Marginal Effects for Probit and Tobit with Endogeneity

Marginal Effects for Probit and Tobit with Endogeneity

Introduction

Probit and Tobit are some of the most popular nonlinear models in applied economics. When a covariate is endogenous, IV-Probit and IV-Tobit models can be used for instrumental variable (IV) estimation of the coefficients ( SmithBlundell1986Ecta, RiversVuong1988JoE).\footnote{ For example, in Stata, these estimators are ivprobit and ivtobit.}

A covariate can be endogenous for two reasons. First, the covariate can be correlated with the individual's unobserved characteristics (unobserved heterogeneity). Second, mismeasurement of the covariate also results in endogeneity (Errors-in-Variables, EiV). We will refer to these two types of endogeneity as the structural\ endogeneity and the EiV. In many empirical settings both sources of endogeneity need to be addressed simultaneously.

In empirical applications of nonlinear models, researchers are often interested in partial effects and related counterfactuals. The goal of this paper is to characterize the partial effects in the classic IV-Probit and IV-Tobit models, allowing for both types of endogeneity, and to emphasize the importance of distinguishing between the two types. We provide the expressions for the partial effects and average partial effects that correctly account for the two kinds of endogeneity. Although the two sources of endogeneity cannot be precisely distinguished using the observed data, we use the constraints of the model to obtain bounds on the amounts of endogeneity that can be attributed to each source. This allows us to characterize sharp bounds on the true partial effects and average partial effects, allowing for both types of endogeneity. We also provide simple estimators of these bounds and corresponding valid confidence intervals that are easy to calculate.

The primary objects of interest of this paper are the partial effects of the covariates, rather than the regression coefficients (the coefficients on the covariates). Denote by $X_{i}^{\ast }$ the potentially endogenous covariate, and by $X_{i}$ its mismeasured observed version. Estimation of the coefficients on $ X_{i}^{\ast }$ and other covariates is a simpler task than estimation of the partial effects of $X_{i}^{\ast }$. In particular, in the IV-Probit, IV-Tobit, and related models, to identify and estimate these coefficients it is sufficient to simply consider $X_{i}$ as the endogenous regressor of interest without needing to distinguish between the types of endogeneity. \footnote{ In particular, SmithBlundell1986Ecta and RiversVuong1988JoE simply consider $X_{i}$ as endogenous. Similarly, in a recent paper, ChesherKimRosen2023JoE provide a sharp identified set for the coefficients on the covariates in a Tobit model with endogeneity under weak assumptions. The approaches of these papers implicitly allow for mismeasured covariates, as long as the focus is only on the regression coefficients.}

In nonlinear models, the need to differentiate between the two kinds of endogeneity arises because structural endogeneity and EiV play different roles. In particular, partial effects of covariates are averaged with respect to the distribution of the individual unobserved heterogeneity. On the other hand, one aims to remove the impact of the measurement errors, since they are not properties of individuals but a deficiency in the measurement process. The textbook treatment of the problem often focuses only on the first type of endogeneity, treating endogeneity as purely structural. When $X_{i}^{\ast }$ is mismeasured, the partial effect of $ X_{i}^{\ast }$ in nonlinear models differs from the effect of $X_{i}$ one would calculate using the standard formulas that assume the endogeneity is purely structural.

To identify the partial effects of $X_{i}^{\ast }$ and other covariates one needs to identify the distribution of the true unobserved heterogeneity not contaminated by the measurement error. It turns out that this distribution is only partially identified. Thus, even though the IV-Probit and IV-Tobit methods consistently estimate the coefficients on all regressors regardless of the sources of endogeneity, the effects of the covariates on the outcomes are only partially identified. The width of the identified set depends on how hard it is to disentangle structural endogeneity and EiV for the data at hand. Importantly, we find that naively ignoring the distinction between the two types of endogeneity can result in both under- and over-estimation of the magnitude of the partial effects by these IV estimators.\footnote{Wooldridge2010Book, page 586, alludes to the potential importance of the sources of endogeneity for the partial effects in IV-Probit, but does not elaborate.}

IV-Probit and IV-Tobit can be interpreted as control variable estimators. Partial effects in general control variable models were considered by BlundellPowell2003Advances, Chesher2003Ecta, ImbensNewey2009Ecta, and Wooldridge2005APE,Wooldridge2015JHR, among others. These control variable methods focus on structural endogeneity exclusively but do not consider EiV. The problems of estimation with structural endogeneity or measurement errors are studied by two large but (mostly) distinct literatures in econometrics, see, e.g., Matzkin2013AnnRev and Schennach2020HB-ME for reviews. In nonlinear models, accounting for both types of endogeneity is challenging, see, e.g., Schennach2022JEcLit. Exceptions include AdusumilliOtsu2018ET,SongSchennachWhite2015QE,SchennachWhiteChalak2012JoE,HahnRidder2017JoE . These papers obtain point identification results when the distribution of the measurement error is either known or can be recovered from repeated measurements using the lemma of Kotlarski1967. Such datasets, however, are relatively rare.

Control variable methods in nonlinear models typically require the endogenous variable to be continuously distributed (e.g., see ImbensNewey2009Ecta, for a discussion). This limitation also applies to our framework: the mismeasured endogenous variable $X_i$ is assumed to be continuously distributed. Other covariates can be discrete. Note that properly accounting for both types of endogeneity of $X_i$ is essential for characterizing ceteris paribus effects of all covariates, including the discrete ones.

The advantage of gaussian nonlinear models is their simplicity and transparency, which makes them a convenient starting point in an empirical analysis. Our approach in particular provides the researchers with a simple way to gauge the importance of properly accounting for the two types of endogeneity, which is essential given the ubiquity of both in economic applications. In addition, for the settings where relaxing gaussianity is important, we develop an extension of our approach that allows both the first stage unobservables and the measurement errors to be non-gaussian.

