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Testing identifying assumptions in Tobit models
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\address{$^1$Universidad ORT Uruguay $^2$Monash University, IZA $^3$Iowa State University}
{ Keywords: Tobit models, hypothesis testing, Testable Implications, Instrumental variables.
JEL subject classification: C12, C24, C26, C34.}
Since the seminal work of tobin_estimation_1958, Tobit models have earned attention in economics, business and social sciences.\footnote{According to Google Scholar Tobin's original paper has more than 10000 citations.} tobin_estimation_1958 analyzed household expenditure on durable goods using a regression model that specifically incorporated that expenditure (the dependent variable) cannot be negative. This approach is related to a broader class of censored or truncated regression models, depending on whether observations outside a specified range are lost or censored. When applied researchers are interested in modelling limited dependent variables, potentially with mass accumulation points, the Tobit family of models provides structure to identify parameters of interest, such as the average treatment effect (ATE). Identification relies on three primary sources: (i) instrument exogeneity (or exogeneity of the variable of interest itself), (ii) normality of the model's latent variables, and, in the case of an instrumental variable approach to endogeneity, (iii) the relevance condition for the instrument. While researchers recognize the model's restrictive nature, it remains a valuable tool in the empirical literature. In this paper, we develop a test for the validity of the Tobit model's structure and assumptions, providing three main contributions to the literature. The first is to provide the set of sharp testable equalities that can detect all possible observable violations of the Tobit model. Second, we propose a test for the validity of the Tobit model's identifying assumptions using the sharp equalities that characterize the model to check its falsifiability. Following recent literature, we convert the equalities into conditional moment inequalities and implement the test by existing inferential methods from chernozhukov_intersection_2013. The Tobit model family is used for continuous outcomes with accumulation points. In the case of household expenditure, the outcome may exhibit a zero accumulation due to censoring. When evaluating causality, researchers may be interested in a treatment variable, not necessarily binary, which would result in a large number of moment equalities to be tested. This creates significant challenges to test implementation. We propose a discretization of the space of the treatment and the outcome that balances the computational requirements and data availability for different parts of their joint support. This simplifies the implementation, making it easy to compute and providing an asymptotically valid testing procedure.
The third contribution is to review and propose alternative approaches that can be used when the model is rejected. We explore an alternative path to partially identify the parameter of interest by assuming the monotonicity of selection into treatment. Finally, we provide an empirical example illustrating the methodology's practical relevance. More generally, the current paper contributes to the growing literature on testing identifying assumptions of econometric models. We focus on two main models: (i) the “classic Tobit” model in which the main variable of interest is assumed to be exogenous and (ii) the instrumental variable (IV) Tobit model. In both cases, the proposed test considers all observable violations of the model structure, such as linear index, normality of the latent errors, independence of the treatment, and homoskedasticity; with the addition of the validity of the instrument for the IV-Tobit. Additional results for variants of the Tobit family of models are presented in the Appendix.
There is a vast literature related to testing the validity of Tobit models and their assumptions, which we contribute to. Most of the preceding work focuses on testing a particular assumption or feature of the model while maintaining other assumptions and structures as valid. nelson_test_1981 constructs a Hausman-type test for misspecification of the classic Tobit model (that is, normality, linear index, and homoskedasticity) where the maximum likelihood estimates are compared with method of moment estimates. Nelson's test compares the sample proportion of non-censored observations with the hypothesized probability of being non-censored in the Tobit model. bera_testing_1984 state that the test is equivalent to the Lagrange multiplier (LM) test of the Tobit model against Cragg's model lin_test_1984 and propose an alternative LM test for the normality assumption against other distributions of the Pearson family of distributions while the remaining assumptions are maintained. newey_specification_1987 considers both exogenous and endogenous explanatory variables cases using symmetrically censored least squares estimators to construct specification tests of normality and homoskedasticity assumptions via a Hausman-type specification test. holden_testing_2004 examines several statistics proposed to test the normality assumption in the Tobit (censored regression) model and reinterprets them as a version of the LM (score) test for a common null hypothesis. reynolds_testing_1991 use an information matrix misspecification test to detect violations of the distributional assumptions of the Tobit model.
Other tests include drukker_bootstrapping_2002, which operationalized conditional moment tests developed by newey_maximum_1985 and tauchen_diagnostic_1985 to the case of misspecification of the distribution of the classic Tobit model. With a similar intuition to our framework, their test writes down conditional moment restrictions, which should have zero conditional expected values under the null. Since the model was estimated by maximum likelihood, the assumed data-generating process specifies the moments of disturbances conditional on the covariates to be the ones of a normal distribution. drukker_bootstrapping_2002 use these moment-based methods based on the third and fourth moments of the normal distribution. While intuitively similar, our procedure detects all possible violations of the model, not only those evident from deviations in the third and fourth moments. smith_exogeneity_1986 propose a test of the treatment variable's exogeneity in the IV-Tobit Model by a control function approach exploiting the joint normality of the latent variables. Most of these approaches focus on testing a particular subset of assumptions or consider a specific class of alternatives.
The developments proposed in this paper consider all observable violations of the general Tobit model structure and its assumptions, serving as a useful test for empirical researchers constructing models for censored/truncated data. This work contributes to the growing literature on the testability of the identifying assumptions in various econometric models. Those include sharp tests of the validity in instrumental variable models\footnote{gunsilius_nontestability_2021 proves that instrument validity cannot be tested in the case where the endogenous treatment is continuously distributed.} and local average effects mourifie_testing_2017,kitagawa_test_2015, huber_testing_2015,kedagni2020generalized, sharp tests of the validity of assumption in the regression discontinuity designs arai_testing_2022, sharp tests of the assumptions of the bivariate probit acerenza_testing_2023 and sharp tests in the context of encouragement designs bai2024sharptestableimplicationsencouragement among others. Our testing procedures are connected to kedagni2020generalized, which provides tests for assumptions related to the instrumental variables in the model, while we also consider the implications of the parametric structure on the outcome and latent error structure, which are inherent to Tobit models.
