, Hispanic, or of another race.
ABBKK02 analyze the impact of the school voucher program PACES in Colombia. We consider their 2SLS specification in Table 7, Column 3, which estimates the effect of private-school scholarship use on academic and marital outcomes, using voucher status as an instrument. The sample consists of 1,147 PACES applicants from the 1995 cohort in Bogot\'{a}, with an average age of 12.6. The outcome $Y$ represents various measures: highest grade completed, grade repetitions since the lottery, or indicators for currently being in school, having finished 8th grade, or being married or living with a companion. The treatment $T$ is a binary indicator for ever having used a private-school scholarship. The instrument $Z$ is a binary indicator for winning a PACES voucher. The vector of controls $X$ includes city, year of application, phone access, age, gender, residence strata, month of interview, and type of survey.
DH20 examine the effect of queen rule on war. We focus on their 2SLS specification in Table 3, Column 3, estimated using an unbalanced panel of 3,586 observations covering 193 reigns in 18 European polities from 1480 to 1913. The outcome $Y$ indicates whether a polity is at war in a given year. The treatment $T$ is a binary indicator for whether a queen is in power. The instrument $Z$ is a binary indicator for whether the previous monarchs had a male firstborn child. The vector of controls $X$ includes indicators for whether the previous monarchs were unrelated corulers, whether they had any legitimate children (with and without missing birth years), whether the gender of the firstborn is missing, as well as polity and decade dummies.
The results for AE98, ABBKK02, and DH20 are presented in Tables (ref), (ref), and (ref), respectively. For each table, we replicate the original 2SLS estimation results in Column 1.\footnote{DH20 use clustered standard errors but report only the $p$-values computed via the wild bootstrap. To facilitate comparison with our logit-based IV estimators, we report the plug-in clustered standard error for their 2SLS estimator.} We present logit-based IV and augmented logit-based IV estimation results in Columns 2 and 3, respectively. As described in Algorithm (ref) above, in order to calculate the logit-based IV estimator, we first estimate a logit regression of the instrument on the controls to obtain the residual, defined as the instrument minus its predicted probability. We then estimate an IV regression of the outcome on the treatment, using this residual as the instrument. As described in Algorithm (ref), In order to calculate the augmented logit-based IV estimator, we first estimate a logit regression of the treatment on the controls---using only the subset of observations where the instrument equals zero---and obtain the predicted probability. Next, we estimate another logit regression of the instrument on the controls and the predicted probability from the first step and obtain the residual as before. Finally, we estimate an IV regression of the outcome on the treatment using the residual from the second step as the instrument. The standard errors of these estimators are computed using plug-in methods based on the asymptotic variance formulas derived in Theorems (ref) and (ref). For DH20, we calculate clustered standard errors using the clustering structure in the original study. The p-values for the Hausman test and for the split-sample Hausman test are given in Columns 4 and 5. In addition, each table contains information on the fraction of observations $i$ with $X_i^\top\widehat\gamma > 1$ and with $X_i^\top\widehat\gamma < 0$.
In both AE98 and ABBKK02, the logit-based IV estimators yield estimates and standard errors that are nearly identical to those of the 2SLS estimator, which is consistent with the fact that there are no observations $i$ in these studies with $X_i^\top\widehat\gamma > 1$ or with $X_i^\top\widehat\gamma < 0$. Moreover, the p-values for both the Hausman test and the split-sample Hausman test are well above 10%. This evidence suggests that both the 2SLS estimator and our logit-based IV estimators are appropriate for these studies and all three have a causal interpretation.
However, in DH20, the logit-based IV estimators yield results that differ substantially from those of the 2SLS estimator. While the 2SLS estimate is positive, the logit-based IV estimates are negative and insignificant. In this study, $6.83\%$ of the observations have $X_i^\top\widehat\gamma > 1$ and $10.15\%$ have $X_i^\top\widehat\gamma < 0$, indicating that the 2SLS estimator may be assigning negative weights to a non-trivial fraction of compliers. In this case, the 2SLS estimator lacks a causal interpretation, making the logit-based IV estimators the preferred approach. Moreover, the p-values for the Hausman tests are also well above 10%, reinforcing the conclusion that our logit-based IV estimators are appropriate for this study.