The rest of the paper is organized as follows. The analysis of partial effects in the Probit and Tobit models is virtually identical, thus we first focus on Tobit in Sections (ref)-(ref). Section (ref) then considers the Probit model. Section (ref) extends the analysis to cover the average partial effects and other counterfactuals. Section (ref) provides some Monte Carlo simulation results. Section (ref) presents an empirical application. Section (ref) relaxes gaussianity assumptions.

The Model

The Tobit model is often used for estimation of economic models with a \textquotedblleft corner solution,\textquotedblright\ i.e., models where the outcome variable $Y_{i}\,$\ is forced to be non-negative. The examples of such dependent variables $Y_{i}$ include the amounts of charitable contributions, hours worked, or monthly consumption of cigarettes.

First, consider the standard Tobit model with exogenous covariates and without EiV:

the individual unobserved heterogeneity $U_{i}^{\ast }$ has a normal distribution $N\left( 0,\sigma _{U^{\ast }}^{2}\right) $ and is independent from the covariates $X_{i}^{\ast }$ and $W_{i}$. We use the asterisk to denote variables that will be affected by the EiV, as we explain in detail below.

We collect the covariates in a vector $H_{i}^{\ast }=\left( X_{i}^{\ast },W_{i}^{\prime }\right) ^{\prime }$, so ((ref)) can be written as

We denote the standard normal cumulative distribution and density functions by $\Phi $ and $\phi $.\footnote{ Most of the analysis in Sections (ref) and (ref) equally applies to the Probit model. For simplicity of exposition, we focus on the Tobit model for the moment, and then discuss Probit in Section (ref).}

In the Tobit model, one is usually interested in the partial effects (marginal effects) of covariates $H_{i}^{\ast }$ on $\mathrm{E}\left( Y_{i}|H_{i}^{\ast }\right) $ and $\mathrm{P}\left( Y_{i}>0|H_{i}^{\ast }\right) $. For concreteness we consider partial effects of the continuously distributed covariates.

The partial effect of the $j^{th}$ covariate on the mean $\mathrm{E}\left( Y_{i}|H_{i}^{\ast }=h\right) $ at a given $h$ is

The partial effect of the $j^{th}$ covariate on the probability $P\left( Y_{i}>0|H_{i}^{\ast }=h\right) $ is

These formulas for the $PE_{j}$ are standard, see, e.g., Wooldridge2010Book, for detailed calculations. Most often one considers the partial effects at the means of the covariates $h=E\left[ H_{i}^{\ast } \right] $ or partial effects averaged with respect to the distribution of $H_i^*$.

When $X_{i}^{\ast }$ is correlated with $U_{i}^{\ast }$ and we observe data $ \left( Y_{i},X_{i}^{\ast },W_{i},Z_{i}\right) $, the IV-Tobit model can be estimated using instrumental variables $Z_{i}$, as proposed by SmithBlundell1986Ecta, Newey1987, and RiversVuong1988JoE. Assume that

where $V_{i}^{\ast }$ is a normal random variable, possibly correlated with $ U_{i}^{\ast }$,

and $\left( U_{i}^{\ast },V_{i}^{\ast }\right) $ is independent from $\left( Z_{i},W_{i}\right) $. In this model $X_{i}^{\ast }$ is continuously distributed.

The IV-Tobit model in ((ref))-((ref)) can be estimated using a random sample of $\left( Y_{i},X_{i}^{\ast },W_{i},Z_{i}\right) $ in two steps, see, e.g., Wooldridge2010Book. First, one estimates $V_{i}^{\ast }$ in equation ((ref)) by the residuals $\hat{V}_{i}^{\ast }$ in the regression of $X_{i}^{\ast }$ on $ \left( W_{i},Z_{i}\right) $. Note that we can write $U_{i}^{\ast }=e_{i}^{\ast }+\theta _{V^{\ast }}V_{i}^{\ast }$, where $\theta _{V^{\ast }}\equiv \sigma _{U^{\ast }V^{\ast }}/\sigma _{V^{\ast }}^{2}$, and $ e_{i}^{\ast }$ is independent of $Z_{i}$, $W_{i}$, and $V_{i}^{\ast }$ (and hence of $X_{i}^{\ast }$). Then, one estimates the standard Tobit model

where $V_{i}^{\ast }$ are replaced by their estimates $\hat{V}_{i}^{\ast }$. (Alternatively, the two steps can be combined and all of the parameters can be estimated simultaneously by the Maximum Likelihood Estimator.) The reason this approach works is that equation ((ref)) creates a control variable $V_{i}^{\ast }$, and the inclusion of $V_{i}^{\ast }$ in the above equation makes $X_{i}^{\ast }$ exogenous.

To estimate the partial effects, one would plug the estimates $\widehat{ \theta }$ and $\widehat{\sigma }_{U^{\ast }}^{2}$ into equations ((ref))-((ref)) in place of $\theta _{0}$ and $\sigma _{U^{\ast }}^{2}$.

So far we were assuming that the data has no measurement errors. We now allow $X_{i}^{\ast }$ to be mismeasured, i.e., that instead of $X_{i}^{\ast } $ we observe its noisy measurement $X_{i}$:

We assume that $\varepsilon _{i}\perp \left( U_{i}^{\ast },V_{i}^{\ast },W_{i},Z_{i}\right) $, i.e., the measurement error is classical. The normality assumption simplifies the analysis but it is not crucial. We relax it in Section (ref).