The two papers closest to ours are acerenza_testing_2023 and goff2024testing. The first focuses on bivariate probit models, leading to a finite set of moment equalities as testable restrictions. As mentioned above, this study builds upon their work by tackling the case of the Tobit family of estimators, which provides additional technical and practical challenges given the continuous nature of outcome and treatment, including accumulation points. goff2024testing considers separable parametric instrumental variable models that induce a different set of sharp moment equalities to be tested while our approach focuses on Tobit and IV-Tobit models, which can potentially be non-separable.\footnote{While goff2024testing briefly mentions the possibility of extending their results to non-separable models, they obtain different equalities from ours and do not discuss identification or sharpness in that context. Implementation would also differ from our approach.}
The remainder of the paper is organized as follows. Section (ref) presents the model and identifying assumptions. Section (ref) discusses heuristically the identification of the model's parameters. Section (ref) derives the sharp testable implications. Section (ref) outlines the testing procedure. Section (ref) include simulation evidence about the test's size, while discussions about power are relegated to the Appendix. Section (ref) discusses how to relax the assumptions in case of rejection. Section (ref) provides an empirical illustration revisiting the study of married women's labor supply from lee1995semi to demonstrate the practical implementation and usefulness of the test. Finally, Section (ref) concludes. Additional results and details are collected in Appendices (ref)-(ref).
This section describes two popular “two-part” models for truncated data for which we derive testable implications in Section (ref): the classic Tobit, and the IV-Tobit.\footnote{We omit conditioning on other exogenous covariates ($X$) for simplicity of exposition. The results including covariates are presented in Section (ref).}
Let $Y=\max(0,Y^*)$ be the observed outcome taking values in $\mathcal{Y}\subset \mathbb{R}^{+}$ with $Y^*$ a latent continuous dependent variable taking values in $\mathcal{Y^*}\subset \mathbb{R}$. Both the treatment of interest, $D$, and the instrumental variable, $Z$, can be discrete or continuous and take values in $\mathcal{D} \subset \mathbb{R}$ and $\mathcal{Z}\subset \mathbb{R}$, respectively. We consider the normalized coefficients by the standard deviation of the unobservable error term, which is convenient for the exposition.\footnote{Identification for the normalized and non-normalized parameters is discussed in section (ref).}
The classic Tobit model considers the case in which the researcher is interested in the effect of an exogenous treatment on a non-negative outcome that has a mass point at zero:
where $U$ is an unobservable (latent) error. In addition to the model structure in system (ref), the classic Tobit restricts the distribution of the error term and its relationship to the treatment.
Assumption (ref) states that the treatment of interest is independent of the model's unobservables. Assumption (ref) imposes that the latent error has a normal distribution. To simplify some of the derivations later in the manuscript, we defined the scaled version the model for which the transformed error term would have distribution $N(0,1)$.
We refer to the standardized parameters without the tilde, thus for example $\alpha_1=\frac{\tilde{\alpha}_1}{\sigma}$.
Despite its simplicity, the classic Tobit model can accommodate continuous or discrete treatments, and the normality assumption for the latent error can be replaced with other distributional assumptions that preserve parameter identification.
The IV Tobit considers the case in which the treatment of interest is endogenous, but an instrumental variable is available to identify its effect. It can be traced back to heckman1977dummy,amemiya1979estimation,nelson1978specification among several approaches to consider censored variables with endogeneity.\footnote{In Section (ref), we discuss alternative models and how identification can still be achieved under different assumptions, including about the distribution of the latent error terms.} Let the IV Tobit model be:\footnote{Results for the IV-Tobit model are presented for continuous $D$ but hold for a binary treatment, $D=1\{\gamma_0+\gamma_1 Z-V>0\}$. In that case, the parameters are identified only up to scale.}
Where $U, \tilde{V}$ are the latent structural error terms. Alternatively, in its reduced form representation:
Where $\tilde{\beta}_0=\tilde{\alpha}_0+\tilde{\alpha}_1\tilde{\gamma}_0, \tilde{\beta}_1=\tilde{\alpha}_1 \tilde{\gamma}_1$ and $\tilde{W}, \tilde{V}$ are the reduce form error terms where $\tilde{W}=\tilde{\alpha}_1\tilde{V}+U$. To that model structure, restrictions on the joint distribution of the latent variables and their relationship to the instrumental variable are added.
Assumption (ref)(i) states that the instrumental variable is independent of the model's structural latent variables. The second part of Assumption (ref) is the usual instrument relevance in determining the treatment. This assumption also implies that $Z$ is independent of $\tilde{W}, \tilde{V}$, since $\tilde{W}=\tilde{\alpha}_1\tilde{V}+U$. Assumption (ref) characterizes the distribution of the latent vector of error terms as bivariate normal. Then, $\tilde{W}, \tilde{V}$ also follows a bivariate normal distribution and $\tilde{\alpha}_0, \tilde{\alpha}_1, \tilde{\gamma}_0, \tilde{\gamma}_1$ could be scale normalized so $(W,V)'$ a normalization of $(\tilde{W},\tilde{V})'$ follows a standard bivariate normal distribution with covariance $\rho$, i.e., $
\sim \mathcal N(\mu,\Sigma)$, where $\mu=
$, and $\Sigma=
$.\footnote{{Note that if $U, \tilde{V}$ follow a bivariate normal distribution, $
\sim \mathcal N(\mu,\Sigma)$, where $\mu=
$, and $\Sigma=
$, then $\tilde{W}, \tilde{V}$ follow a bivariate normal distribution, $
\sim \mathcal N(\mu_1,\Sigma_1)$, where $\mu_1=
$, and $\Sigma_1=
\equiv
$. Then the reduced form system can be scale normalized to $W=\frac{\tilde{W}}{\sigma_{\tilde{W}}}, V=\frac{\tilde{V}}{\sigma_{\tilde{V}}}$ which follows a bivariate normal distribution, $
\sim \mathcal N(\mu_2,\Sigma_2)$, where $\mu_2=
$, and $\Sigma_2=
\equiv
$.Define analogously, $\beta_0=\frac{\tilde{\beta}_0}{\sigma_{\tilde{W}}}, \beta_1=\frac{\tilde{\beta}_1}{\sigma_{\tilde{W}}}, \gamma_0=\frac{\tilde{\gamma}_0}{\sigma_{\tilde{V}}}, \gamma_1=\frac{\tilde{\gamma}_1}{\sigma_{\tilde{V}}}$.} } The latter characterization aids the development of the sharp testable implications.