Note that the researcher's object of interest has not changed: the goal is to estimate the partial effects defined in equations ((ref))-( (ref)). The structural endogeneity and measurement errors are difficulties that an estimation procedure needs to overcome. In particular, note that we are interested in estimation of the effect of $X_{i}^{\ast }$ and not in the effect of the error-laden $X_{i}$.\footnote{ This is similar to the linear regression settings, where one would be interested in the effect of $X_{i}^{\ast }$ on $Y_{i}$. The slope coefficient in the OLS\ regression of $Y_{i}$ on $X_{i}$ is not the object of interest because it is subject to the attenuation bias due to the EiV (and also possibly due to the endogeneity of $X_{i}^{\ast }$).}

Analysis of the Model

First, we use the model in equations ((ref))-((ref)) to obtain the model in terms of the observable $X_{i}$. Since $X_{i}^{\ast }=X_{i}-\varepsilon _{i}$, we can rewrite ((ref)) as

where $U_{i}\equiv U_{i}^{\ast }-\theta _{01}\varepsilon _{i}$. Let $ V_{i}\equiv V_{i}^{\ast }+\varepsilon _{i}$. The model in equations ((ref))-((ref)) can be written as

The definitions of $U_{i}$ and $V_{i}$ imply that

Note that variables $X_{i},U_{i},V_{i}$ are the analogs of the true variables $X_{i}^{\ast },U_{i}^{\ast },V_{i}^{\ast }$ that arise due to the measurement errors $\varepsilon _{i}$. In the absence of measurement errors, i.e., when $\varepsilon _{i}=0$, we have $X_{i}=X_{i}^{\ast }$, $ U_{i}=U_{i}^{\ast }$, $V_{i}=V_{i}^{\ast }.$

The model in equations ((ref))-((ref)) can be estimated by MLE or using the control variable two-step approach described earlier. Specifically, both approaches will consistently estimate parameters $\theta _{0}$ and the covariance matrix of the unobservables in equation ((ref)), i.e., $\sigma _{U}^{2}$, $\sigma _{UV}$, and $\sigma _{V}^{2}$.

Note that because the model is nonlinear, the marginal effects defined in equations ((ref))-((ref)) depend not only on $\theta _{0}$ but also on $\sigma _{U^{\ast }}^{2}$. Thus, even though the available data $ \left(Y_{i},X_{i},W_{i},Z_{i}\right) $ allows immediately estimating $\theta _{0}$, we cannot obtain the marginal effects because we do not know $ \sigma_{U^{\ast }}^{2}$. Naively using an estimate of $\sigma _{U}^{2}$ in place of $\sigma _{U^{\ast }}^{2}$ would lead to a biased estimate of the partial effects, since $\sigma _{U}^{2}\geq \sigma _{U^{\ast }}^{2}$, as implied by equation ((ref)).

The problem with identifying $\sigma _{U^{\ast }}^{2}$ is that the data only allows identification of the $3$ parameters $\sigma _{U}^{2}$, $\sigma _{UV}$ , and $\sigma _{V}^{2}$. However, the distribution of the true $\left( U_{i}^{\ast },V_{i}^{\ast },\varepsilon _{i}\right) $ is governed by $4$ parameters: $\sigma _{U^{\ast }}^{2}$, $\sigma _{U^{\ast }V^{\ast }}$, $ \sigma _{V^{\ast }}^{2}$, and $\sigma _{\varepsilon }^{2}$. Thus, one cannot uniquely determine these $4$ parameters from the $3$ equations ((ref)). In other words, models with different values of $\sigma _{\varepsilon }^{2}$ are observationally equivalent: they correspond to identical distributions of the observables $\left( Y_{i},X_{i},W_{i},Z_{i}\right) $ even though they imply different values of true $\sigma _{U^{\ast }}^{2}$. Thus, one cannot uniquely determine (i.e., point-identify) $\sigma _{U^{\ast }}^{2}$ from the data $\left( Y_{i},X_{i},W_{i},Z_{i}\right) $. Correspondingly, one cannot point-identify the partial effects, which depend on $\sigma _{U^{\ast }}^{2}$.

Equations ((ref)) provide restrictions on $\sigma _{U^{\ast }}^{2}$, which we will use to provide bounds on the possible values of true $\sigma _{U^{\ast }}^{2}$, and hence on the values of the partial effects.

\paragraph{Bounds on $\protect\sigma _{U^{\ast }}^{2}$ }

From equations ((ref)) the upper bound on $\sigma _{U^{\ast }}^{2}$ is $\sigma _{U^{\ast }}^{2}\leq \sigma _{U}^{2}$. We now obtain the lower bound on $\sigma _{U^{\ast }}^{2}$. In particular, we look to find the smallest $\sigma _{U^{\ast }}^{2}$ that satisfies equations ((ref)), Cauchy-Schwarz inequality $\sigma _{U^{\ast }V^{\ast }}^{2}\leq \sigma _{U^{\ast }}^{2}\sigma _{V^{\ast }}^{2}$, and the non-negativity constraints $\sigma _{U^{\ast }}^{2}\geq 0$, $\sigma _{V^{\ast }}^{2}\geq 0$, and $\sigma _{\varepsilon }^{2}\geq 0$. Let $\rho _{UV}=\mathrm{corr}\left( U_{i},V_{i}\right) $.

Proposition (ref) provides the bounds in terms of the quantities that can be estimated using the data $\left( Y_{i},X_{i},W_{i},Z_{i}\right) $. Condition $\left\vert \rho _{UV}\right\vert <1$ guarantees that the denominator in the fraction above is positive. The proof of Proposition (ref) also provides bounds on $\sigma _{U^{\ast }V^{\ast }}$ and $\sigma _{\varepsilon }^{2}$.

\paragraph{Correct Partial Effects}

We now use the bounds on $\sigma _{U^{\ast }}^{2}$ from Proposition (ref) to obtain the bounds on the partial effects, in terms of the parameters that can be recovered from data. For simplicity and concreteness of exposition, we first consider partial effects evaluated at some fixed values of covariates. We consider average partial effects and other counterfactuals of interest in Section (ref).

For a given $\sigma _{U^{\ast }}^{2}$, the partial effects for the $j^{th}$ covariate are defined as in equations ( (ref))-((ref)),

The lower and upper bounds for partial effects for the $j^{th}$ covariate, $ PE_{j}\left( h\right) $, are computed as

Function $PE_{j}^{\text{Tob}}\left( h,\sigma _{U^{\ast }}^{2}\right) $ in ( (ref)) is a monotone function of $\sigma _{U^{\ast }}^{2}$, so the minimum and maximum in equation ((ref)) are achieved on the boundaries of interval $\left[ \underline{\sigma } _{U^{\ast }}^{2},\sigma _{U}^{2}\right] $.

Function $PE_{j}^{\Pr }\left( h,\sigma _{U^{\ast }}^{2}\right) $ in equation ((ref)) is not monotone in $\sigma _{U^{\ast }}^{2}$, but the bounds in equation ((ref)) for $PE_{j}^{\Pr }\left( h\right) $ can also be simplified. The minimum and maximum over $\sigma _{U^{\ast }}^{2}\in \left[ \underline{\sigma }_{U^{\ast }}^{2},\sigma _{U}^{2}\right] $ can be attained only at $\sigma _{U^{\ast }}^{2}=\underline{\sigma }_{U^{\ast }}^{2}$, at $\sigma _{U^{\ast }}^{2}=\sigma _{U}^{2}$, and, when $\left( \theta _{0}^{\prime }h\right) ^{2}\in \left[ \underline{\sigma }_{U^{\ast }}^{2},\sigma _{U}^{2}\right] $, at $\sigma _{U^{\ast }}^{2}=\left( \theta _{0}^{\prime }h\right) ^{2}$. Thus, one only needs to evaluate $PE_{j}^{\Pr }\left( h,\sigma _{U^{\ast }}^{2}\right) $ at these $2$ or $3$ points to calculate the minimum and maximum in equation ((ref)).

Since $\sigma _{U}\geq \sigma _{U^{\ast }}$, naively using $\sigma _{U}$ instead of $\sigma _{U^{\ast }}$ when calculating $PE_{j}^{\text{Tob}}\left( h\right) $, would lead to attenuation bias when $\theta _{0}^{\prime }h>0$, but would bias $PE_{j}^{\text{Tob}}\left( h\right) $ away from zero when $ \theta _{0}^{\prime }h<0$, i.e., the EiV would make naive $PE_{j}^{\text{Tob} }\left( h,\sigma _{U}^{2}\right) $ over-estimate the partial effects $ PE_{j}^{\text{Tob}}\left( h\right) $ in the latter case. Likewise, for the probability, naively using $PE_{j}^{\Pr }\left( h,\sigma _{U}^{2}\right) $ can both under- and over-estimate the true partial effect $PE_{j}^{\Pr }\left( h\right) $.

\paragraph{Estimation}

Using the standard two-step or MLE approaches described in Section (ref), one obtains the estimates of $\theta _{0}$, $\sigma _{U}^{2}$, $ \sigma _{V}^{2}$, and $\sigma _{UV}$ (and of their variance-covariance matrix for inference). Then, from equation ((ref)) one obtains the estimate of $\underline{\sigma }_{U^{\ast }}^{2}$.

For a given value of $\sigma _{U^{\ast }}^{2}$, the estimated partial effects would be

Then, the estimated bounds on $ PE_{j}\left( h\right) $ are

where the minimum and maximum are easily computed using univariate numerical optimization. For the partial effects in equation ((ref)), these extrema can also be computed as described under equation ((ref)).

For example, one often considers the partial effects at the mean values of covariates taking $h=(\overline{X},\overline{W}^{\prime })^{\prime }$, where $\overline{X}$ and $\overline{W}$ are the sample averages. Note that $\mathrm{E}\left[ X_{i}\right] =\mathrm{E}\left[ X_{i}^{\ast }\right] $.

\paragraph{Inference}

To provide a simple method for inference about the partial effects, we adopt a Bonferroni approach (e.g., McCloskey2017JoE). This approach allows us to avoid computational challenges that often arise in the context of subvector inference in partially identified models. The construction of a $1-\alpha $ confidence interval for a partial effect $PE_{j}(h)$ proceeds in two steps:

We now provide the implementation details for each step.