In this section, we provide heuristic arguments for the identification of the parameters of the respective models, expanding on han_identification_2017 and acerenza_testing_2023. These are known in the literature and discussed here for completeness and intuition. The testable implications that are our main result are presented in Section (ref).
Note that,
The equality follows from the model's structure in Equation (ref) and assumptions (ref) and (ref).
Since the standard normal CDF, $\Phi(.)$, is monotonic, we use its inverse to obtain:
Thus, $\alpha_1=\frac{Cov(\Phi^{-1}(1-P(Y=0|D)),D)}{Var(D)}$, and $\alpha_0=E(\Phi^{-1}(1-P(Y=0|D)))-\alpha_1 E(D)$. The coefficients have been normalized by dividing by the latent variable's ($U$) square root of the variance, denoted by $\sigma$. This implies that the ratio of the original parameters is identified since $\frac{\alpha_1}{\alpha_0}=\frac{\frac{\tilde{\alpha}_1}{\sigma}}{\frac{\tilde{\alpha}_0}{\sigma}}= \frac{\tilde{\alpha}_1}{\tilde{\alpha}_0}$. { Recall that,
Then, $f_{U|U<c}(u|U<c)=
$, and $f_{U|U>c}(u|U>c)=
$. Note then that,
where the fourth equality uses independence, and the fifth equality uses the properties of standard truncated normal distributions. Since $\alpha_0,\alpha_1$ are identified, then the inverse Mills ratio is identified,
Thus, }
Which in turn implies (by the property that we can express $X=E(X|W)+e$, where $E(e|W)=0$):
with $e$ mean independent of $D, \lambda(\alpha_0+\alpha_1 D)$. Since $D$ is known and $\lambda(\cdot)$ is identified, we get a three-by-three linear system with a unique solution for $\tilde{\alpha}_0, \tilde{\alpha}_1, \sigma$. In other words, the normality assumption allows identification by leveraging the relationship of the variable of interest ($D$) and the outcome at the accumulation point ($Y=0$) to obtain the scaled parameters, and the relationship of treatment with the distribution of $Y$ (when $Y>0$) to overcome the normalization. However, the reliance on the normality and linearity assumptions underlines the importance of adequately testing the model's structure and assumptions in empirical applications.
Turning the focus to the IV-Tobit model, the first stage is readily identified from the linear structure, $\tilde{\gamma}_1=\frac{Cov(Z,D)}{Var(Z)}$, $\tilde{\gamma}_0=E(D)-\tilde{\gamma}_1 E(Z)$. { Furthermore, $\tilde{V}=D-E(D)+\frac{Cov(Z,D)}{Var(Z)} E(Z)-\frac{Cov(Z,D)}{Var(Z)}Z$, thus $\tilde{V}$ and $\sigma_{\tilde{V}}^2$ are identified. Combining these results implies that we can identify $\gamma_0=\frac{\tilde{\gamma}_0}{\sigma_{\tilde{V}}}$ and $\gamma_1=\frac{\tilde{\gamma}_1}{\sigma_{\tilde{V}}}$.} Now note that calculating $P(Y=0|Z)$ from the reduce form:
{ Where $\beta_0=\frac{\tilde{\beta}_0}{\sigma_{\tilde{W}}}$ and $\beta_1=\frac{\tilde{\beta}_1}{\sigma_{\tilde{W}}}$.}
Similarly to subsection (ref), by inverting the normal CDF we obtain $-\beta_1=\frac{Cov(\Phi^{-1}(1-P(Y=0|Z)),Z)}{Var(Z)}$, $-\beta_0=E(\Phi^{-1}(1-P(Y=0|Z)))+\beta_1 E(Z)$. { The relationship between $\beta_0, \beta_1, \gamma_0, \gamma_1, \sigma_{\tilde{V}}$ identifies $\tilde{\alpha}_0, \tilde{\alpha}_1$ up to scale $\left(\alpha_0=\frac{\tilde{\alpha}_0}{\sigma_{\tilde{W}}}, \alpha_1=\frac{\tilde{\alpha}_1}{\sigma_{\tilde{W}}}\right)$.} To identify $\rho$, let $s=c-\beta_0-\beta_1 z$ and $t=d-\gamma_0-\gamma_1 z$. Then, by using
We can obtain
which by classic results of bivariate normal random variables if we differentiate w.r.t. $\rho$ is:
Where $\phi_{a,b}$ is the bivariate normal probability density, which is positive for any $s,t$. Hence, $\Phi_{W,V}(.,.; \rho)$ is monotonic in $\rho$ and thus invertible. So $\rho$ can be identified.