Step 1. The confidence interval for $\sigma _{U^{\ast }}^{2}$ is constructed based on the bounds given in Proposition (ref). As the upper bound, we take $\hat{\sigma} _{U}^{2}+z_{1-\alpha _{1}/2}\times s_{\hat{\sigma}_{U}^{2}}$, where $s_{\hat{ \sigma}_{U}^{2}}$ is the standard error of $\hat{\sigma}_{U}^{2}$, and $ z_{1-\alpha _{1}/2}$ is the $1-\alpha _{1}/2$ quantile of the standard normal distribution. The lower bound is based on $\underline{\hat{\sigma}} _{U^{\ast }}^{2}=\max \{\hat{\xi}_{1},\hat{\xi}_{2}\}$, where $\hat{\xi}_{1}$ and $\hat{\xi}_{2}$ are the plug-in estimators of the two terms on the right hand side of equation ((ref)). Note that $\hat{\xi}_{1}$ and $\hat{\xi}_{2}$ are (generally) jointly asymptotically normal and their asymptotic variance-covariance matrix can be computed using the delta method. Then, as the lower bound of $CI_{1-\alpha _{1}}^{\sigma _{U^{\ast }}^{2}}$, we take $\max \{\hat{\xi}_{1}-c_{1-\alpha _{1}/2}\times s_{\hat{\xi }_{1}},\hat{\xi}_{2}-c_{1-\alpha _{1}/2}\times s_{\hat{\xi}_{2}}\}$. Here $ s_{\hat{\xi}_{1}}$ and $s_{\hat{\xi}_{2}}$ are the standard errors of $\hat{ \xi}_{1}$ and $\hat{\xi}_{2}$, and $c_{1-\alpha _{1}/2}$ is the $1-\alpha _{1}/2$ quantile of $\max \{\eta _{1},\eta _{2}\}$, where $\left( \eta _{1},\eta _{2}\right) $ are jointly normal with unit variances and correlation $\hat{\rho}_{\hat{\xi}_{1},\hat{\xi}_{2}}$, and $\hat{\rho}_{ \hat{\xi}_{1},\hat{\xi}_{2}}$ is an estimator of the correlation between $ \hat{\xi}_{1}$ and $\hat{\xi}_{2}$ (e.g., see RomanoWolf2005Ecta). By a standard argument, the confidence interval for $\sigma _{U^{\ast }}^{2}$ given by

has asymptotic coverage at least $1-\alpha _{1}$ for the true $\sigma _{U^{\ast }}^{2}$. In the numerical illustrations we take $\alpha _{1}=\alpha /10$.

Step 2. First, the standard $CI_{1-\left( \alpha -\alpha _{1}\right) }^{PE_{j}(h)}\left( \sigma _{U^{\ast }}^{2}\right) $ is $\left[ l_{1-\left( \alpha -\alpha _{1}\right) }^{PE_{j}(h)}\left( \sigma _{U^{\ast }}^{2}\right) ,u_{1-\left( \alpha -\alpha _{1}\right) }^{PE_{j}(h)}\left( \sigma _{U^{\ast }}^{2}\right) \right]$ constructed by adding and subtracting $z_{1-(\alpha -\alpha _{1})/2}\times s_{\widehat{PE} _{j}(h,\sigma _{U^{\ast }}^{2})}$ from $\widehat{PE}_{j}(h,\sigma _{U^{\ast }}^{2})$. The standard error $s_{\widehat{PE}_{j}(h,\sigma _{U^{\ast }}^{2})} $ of $\widehat{PE}_{j}(h,\sigma _{U^{\ast }}^{2})$ can be computed using the delta method. Then we can construct $CI_{1-\alpha }^{PE_{j}(h)}$ as

where the minimum and maximum are easily calculated using univariate numerical optimization over $\sigma _{U^{\ast }}^{2}$. By the standard Bonferroni argument, the confidence interval $CI_{1-\alpha }^{PE_{j}(h)}$ has asymptotic coverage of at least $1-\alpha $ for the true partial effect $ PE_{j}(h)$.

The constructed confidence interval is asymptotically valid as long as (i) the first step confidence interval $CI_{1-\alpha _{1}}^{\sigma _{U^{\ast }}^{2}}$ covers the true $\sigma _{U^{\ast }}^{2}$ with probability at least $1-\alpha _{1}$ asymptotically, and (ii) the delta method applies to $ \widehat{PE}_{j}(h,\sigma _{U^{\ast }}^{2})$ for the true $\sigma _{U^{\ast }}^{2}$. Both conditions are satisfied provided that the true $\sigma _{U^{\ast }}^{2}$ is bounded away from zero. Note that in this case $ CI_{1-\alpha _{1}}^{\sigma _{U^{\ast }}^{2}}$ is valid even if $\theta _{01}\sigma _{UV}+\sigma _{U}^{2}$ is equal to (or local to) zero, which implies that $CI_{1-\alpha }^{PE_{j}(h)}$ is also valid.

Probit

IV-Probit model is the same as IV-Tobit except $m\left( s\right) =1\left\{ s>0\right\} $ in equation ((ref)). Since Probit is a binary outcome model, in equation ((ref)) one needs to impose a scale normalization, for example, $\left\Vert \theta _{0}\right\Vert =1$ or $\sigma_{U}^2 = 1$. For both of those normalizations, $\theta_0$ and the other reduced form parameters are point identified, and the sharp bounds for $\sigma_{U^*}^2$ are given by Proposition (ref) as before.

For Probit, we are interested in the partial effects of covariates on the probability of $Y_{i}=1$, which are given by $PE_{j}^{\Pr }\left( h\right) $ in equation ((ref)).\ Similarly to the IV-Tobit model, the IV-Probit model can be estimated by MLE or by the two-step approach identical to the one described in Section (ref), except the second step uses the standard Probit estimator in place of the Tobit estimator. Then the bounds on $PE_{j}^{\Pr }\left( h\right) $ are estimated as in equation ((ref)). Confidence intervals for $PE_{j}^{\Pr }(h)$ can be computed exactly as described above.