Recall the previous coefficients are normalized, as noted by wooldridge_econometric_2010 we can identify the original and reduced form covariance matrices, as well as the true coefficients without the normalization. { To explain this in detail, recall that we have identified $\rho$, the covariance between $W,V$ the parameters $\gamma_1,\gamma_0,\beta_1,\beta_0$ and the variance of $\tilde{V}$, $\sigma^2_{\tilde{V}}$. Similarly to the classic tobit model, we can leverage normality of $W$, independence of $Z$, and exploit the reduced form representation of $Y$ to obtain:
Which in turn implies
with $e$ mean independent of $(Z, \lambda(\beta_0+\beta_1 Z))$. Since $Z$ is known and $\lambda(\beta_0+\beta_1 z)$ is identified for any given $Z=z$, we can get a solution for $\sigma_{\tilde{W}}$ as one of the coefficients for the linear projection of $Y$ on $(Z, \lambda(\beta_0+\beta_1 Z))$, identifying the non-standardized reduced form error for $Y$. Finally, since $\beta_0=\frac{\tilde{\beta}_0}{\sigma_{\tilde{W}}}=\frac{\tilde{\alpha}_0+\tilde{\alpha}_1 \tilde{\gamma}_0}{\sigma_{\tilde{W}}}$, $\beta_1=\frac{\tilde{\beta}_1}{\sigma_{\tilde{W}}}=\frac{\tilde{\alpha}_1 \tilde{\gamma}_0}{\sigma_{\tilde{W}}}$ and we have identified $\sigma_{\tilde{V}}, \gamma_0, \gamma_1, \tilde{\gamma}_0, \tilde{\gamma}_1$ and $\sigma_{\tilde{W}}$ we can recover $\tilde{\alpha}_0, \tilde{\alpha}_1$. It now only remains to recover the original variance-covariance structure. Under bivariate normality of the latent variables in the structural model, we can express the following relationships between $U,\tilde{V}$ with $\tilde{W}=\tilde{\alpha}_1U+\tilde{V}, V$ and $W=\frac{\tilde{\alpha}_1U+\tilde{V}}{\sigma_{\tilde{W}}}, V=\frac{\tilde{V}}{\sigma_{\tilde{V}}}$ $
\sim \mathcal N\Bigg(
,
\Bigg)$ \par $
\sim \mathcal N\Bigg(
,
\Bigg)$ \par $
\sim \mathcal{N}\left(
,
\right)$ \par Since we have identified $\tilde{\alpha}_1, \sigma_{\tilde{V}}, \sigma_{\tilde{W}}, \rho$ from the following equations: $$\rho=\frac{\tilde{\alpha}_1 \sigma_{\tilde{V}}^2+\rho_{U\tilde{V}} }{\big(\tilde{\alpha}_1^2\sigma^2_{\tilde{V}}+\sigma^2_U+2\rho_{U\tilde{V}}\tilde{\alpha}_1\sigma_{\tilde{V}}\sigma_U \big)\sigma_{\tilde{V}}} $$ $$\sigma^2_{\tilde{W}}=\tilde{\alpha}_1^2\sigma^2_{\tilde{V}}+\sigma^2_U+2\rho_{U\tilde{V}}\tilde{\alpha}_1\sigma_{\tilde{V}}\sigma_U $$ We can identify $\sigma_{U}$ and $\rho_{U\tilde{V}}$, and thus, recover all the structural parameters. }
This section presents the sharp testable equalities for the models described in Section (ref). Deviations from these equalities imply violations of the “null hypothesis” that the Tobit model (linear latent index, treatment/instrument independence, instrument relevance, and normality) is valid. The equalities are conditional on parameters being identified under the model's assumptions, and a discussion on identification is postponed to Section (ref).
A defining characteristic of Tobit and similar models is the mass accumulation at zero for the distribution of the non-negative outcome, $Y$. Thus, the model's testable implications require characterizing the distribution at both the mass point and beyond it. Starting at the continuous part of the distribution of $Y$, the conditional probabilities of $c_0 \leq Y\leq c_1$ are observed for $c_1,c_0>0$. For any value of the treatment variables $d \in \mathcal{D}$:
where $\Phi(.)$ is the standard normal CDF. The first equality follows from the law of total probability. The second through fourth equalities follow from the model's structure described in Equation (ref), namely $P(Y>0, Y^*< 0|D=d)=0$ and $Y^{*}=Y$ for $Y^{*}>0$. The fifth one is given by the latent linear model structure of $Y^*$, and finally, the last step follows from assumptions (ref) and (ref). Recalling the accumulation point, the observed event of $Y=0$ has a probability,
The equalities described in equations (ref)-(ref) fully characterize the distribution of $Y$ conditional on $D$, connecting the probabilities in the observed data to those implied by the Tobit model. We collect these results more formally in Theorem (ref).
The proof for the equalities follows from the discussion above, and further details about sharpness are presented in Appendix (ref). Then, the equalities ((ref))-((ref)) are sharp testable implications for the validity of the classic Tobit Model in (ref) coupled with assumptions (ref)-(ref). They serve as the basis for the test procedure described in Section (ref).
We turn our attention to the IV Tobit case and propose testable implications that can be used to test the model described in (ref) and the associated assumptions (ref) and (ref). The observed data includes $(Y,D,Z)$, and we characterize the joint distribution of $(Y,D) \in \mathcal{Y} \times \mathcal{D}$ conditional on the instrument, $Z$, to obtain the model's testable implications. The mapping from observed probabilities to their corresponding quantities implied by the IV Tobit model requires jointly evaluating the continuous (interior) support for the outcome and treatment variables, as well as the accumulation points and distribution tails conditional on $Z$. Consider any $0<c_0<c_1,d_{0}<d_{1}$, and $z \in \mathcal{Y} \times\mathcal{D} \times \mathcal{Z}$,
where $\Phi_{W,V}(w,v;\rho)$ denotes the joint c.d.f. of $(W,V)$, a standard bivariate normal with coefficient of correlation $\rho$. Similarly, considering the case that the observed $Y$ equals zero,
Then, the equalities ((ref))-((ref)) are sharp testable implications for the validity of the instrumental variable Tobit Model in equations ((ref)) coupled with assumptions (ref)-(ref). We collect these results more formally in Theorem (ref).
The proof for the equalities follows from the discussion above, and further details about sharpness are presented in Appendix (ref). They serve as the basis for the test procedure described in Section (ref).
Similarly to the discussion in Proposition (ref), we conjecture that the approaches proposed in Theorem (ref) can be adapted to models in which the researcher is willing to assume a known joint distribution for the latent errors, $(U,V)$, replacing the normal distribution, as long as the model is identified.
To test the sharp equalities, we rewrite each of these equalities as two inequalities mourifie_testing_2017, acerenza_testing_2023. For a concrete example, we first note that $$P(c_1 \leq Y|D=d) =1-\Phi\left(\frac{c_1}{\sigma}-\frac{\tilde{\alpha}_0}{\sigma}-\frac{\tilde{\alpha}_1}{\sigma} d\right) \iff E\left[1\{c_1 \leq Y \}-1+\Phi\left(\frac{c_1}{\sigma}-\frac{\tilde{\alpha}_0}{\sigma}-\frac{\tilde{\alpha}_1}{\sigma} D\right)|D\right]=0.$$ We rewrite each of these moment equalities implied by the restrictions on the empirical distribution as moment inequalities $$E\left[1\{c_1 \leq Y \}-1+\Phi\left(\frac{c_1}{\sigma}-\frac{\tilde{\alpha}_0}{\sigma}-\frac{\tilde{\alpha}_1}{\sigma}D\right)|D\right]\leq 0, E\left[-1\{c_1 \leq Y \}+1-\Phi\left(\frac{c_1}{\sigma}-\frac{\tilde{\alpha}_0}{\sigma}-\frac{\tilde{\alpha}_1}{\sigma} D\right)|D\right]\leq 0.$$ In doing this for all the restrictions on the empirical distribution, we can implement a test relying on existing intersection bounds inferential methods such as chernozhukov_intersection_2013, which is specifically suited to test conditional moment inequalities.
Note that the equalities for the classic Tobit hold for any pair of constants $(c_{0},c_{1})$, and the ones from the IV Tobit hold for pairs of $(c_{0},c_{1})$ and $(d_{0},d_{1})$. We propose a partition of $\mathcal{Y}\times\mathcal{D}$ to test sufficient conditions of these sharp equalities. Rejection of any of the null hypotheses that the equalities hold at these particular levels implies violations of the model.
The sharp testable equalities for every $c_{0},c_{1} \in \mathcal{Y}$ are given by equations ((ref)) and ((ref)). For a partition of the support of $Y$ into $K$ arbitrary chosen sets $C_k=(0,c_k)$ such that $C_k \in C_{k+1}$, the following set of sufficient inequalities are, for some chosen values of $c_{k},c_{k+1} \in \mathcal{Y}$, related to the components of equations ((ref))-((ref)).
The formulation of the inequalities considered will depend on each partition's location on the support of the outcome variable $Y$. Let $c_{1}=0$ and $W_{k}$ be
The intersection bounds framework considers the following $2(K+1)$ inequalities
We can write more compactly as
where $\theta_k(D)$ collects all the inequalities being tested. The decision rule for the test is given by chernozhukov_intersection_2013, we reject $H_{0}$ if
where $\hat{\theta}(D,k)$ is a nonparametric estimator for $\theta_{k}(D)$, $\hat{s}(D,k)$ its standard error, and $\kappa_{1-\alpha}$ is a critical value at the significance level $\alpha$.
For the instrumental variable Tobit model, the continuous support for both the outcome and treatment poses challenges to the implementation of the test.\footnote{Note that a discrete treatment can also be accommodated as an intermediate step between the current derivation and the derivations from acerenza_testing_2023.} The sharp testable equalities for every $c_{0},c_{1} \in \mathcal{Y}$ and $d_{0},d_{1} \in \mathcal{D}$ are given by equations ((ref))-((ref)).
Consider a partition of the support of $Y$ into $K$ arbitrary chosen sets $C_k=(0,c_k)$ such that $C_k \in C_{k+1}$ and of the support of $D$ into $Q$ arbitrary chosen sets $D_q=(0,d_q)$ such that $D_q \in D_{q+1}$, the following set of sufficient inequalities are related to the components of ((ref))-((ref)). Analogous to the classic Tobit case, the formulation of the inequalities considered will depend on each partition's location on the joint support of the outcome and treatment variables. Let $W_{kq}$ be given by,
Where we used the simplifying notation defined in equations ((ref))-((ref)) for each partition of the support for $Y$ and $D$. The intersection bounds framework considers the following $2(K+1)(Q+1)$ inequalities
We can write more compactly as
where $\theta_kq(Z)$ collects all the inequalities being tested. The decision rule for the test is given by chernozhukov_intersection_2013, we reject $H_{0}$ if
where $\hat{\theta}(Z,k,q)$ is a nonparametric estimator for $\theta_{kq}(Z)$, $\hat{s}(D,k,q)$ its standard error, and $\kappa_{1-\alpha}$ is a critical value at the significance level $\alpha$.
To implement the test within the chernozhukov_intersection_2013 intersection bounds inferential method, we use the CLR Stata package described in chernozhukov_implementing_2015. The parameters in the relevant model, for example, $\beta_{0}, \beta_{1}, \gamma_{0}, \gamma_{1}, \rho$, are replaced by their maximum likelihood estimators (MLE), and asymptotic validity of this “plug-in” test follows from the argument described by acerenza_testing_2023. Some additional details on the test implementation are discussed in Section (ref).
The proposed procedure can be extended to include exogenous covariates, $X$, within the linear index model in (ref). Then, the testable equalities for classic and IV Tobit models can be derived with the additional conditioning on $X$. The test with covariates could be implemented by generating a partition of the covariate space, say in $J$ grids, similar to the partition of the exogenous variable used in Section (ref). For every grid in the partition, one computes the test statistic (ref) or (ref) for the classic or the IV-tobit, respectively. Then obtain critical values that account for multiple testing via a Bonferroni correction. In particular, for every grid, set the critical value at the significance level $\frac{\alpha}{J}$, namely $\kappa_{1-\frac{\alpha}{J}}$. \footnote{A theoretically interesting approach would partition the joint support of all the exogenous variables and compute the test statistics across all parts of the grid. However, this approach would entail significant implementation challenges since the statistical package for chernozhukov_implementing_2015 allows for only one exogenous covariate.} This procedure can be cumbersome when there are many covariates or when they are continuous.
Another route that is less computationally intensive follows acerenza_testing_2023. We illustrate it for the $IV$-tobit case, and a similar logic could be applied to the classic Tobit case. Thus, with covariates added to the linear index function, we have,
We extend Assumption (ref) to formalize covariates' exogeneity:
Under Assumptions (ref)-(ref), by a similar argument to that used in Section (ref), any given generic testable implication becomes
for all $z \in \mathcal Z$ and $x \in \mathcal X$, where $x$ can be a vector. Equivalent conditions for the case that $Y=0$ and other parts of the support of $Y$ and $D$ can be similarly obtained.
Since the test is nonparametric, we face challenges when $X$ is high-dimensional, especially with continuous covariates. Another operational limitation is that the Stata's clrbound package only allows for one conditioning variable at a time. For those practical reasons, we propose implementing a weaker, non-sharp, version of the testable equalities. Continuing with the example of Equality (ref), we can integrate over the covariates $X$ (or alternatively, $Z$), taking advantage of the fact that $\mathbb E[W\vert Z=z, X=x]=0$ implies $\mathbb E[W\vert Z=z]=0$ and $\mathbb E[W\vert X=x]=0$ for all random variables $W$.