Average Partial Effects and Other Counterfactuals

In addition to the partial effects at a given $h$, researchers are often interested in the Average Partial Effects

which are the partial effects $PE_{j}\left( h\right) $ averaged with respect to the distribution of $H_{i}^{\ast }=\left( X_{i}^{\ast },W_{i}^{\prime }\right) ^{\prime }$. Define

Note that the distribution of $X_{i}^{\ast }$ is not directly observable due to the EiV. Averaging $PE_{j}\left( h\right) $ with respect to the distribution of the observed $H_{i}=\left( X_{i},W_{i}^{\prime }\right) ^{\prime }$ would result in biased estimators of the APEs. To account for this, in the Appendix we show that these APEs can be calculated as

Hence, for any given value of $\sigma _{U^{\ast }}^{2}$, these APEs can be estimated by

Finally, the estimated bounds on the APEs are obtained by finding the minimum and maximum over $\sigma _{U^{\ast }}^{2}\in \left[ \hat{\underline{ \sigma }}_{U^{\ast }}^{2},\hat{\sigma}_{U}^{2}\right] $. These can be easily computed numerically, since $\widehat{APE}_{j}\left( v\right) $ are smooth functions of a scalar argument $v$. Our two-step approach to inference also applies to the APEs with a minimal modification. The only difference is that in Step 2 the construction of the standard error $s_{\widehat{APE} _{j}}\left( \sigma _{U^{\ast }}^{2}\right) $ as usual needs to account for the sampling variability in both the parameter estimators and the data entering the expressions for the APEs directly.

It is also straightforward to apply our analysis to other counterfactuals, including partial effects and APEs of discrete covariates, as well as to the ordered Probit and two-sided Tobit models. Proposition (ref) and the bounds on $\sigma _{U^{\ast }}^{2}$ remain the same, and hence the estimation and inference procedures remain unchanged, except for different formulas in equations ((ref))-((ref)) corresponding to the counterfactuals of interest.

Numerical Illustration

We simulate a Tobit model with endogenous and mismeasured $X_{i}^{\ast }$, as in equations ((ref))-((ref)), with $W_{i}=1$, $ Z_{i}\sim N\left( 0,1\right) $, $\left( \theta _{01},\theta _{02},\sigma _{V^{\ast }},\sigma _{U^{\ast }},\sigma _{\varepsilon },\pi _{01},\pi _{02}\right) =\left( 2,1,1,1,1,1,0\right) $, and $n=1000$. Figure (ref) plots the results for the Partial Effects (PEs) of $X_{i}^{\ast }$ at the population mean values of the covariates.

We consider a range of designs corresponding to the true values of $\rho _{U^{\ast }V^{\ast }}\in \left[ -0.95,0.95\right] $ on the horizontal axis. For each $\rho _{U^{\ast }V^{\ast }}$, the figure shows the true PE (\textquotedblleft true\textquotedblright ), the true (population) bounds for the PE obtained using Proposition (ref) (\textquotedblleft true bounds\textquotedblright ), as well as the medians over the Monte Carlo replications of the estimated lower and upper bounds on the PE (\textquotedblleft LB\textquotedblright\ and \textquotedblleft UB\textquotedblright ) and the corresponding $95\%$ confidence intervals (\textquotedblleft CI\textquotedblright ) based on the two-step IV-Tobit estimator. The true bounds for the PE are calculated using the point identified parameters $\theta _{0}$, $\sigma _{U}^{2}$, $\sigma _{UV}$, and $ \sigma _{V}^{2}$, see equations ((ref))-((ref)). For comparison, we also include the results for the PE calculated using the standard naive IV-Tobit estimator (\textquotedblleft naive\textquotedblright ) and the corresponding confidence intervals (\textquotedblleft CI naive\textquotedblright ). The \textquotedblleft naive\textquotedblright\ estimators of the partial effects are $\widehat{PE} _{1}^{\text{Tob}}\left( h,\widehat{\sigma }_{U}^{2}\right) $ and $\widehat{PE }_{1}^{\Pr }\left( h,\widehat{\sigma }_{U}^{2}\right) $, i.e., they replace $ \sigma _{U^{\ast }}^{2}$ with $\widehat{\sigma }_{U}^{2}$ in equation ((ref)).

Figure (ref) (a) shows the bounds on $PE_{h_{1}}^{\text{Tob}}\left( h\right) $, while Figure (ref) (b) considers $PE_{1}^{\text{Pr} }\left( h\right) $. As expected, the true PE is between the lower and upper bounds for all values of $\rho _{U^{\ast }V^{\ast }}$. By construction, the \textquotedblleft naive\textquotedblright\ \ IV-Tobit estimator of $PE_{1}^{ \text{Tob}}\left( h\right) $ coincides with one of the bounds. In both panels, the \textquotedblleft naive\textquotedblright\ estimates are below the true values for every $\rho _{U^{\ast }V^{\ast }}$, and the \textquotedblleft naive\textquotedblright\ IV-Tobit confidence intervals do not include the true PE. In this design, the identified set and the confidence intervals for the true partial effects are much wider than those of the \textquotedblleft naive\textquotedblright\ estimator. The relative width of the identified set and the confidence intervals depends on the specific parameter values. In contrast to these simulations, in the example of the next section, the identified set is very narrow and the confidence intervals for the partial effects have width similar to those of the naive estimator.

To gain some intuition about the shape of the bounds in Figure (ref), notice that when $\theta _{01}>0$, measurement error in $X_{i}^{\ast }$ introduces a negative correlation between $X_{i}$ and $U_{i}$. Thus, observing $\rho _{UV}>0$ is only consistent with $\rho _{U^{\ast }V^{\ast }}>0$, but not with $\rho _{U^{\ast }V^{\ast }}\leq 0$, i.e., implies positive correlation due to the structural endogeneity. On the other hand, observing $\rho _{UV}<0$ can be explained both by the effect of EiV combined with $\rho _{U^{\ast }V^{\ast }}\gtrless 0$, and by $\rho _{U^{\ast }V^{\ast }}<0$ without any EiV. Thus, when $\rho _{UV}<0$ it is harder to disentangle structural endogeneity and EiV. As a consequence, the true and the estimated correct bounds on the PEs in Figure (ref) are wider for the negative values of $\rho _{U^{\ast }V^{\ast }}$.