Implementation becomes very similar to the IV-Tobit discussed above by redefining the simplifying notation in equations ((ref))-((ref)), with the only difference being the inclusion of $X$ on the linear indexes in $\Phi_{W}(\cdot)$ and $\Phi_{W,V}(\cdot,\cdot,\cdot)$. For example, $\Phi^{1}_{c_{1},c_{0},d_{1},d_{0}}$ is given by equation (ref) and similarly for the other terms. Then, compute the new $W_{kq}$ in the same manner as in Section (ref), and the intersection bounds framework considers the similar $2(K+1)(Q+1)$ inequalities, based on the partition of the support of $Y$ and $D$.
This is the test procedure that we implement for the empirical application in Section (ref).
Under Assumption (ref), one could base the test on conditioning on a particular covariate $X_{C}$ instead of $Z$ by integrating the sharp Equality ((ref)) over $Z$ and then implementing the intersection bounds procedure using $X_{C}$ as the sole conditioning variable.\footnote{Assumption (ref) constraints the relationship between the covariates and the latent error terms in such a way that one could combine the information on all $X$ and $Z$ in an index and construct non-sharp testable equalities that would hold conditional on this index. For example, having $\sup_{t}E[W_{kq}|Z+X_1+...+X_C=t]= 0$, for $k=0,\dots,K;q=0,\dots,Q$.}
In this section, we provide simulation exercises for the proposed tests. The testing procedure described relies on testing sharp equalities that should hold for any arbitrary partition of the outcome, treatment and instrument support. When both the outcome and treatment are continuous, evaluating all possible equalities is technically challenging. We focus on a non-sharp set of the equalities by evaluating them at different grid partitions of their support, in a similar spirit to honore2020selection which is particularly well suited for continuous supports. Naturally, this choice makes the test less powerful as we don't consider the continuum of equalities derived in Section (ref), but is justified by the ease of implementation of the procedure based on intersection bounds and the performance of the test on the simulations below.
For the simulations related to the classic Tobit, we partition the support of the observed outcome variable into the accumulation point ($Y=0$) and four quartiles on the (untruncated) positive range, while for the IV-Tobit case we also partition the support of the treatment variable ($D$) to create a grid based on both the outcome and treatment.\footnote{Note that if the treatment variable was discrete, the respective grid points could be set naturally to the possible countable values $D$ takes.} The choice of the number and location of the partitions/evaluation points balances the implementation computational requirements, data availability for different parts of the joint support of the outcome, treatment and exogenous instrumental variable. For the procedure in Section (ref) to be feasible, we must have data on both the outcome and the exogenous variables within each partition. Using the empirical quantiles of the non-truncated outcomes to determine the partitions guarantees a reasonable number of observations for each grid part. Larger sample sizes might allow finer partitions for the outcome support. Still, the added computational requirements created by an increased number of equalities being checked, coupled with the larger datasets, can substantially increase computing time.\footnote{The simulations presented in Section (ref) have limited sample sizes and a relatively small number of equalities being tested due to the long-running time and computational constraints when repeating the test procedure thousands of times.}
When implementing the intersection bounds in STATA using the package clrbound chernozhukov_implementing_2015, the researcher must determine the range of values of the exogenous variable for which each equality will be evaluated. To guarantee the feasibility of the procedure, we adjust the evaluation points for $D$ ($Z$) to the first and $99^{th}$ percentiles of the exogenous variable in each partition of the support for the outcome $Y$.\footnote{See details on implementation on the simulation replication STATA code.}
In evaluating the finite sample performance of the proposed tests, we consider a continuous treatment $D$ following the data generating process described in Equation (ref).
Where $\mathbf{0}_{p}$ is a $p \times 1$ vector of zeroes. The parameter $\rho$ determines the intensity of dependence between the latent variables jointly determining the treatment and outcome. Under the treatment exogeneity condition described in Assumption (ref), $\rho=0$. Table (ref) presents the empirical test sizes for the Classic Tobit test this scenario, for different significance levels $\alpha$. The results indicate that while the test over rejects the null hypothesis for small to mid-sized samples, the test's empirical coverage approaches its desired nominal benchmark as samples larger than 5,000 are used.
To consider the test's performance under violations of the exogeneity assumption, we modify the DGP in Equation (ref) with different values for $\rho$, reflecting various degrees of treatment endogeneity. Larger values of $\rho$ produce more acute violations of the null hypothesis. Table (ref) presents the results for the Classic Tobit test. As expected, the power of the test increases with larger $\rho$ and bigger sample sizes.
To consider a violation with respect to the violation of the normality of errors. We consider a modified DGP from Equation (ref), by seting $(V, Z)^\top \sim \mathcal{N}(\mathbf{0}_2, \mathbf{I}_2)$ and $U\sim F(\cdot)$. Table (ref) presents the power results when $F(\cdot)$ is uniform, lognormal, and t-student distributions. As expected, the rejection rate for the t-student test with 80 degrees of freedom is close to the nominal size of the test, since it represents a very mild violation of the normality assumption. In the other three cases, which deviate significantly from the normality assumption, the test rejects the null hypothesis in most instances. This example demonstrates that the proposed test is effective in identifying violations of the distributional assumptions in the classic Tobit model.
Naturally, researchers concerned about treatment endogeneity should consider the IV-Tobit model and implement the test of its identifying assumptions proposed in Section (ref). Table (ref) presents the empirical coverage for the test of the IV-Tobit model for different levels of treatment endogeneity ($\rho=\{0, 0.5, 0.8\}$) for sample sizes 5,000 and 8,000. The test produces adequate empirical coverage, in line with the results for the classic Tobit test.
When the test proposed in Section (ref) rejects the null hypothesis of the model's validity, researchers must pursue alternative models and less restrictive assumptions to learn confidently about the parameters of interest.
There is a vast literature on alternatives to the Tobit Model that can be implemented in the presence of censored dependent variables. Most approaches consider changes or relaxations of one of the two main assumptions associated with the model. The first assumption is the parametric distribution of the error terms and latent index form connecting treatment (and covariates) to the outcome. The second assumption is the exogeneity of the treatment or potential instrument. Here, we provide a non-exhaustive survey of existing work.
cragg1971some maintains the normality of the errors, linearity of the index and exogeneity but relaxes the way censoring occurs in comparison to the latent structure of the censored outcome. Specifically, while the latent outcome is still modelled by $Y^*=\alpha_0+\alpha_1 D+U$, they allow for the censoring to depend on a different linear index, $P(Y^*>0)=P(\gamma_0+\gamma_1 D+e)$, increasing the model's flexibility.
POWELL1984303 relaxes the parametric structure of the errors while maintaining the latent linear index and treatment exogeneity, and estimates the parameters of interest by least absolute deviations. powell1986symmetrically also maintains linearity of the latent index and treatment exogeneity, but relaxes normality by imposing symmetrical distributions to the latent errors, which leads to estimation by symmetrically censored least squares. newey1987efficient relaxes exogeneity of the treatment, relying on normality and an instrumental variable to identify the model, which is estimated by generalized least squares.
honore1994pairwise relax linearity and mean independence of the unobservable with respect to the treatment to exploit the idea that, although $Y^*_i-\alpha_0-\alpha_1D_i$ is not mean-independent of $D_i$, one can trim any pair of residuals $Y^*_i-\alpha_0-\alpha_1D_i$ and $Y^*_j-\alpha_0-\alpha_1D_j$, and the trimmed residuals are independent and identically distributed conditional on $D_i,D_j$. They estimate the model by identically censored least absolute deviations and identically censored least squares (ICLS). das2002estimators estimates a model using symmetrically censored least squares that relaxes exogeneity of the treatment and normality of the errors. To achieve that they rely on instrumental variables, linearity of the mean of the structural error conditional on the reduced form error, and mean independence of the reduced form error.
blundell2007censored proposes a control variable approach that relaxes exogeneity and normality but maintains the latent linear structure ($\alpha_0+\alpha_1 D+U$). Crucially, they impose that the distribution (or quantiles) of the latent error conditional on the treatment and instrument is only a function of the control variable $V=D-\pi(Z)$, which isolates the endogenous variation on the treatment.\footnote{This assumption is weaker than independence of all errors and instruments since it does not impose $V$ independent of $Z$ but is neither stronger nor weaker than independence of $U$ and $Z$, since it permits $Z$ to affect $U$ through $V$.} This allows them to estimate the effect of the treatment by censored quantile instrumental variable regression augmented by a control variable given by the quantiles of $U$ conditional on $V$ at the quantile of interest. In a similar spirit, chernozhukov2015quantile focuses on conditional quantile functions and flexible approaches to estimate the control variable in the first stage.
Finally, chesher2023iv provides partial identification results for a general alternative by relaxing the exogeneity of the treatment and instrument, linearity of the latent index and imposing no parametric structure of the error term. They characterize the identified set for the parameters of interest following the Generalized Instrumental Variables framework chesher2017generalized, relying on the assumption that the relationship of $Y^*$ to treatment and errors is continuous and monotonic in the errors. Their approach uses the residual sets associated with the structure of the latent function and conditional probability of the error term given potential instruments.
In this subsection, we present an approach that partially identifies the effect of an endogenous treatment variable by replacing the normality and exogeneity assumptions with a monotonicity in treatment selection constraint. While less general than chesher2023iv, this approach is easy to implement and could be useful to empirical researchers. Consider the model that maintains linearity (or a known structure of $Y^*$ up to a finite number of parameters),
As an alternative to treatment exogeneity and normality, consider a constraint on the direction of the endogenous relationship between the treatment and the unobservables that affect the outcome.
Assumption (ref) is common in the partial identification literature jiang2014monotone,manski1997monotone,manski2000amonotone, okumura2014concave. In this context, we restrict the latent selection to be monotonic with respect to the treatment. This assumption is embedded in the classic Tobit model since the inverse mills ratio, $\lambda(\cdot)$, is monotonic (and decreasing) in the treatment variable. Furthermore, independence between $D$ and $U$ restricts the sign of the coefficient of the selection term directly, as the derivative of the inverse mills-ratio is $\lambda^{\prime} (\alpha_0+\alpha_1 D) \alpha_1$. Thus, without imposing independence or a parametric latent structure, we maintain a relevant property of the Tobit model that aids identification. Since it is not as restrictive as imposing a parametric structure and independence, we can only partially identify the parameter of interest. Under Assumption (ref) and the model described in equations ((ref)), treatment and outcome are not independent. Note that,
Then, for any two $d>d^*$ we have, by Assumption (ref):
which implies a lower bound on the parameter interest. For a binary treatment $D\in \{0,1\}$ the lower bound is, intuitively, the difference in average outcomes between treated and untreated individuals away from the mass point at zero:
The bound can be more informative in the case of a continuous or multi-valued treatment. If $\Gamma(D)$ is differentiable we have:
Since the inequality holds for any $d$ in the continuous case or for any $d,d^{*}$ for multi-valued discrete treatment, the linear index structure with constant parameters implies that tighter bounds for $\alpha_{1}$ are given by the maximum value of $\frac{\partial E(Y|D=d,Y>0)}{\partial d}$ across all possible points in the support for $D$. Analogous results with the inequalities reverted can be derived for any $d>d^*$, as we have $\Gamma(d)>\Gamma(d^*)$.
One-sided simple confidence regions can be computed based on these outer sets of the treatment effect. One can estimate $ E(Y|D=d,Y>0)$ using its sample analogs and rely on their asymptotic normality. For example, let the estimators be given by $\widehat{E}(Y|D=d,Y>0)$. By the continuous mapping theorem, $ \frac{\widehat{E}(Y|D=d^*,Y>0)-\widehat{E}(Y|D=d,Y>0)}{d^*-d}$ is asymptotically normal. Thus, a one-sided confidence interval for $ \frac{E(Y|D=d^*,Y>0)-E(Y|D=d,Y>0)}{d^*-d}$ can be computed via bootstrap, which implies a conservative estimate for the lower bound for $\alpha_1$.