Empirical Application

We illustrate the proposed methods in the classic application estimating the Tobit and Probit models for women's labor force participation. Using NLSY97 we construct an up-to-date dataset similar to the well known dataset of Mroz1987Ecta, who estimates Tobit and related models to explain married women's hours of work. The data contains 1138 women continuously married in 2018, 79% of which report working non-zero hours. For Tobit, the dependent variable is the number of hours worked (hours), and for Probit, the dependent variable is working at some point during the year ($1\{hours>0\}$). In both models, the covariates are age, education, experience, experience squared, nonwife income in thousands (nwifeinc), number of children less than six years of age, number of children between 6 and 18 years of age, race-ethnicity indicator variables, and an intercept. Nonwife income can be correlated with the unobserved characteristics (structural endogeneity), and income variables are also known to be frequently mismeasured. Spouse's education (speduc) is used as an instrument for nwifeinc. This specification is used in, e.g., Wooldridge2010Book.

Tables (ref)-(ref) contain the results for average partial effects for Tobit. Table (ref) contains the results on average partial effects on expectation, $APE^{\text{Tob}}$, while Table (ref) contains average partial effects on probability, $APE^{\text{Pr}}$. In both tables, the first column (\textquotedblleft Tobit") provides the average partial effects for different covariates estimated by the standard Tobit MLE, where all covariates are assumed to be exogenous. The remaining columns are based on the two-step IV-Tobit estimator, where speduc is used as an instrument for the endogenous nwifeinc. The second column (\textquotedblleft IV-Tobit") contains the naive estimators of the average partial effects, followed by the $95\%$ confidence intervals (column \textquotedblleft CI for IV-Tobit"). Column “[LB, UB] for APE” provides the proposed estimated bounds for the average partial effects that account for both types of endogeneity. The last column contains the corresponding confidence intervals for the average partial effects.

In both Tables (ref) and (ref), we observe that the confidence intervals for the correct average partial effects are only slightly wider than the naive ones of the IV-Tobit. In particular, using the correct inference approach does not change any of the conclusions about the effects of the variables being statistically significant.

Table (ref) contains the corresponding results for Probit. Again, the confidence intervals for the correct partial effects are not much wider than the naive ones, and as before, using the correct inference approach does not change any of the conclusions about the effects of the variables being statistically significant.

Overall, we find that the proposed valid confidence intervals for the APEs are only slightly wider than those obtained from the naive (and generally invalid) IV-Tobit/IV-Probit estimators. The point estimates of the valid bounds in columns “[LB, UB] for APE” are narrow. We interpret these results as (strong) evidence in favor of practical usefulness of our method. Practitioners often worry that methods providing partial identification of parameters of interest achieve robustness at the cost of producing bounds that are too wide to be useful in practice. Our empirical application shows that this concern need not be an issue for our method.

Relaxing Distributional Assumptions

We have considered the classic IV-Tobit and IV-Probit settings, which require $V_{i}^{\ast }$ and $\varepsilon _{i}$ to be Gaussian in order for $ V_{i}$ and $U_{i}$ to be Gaussian, as in SmithBlundell1986Ecta, RiversVuong1988JoE, and Wooldridge2010Book. However, at least in the model without EiV, the control variable approach does not require $ V_{i}^{\ast }$ to be Gaussian. Likewise, we would like to relax the assumption of Gaussianity on $\varepsilon _{i}$. In this section we propose an approach that weakens the distributional assumptions, while still providing a simple method for computing the identified set for the partial effects of interest. We focus on the IV-Tobit settings.

We model the joint distribution of $U_{i}^{\ast }$ and $V_{i}^{\ast }$ as a mixture of $J$ bivariate normals. Specifically, the joint p.d.f. of $U^{\ast }$ and $V^{\ast }$ takes the form

where $p_{V^{\ast },j}>0$ are the mixing weights, and

and $\phi (\cdot ,\cdot ;\mu ,\Sigma )$ stands for the p.d.f. of a bivariate normal with mean $\mu $ and variance-covariance matrix $\Sigma $. Notice that in this parameterization the marginal distribution of $U_{i}^{\ast }$ is $N(0,\sigma _{U^{\ast }}^{2})$ as in the standard Tobit model, whereas the marginal distribution of $V_{i}^{\ast }$ is flexibly modelled as a mixture of $J$ normals. Since $\sigma _{U^{\ast }V^{\ast },j}$ can vary over $j$, this model also allows rich patterns of dependency between $U_{i}^{\ast }$ and $V_{i}^{\ast }$. Similarly, we model the distribution of $\varepsilon _{i}$ by a mixture of $L$ normals:

where $p_{\varepsilon ,\ell }>0$ are the mixing weights, and $\phi (\cdot ;\mu ,\sigma ^{2})$ stands for the p.d.f. of a $N\left( \mu ,\sigma ^{2}\right) $ distribution, and we denote the corresponding c.d.f. by $\Phi (\cdot ;\mu ,\sigma ^{2})$.