A bootstrap procedure could be used for a nonparametric estimator of $\frac{\partial E(Y|D=d,Y>0)}{\partial d}$ using a local polynomial regression. In this case, the estimator is asymptotically normal and converges at a nonparametric rate that depends on the bandwidth $h$. Recent developments in calonico2018effect, calonico2022coverage for estimation and optimal coverage error bandwidth and kernel selection methods that are nonparametric robust bias-corrected (RBC) can be used through their convenient implementation using the package nprobust calonico2019nprobust. Alternatively, since $\alpha_1 > sup_{d} \frac{\partial E(Y|D=d,Y>0)}{\partial d}$, one could consider obtaining confidence regions for $\alpha_1$ using a CLR approach similar to that described in Section (ref) by considering nonparametric estimates of $\frac{\partial E(Y|D=d,Y>0)}{\partial d}$ or its discrete counterpart.\footnote{We thank an anonymous referee for this suggestion.}
In this section, we implement the proposed test to the data from lee1995semi.\footnote{Data availability statement: The data that support the findings of this study are openly available in the Journal of Applied Econometrics Data Archive at http://dx.doi.org/10.15456/jae.2022313.1130270920.} Using the 1987 cross-section of the Michigan Panel Study of Income Dynamics, the authors study the responses of married women's labor supply ($Y$) - measured in hours per year - to hundreds of dollars in “other” household income ($D$), which is endogenous. The instrumental variables explored are dummy variables for the husband's occupation ($Z$), which implies the restrictive identifying assumption that the wife's labor supply is affected by the husband's occupation only through their income. Following the original study, we add other covariates ($X$) in the linear index for both the outcome and treatment equations, controlling for factors that could impact women's labor supply. Those include a quadratic on the person's age, their years of completed education, the number of children coded in three categories (children up to 5 years old, ages 6 to 13, ages 14 to 17), the local unemployment rate in percentage points, and indicators for race (0 if white, 1 otherwise), homeownership (1 if owner, 0 otherwise) and if the couple has a mortgage on their home (1 if yes, 0 otherwise).
Our empirical illustration considers as an instrument the binary variable indicating if the husband's occupation is classified as manager or professional.\footnote{The 1987 PSID uses 3-digit occupation codes from the 1970 U.S. Census, and this dummy variable seems to include workers listed in the categories “1-195 Professional, Technical, and Kindred Workers,” and “201–245 Managers and Administrators, except Farm.”}
Following lee1995semi, we proceed with the analysis focusing on the data for married couples with non-negative family total income or “other” household income and where the wife was of working age (18-64) and not self-employed. These selections results in 3,277 observations, for which 26 percent of wage observations are censored. Table (ref) presents the estimates obtained using the IV Tobit model.
The first column presents the first-stage estimates indicating the relevance of the potential $IV$. The second column presents the structural equation reflecting the effect the treatment variable has on the outcome equation. The parameter $\rho$ shows evidence of no correlation between the unobservables driving the “other” household income and hours worked after controlling for covariates. The estimated coefficient of interest indicates that other household income negatively affects the wife's labor supply after conditional on the household characteristics. In particular, for women working positive hours, an increase of one thousand dollars in household income from other sources is estimated to reduce hours worked by 9.7 hours per year. The direction of the impacts at the intensive margin of hours worked follows intuitive patterns and is qualitatively similar to those in lee1995semi. However, the model is rejected at conventional significance levels when we test for the IV-Tobit model's validity, ($\hat{\theta}_{0.99}= 0.1335>0$, $\hat{\theta}_{0.95}= 0.1368>0$, and $\hat{\theta}_{0.90}=0.1385>0$). This indicates that the assumptions underlying the IV-Tobit model are not compatible with the empirical distribution of the data, and caution is needed when relying on the results.
Given the rejection of the IV-Tobit model in this case, we relax the distributional and exogeneity of treatment assumptions, and construct lower bounds on the treatment effect under the MTS assumption and latent linear index only. Assumption (ref) imposes that average unobservables affecting women's preferences related to hours worked away from home, are monotonically decreasing in characteristics leading to higher income of other sources in the household, such as husband's income. In other words, households that prefer flexible schedules for women might similarly prioritize partner jobs that provide higher income.
The first column of Table (ref) repeats the estimate for $\alpha_{1}$ from Table (ref). The second column reports the estimated lower bound for $\alpha_{1}$, obtained under Assumption (ref) and latent linear index only based on the nonparametric estimate of the average derivative of the conditional expectation for yearly hours worked with respect to the household income from other sources. As described in Remark (ref), we opt to use the estimate for the average derivatives due to instability of the estimated derivatives across the different values of covariates and treatment. This is a conservative approach regarding the bounds for $\alpha$. Inference for the average derivative can be obtained by bootstrapping. Even after relaxing the normality of errors and treatment exogeneity, the lower bound for the size of the effect of having higher household income from sources other than the wife's labor on their labor supply indicates an effect larger than -4.19 hours worked per year for married women. Hence, we can rule out annual reductions of more than 4.2 hours in female labor supply for every one thousand dollars in other household income, but cannot reject that the effect is zero or positive.
In this paper, we develop sharp testable equalities for the classic Tobit and IV-Tobit models that can detect all observable violations of the model's assumptions. The results are shown to extend to many other popular Tobit-type “two-part” models. By converting these sharp equalities into conditional moment inequalities, we propose a testing procedure that detects violations of the Tobit model assumptions on a grid on the joint support of the outcome (and treatment) variables, leveraging inference results from chernozhukov_intersection_2013 and the implementation from chernozhukov_implementing_2015. Simulation results suggest the test performs well for reasonably sized samples (larger than 5000 observations). The test is conservative for smaller samples, over-rejecting the null hypothesis of model validity. Simulations indicate that the test is powerful to detect violations of the exogeneity assumption for the treatment/instrument that affect the point estimates and inference. Finally, the proposed test exhibits good performance for violations of the distributional assumptions about the error structure. We provide a brief review of existing models that could be implemented under weak/alternative assumptions when the Tobit model is rejected. Furthermore, we propose a simple model that partially identifies the parameter of interest by relying solely on linear index and monotone treatment selection restrictions, a standard assumption from the partial identification literature manski2000amonotone. We illustrate our methods on data from lee1995semi. We replicate qualitatively the results in the original paper and the proposed test for validity of the IV-Tobit model rejects the null hypothesis in this empirical application. We estimate our proposed lower bound for the effect of household income from sources other than the wife's labor on their labor supply, which does not rely on the normality of latent errors or treatment exogeneity. While we can rule out that an extra 1,000 dollars in other household income reduce female labor supply by more than 4.2 hours per year, we cannot rule out that the effect is zero.