Since $\varepsilon _{i}$ is independent from $\left( U_{i}^{\ast },V_{i}^{\ast }\right) $, the joint distribution of $\left( U_{i},V_{i}\right) =\left( U_{i}^{\ast }-\theta _{01}\varepsilon _{i},V_{i}^{\ast }+\varepsilon _{i}\right) $ is a mixture of $J\times L$ bivariate normals, and its p.d.f. is given by

We impose standard constraints $\sum_{j=1}^{J}p_{V^{\ast },j}=1$ and $ \sum_{\ell =1}^{L}p_{\varepsilon ,\ell }=1$. In addition, we need location normalizations on the distributions of $V_{i}^{\ast }$ and $\varepsilon _{i}$ . We follow the standard approach and assume that $E\left[ V_{i}^{\ast } \right] =0$ and $E\left[ \varepsilon _{i}\right] =0$, which are imposed by the restrictions $\sum_{j=1}^{J}p_{V^{\ast },j}\mu _{V^{\ast },j}=0$ and $ \sum_{\ell =1}^{L}p_{\varepsilon ,\ell }\mu _{\varepsilon ,\ell }=0$. Alternatively, it is possible to normalize the medians instead of the means. For example, the restriction $\sum_{\ell =1}^{L}p_{\varepsilon ,\ell }\Phi (0;\mu _{\varepsilon ,\ell },\sigma _{\varepsilon ,\ell }^{2})=1/2$ imposes normalization $\limfunc{median}\left( \varepsilon _{i}\right) =0$, which in particular allows $\varepsilon _{i}$ to be a non-classical measurement error. \footnote{ It is also possible to impose a normalization on the modes of the distributions of $\varepsilon _{i}$ and/or $V_{i}^{\ast }$, although this is less convenient computationally.}

This model naturally generalizes the classic Gaussian model considered in the previous sections. Taking $J=1$ restricts $V_{i}^{\ast }$ to be Gaussian. Taking $L=1$ corresponds to $\varepsilon _{i}$ being Gaussian.

Thus, we consider the IV-Mixed-Tobit model that consists of equations ((ref))-((ref)) and replaces equation ((ref)) with the assumption that

where $\sum_{k=1}^{K}p_{k}\mu _{UV,k}=\left( 0,0\right) ^{\prime }$ and $ \sum_{k=1}^{K}p_{k}=1$.

First, we discuss identification of the parameters of the above IV-Mixed-Tobit model. As in Section (ref), $\pi _{0}$ is immediately identified by the first stage regression of $X_{i}$ on $Z_{i}$ and $W_{i}$. Identification of the remaining parameters is less straightforward and is established by the following proposition. This identification result appears to be new.

Proposition (ref) establishes identification of parameters $\theta _{0},\pi _{0},\left\{ p_{k},\mu _{UV,k},\Sigma _{UV,k}\right\} _{k=1}^{K}$. These parameters can be estimated by the MLE. The log-likelihood can be optimized using the EM algorithm commonly employed for estimation of mixture models. Alternatively, for moderate $K$, it is also feasible to optimize the log-likelihood directly since both the log-likelihood and its Jacobian are available in closed form. From a practical perspective, we do not recommend models with large $K$, because Gaussian mixture models are known to be highly flexible even with relatively small $K$.

Our goal is to provide bounds on the partial effects. To do this, as in Section (ref), the key is to construct the identified set for $ \sigma _{U^{\ast }}^{2}$. Notice that, for every $k$, $\Sigma _{UV,k}$ is equal to $\Sigma _{UV,j\ell }$ for some $j$ and $\ell $ satisfying the restrictions given by equation (ref). Since these restrictions have the same structure as the ones in equation (ref), we can apply the result of Proposition (ref) with a given $\Sigma _{UV,k}$ to construct an identified set for $\sigma _{U^{\ast }}^{2}$ given by

where $\underline{\sigma }_{U^{\ast },k}^{2}$ is computed as in equation ( (ref)) with $\sigma _{U,k}^{2}$, $\sigma _{V,k}^{2}$ and $ \sigma _{UV,k}$ in place of $\sigma _{U}^{2}$, $\sigma _{V}^{2}$, and $ \sigma _{UV}$, respectively. Hence, we can construct an identified set for $ \sigma _{U^{\ast }}^{2}$ by intersecting $\mathcal{I}_{k}$ for $k\in \{1,\ldots ,K\}$, i.e., the bounds for $\sigma _{U^{\ast }}^{2}$ can be constructed as

Once the identified set for $\sigma _{U^{\ast }}^{2}$ is constructed, the rest of the analysis follows as in the Gaussian model in Sections (ref)-(ref). In particular, sharp bounds for the partial effects are constructed as in equation (ref) with $ \mathcal{I}$ as the identified set for $\sigma _{U^{\ast }}^{2}$. That is, the lower and upper bounds for partial effects for the $j^{th}$ covariate, $ PE_{j}\left( h\right) $, are given by

Note that set $\mathcal{I}$ is a closed interval, as in the original IV-Tobit model. Thus, the discussion concerning the implementation of the identified sets for the partial effects that follows equation ((ref)) also applies to the IV-Mixed-Tobit model of this section.

Conclusion

Both structural endogeneity and mismeasurement of covariates are pervasive in economic data. Estimating partial effects and other counterfactuals using such data is an important practical problem. This paper addresses the problem in the classic Probit and Tobit models. The relative simplicity of these nonlinear models allows for transparent analysis and practical solutions. The paper provides simple estimators and confidence intervals that are easy to implement. The paper also shows how the proposed solutions can be extended to settings with non-Gaussian unobservables.

Acknowledgements

Evdokimov gratefully acknowledges the support from the Spanish MICIU/AEI via grants RYC2020-030623-I, funded by MICIU/AEI/10.13039/501100011033 and by “ESF Investing in your future”, PID2019-107352GB-I00, PID2022-140825NB-I00, and Severo Ochoa Programme CEX2019-000915-S. Zeleneev gratefully acknowledges the generous funding from the UK Research and Innovation (UKRI) under the UK government’s Horizon Europe funding guarantee (Grant Ref: EP/X02931X/1).

\nocite{NLSY97}