Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
17,327 characters · 4 sections · 7 citation commands
Some Finite-Sample Results on the Hausman Test
This paper investigates the endogeneity test for potentially endogenous regressors in linear instrumental variable (IV) models that utilizes the control function (CF) approach. Specifically, we focus on the following model:
where $x\equiv(y_{1}^{\top},z_{1}^{\top})^{\top}$, $y_{1}$ represents potentially endogenous regressors, $\varepsilon$ and $v$ are unobserved error terms in the structural and reduced-form equations respectively, and $z\equiv(z_{1}^{\top},z_{2}^{\top})^{\top}$ represents exogenous variables which satisfy:
Since $z_{2}$ is excluded from the structure equation ((ref)) and satisfies the orthogonality conditions in ((ref)), it serves as instruments for $y_{1}$.
One widely used method for testing the endogeneity of $y_{1}$ is the Hausman test hausman1978 . This test compares the ordinary least squares (OLS) estimator $\hat{\beta }_{ols}$ of $\beta$ against the two-stage least squares (2SLS) estimator $\hat{\beta}_{2sls}$. The test rejects the null hypothesis that $y_{1}$ is exogenous if the difference between $\hat{\beta}_{ols}$ and $\hat{\beta }_{2sls}$ exceeds a certain threshold determined by the significance level of the test.
As an alternative to the Hausman test in the IV approach, we can also apply the CF approach to obtain a consistent estimator of $\beta$ and test for the endogeneity of $y_{1}$. Specifically, we run an OLS regression of the following model
where we replace $v$ by the estimated residual $\hat{v}$ in the OLS regression of the reduced-form equation ((ref)), and obtain estimators $\hat{\theta }_{cf}$ and $\hat{\rho}_{cf}$. Then we test the endogeneity of $y_{1}$ using the Wald test for the null hypothesis $H_{0}:\rho=0$.
This paper makes several contributions to the existing literature. First, we demonstrate that $\hat{\rho}_{cf}$ is a linear transformation of $\hat{\beta }_{ols}-\hat{\beta}_{2sls}$, thereby elucidating the connection between the Wald test for $H_{0}:\rho=0$ in the CF approach and the Hausman test in the IV approach. Second, we show that the Wald test differs from the Hausman test primarily in how the asymptotic variances of $\hat{\beta}_{2sls}$ and $\hat{\beta}_{ols}$ are estimated, thus representing a variant of the Hausman test. Third, our analysis reveals that the Wald test statistic is numerically larger than the Hausman test statistics, which rely on $\hat{\beta}_{ols}$ or $\hat{\beta}_{2sls}$ to obtain estimators of\ the asymptotic variances of $\hat{\beta}_{ols}\ $and $\hat{\beta}_{2sls}$. Since the Wald test employs the same critical value as the Hausman test, it exhibits larger statistical power in finite samples. These findings are established without imposing restrictive assumptions on either the null or alternative hypotheses.
The remainder of the paper is structured as follows. Section (ref) introduces the test statistics employed in both the Hausman test and the Wald test. Section (ref) establishes a numerical order among the test statistics introduced in Section (ref) and discusses its implications for relative power properties of the tests in finite samples. Section (ref) concludes the paper. The proofs are presented in the Appendix.
Notation. We use $a\equiv b$ to indicate that $a$ is defined as $b$. For any real vector $a$, $d_{a}$ denotes the dimension of $a$. For any positive integer $k$, $I_{k}$ denotes the $k\times k$ identity matrix. For any $k_{1}\times k_{2}$ matrix $A$, $A^{\top}$ denotes the transpose of $A$, $P_{A}\equiv A(A^{\top}A)^{-1}A^{\top}$ and $M_{A}\equiv I_{k_{1}}-A(A^{\top }A)^{-1}A^{\top}$ as long as $A^{\top}A$ is non-singular. For any square matrix $A$, $A>0$ means $A$ is positive definite.
We begin by formulating the Hausman test for assessing the endogeneity of $y_{1}$ in the model specified by ((ref)) and ((ref)). Suppose we have a random sample $\left \{ y_{1,i},y_{2,i},z_{i}\right \} _{i=1}^{n}$, where $z_{i}\equiv(z_{1,i}^{\top},z_{2,i}^{\top})^{\top}$ for $i=1,\ldots,n$. Recall that $x_{i}\equiv(y_{1,i}^{\top},z_{1,i}^{\top})^{\top}$, $i=1,\ldots,n$, denotes the regressors in ((ref)). Let $X\equiv(x_{1},\ldots,x_{n})^{\top }$, $Y_{1}\equiv(y_{1,1},\ldots,y_{1,n})^{\top}$, $Y_{2}\equiv(y_{2,1} ,\ldots,y_{2,n})^{\top}$, and\ $Z\equiv(z_{1},\ldots,z_{n})^{\top}$. The OLS and 2SLS estimators of $\theta$ in ((ref)) are given by \[ \hat{\theta}_{ols}\equiv(X^{\top}X)^{-1}X^{\top}Y_{2}\ \ \ \ \ \ \ \text{and} \ \ \ \ \ \ \ \hat{\theta}_{2sls}\equiv(X^{\top}P_{Z}X)^{-1}X^{\top}P_{Z}Y_{2} \] respectively. Let $\hat{\beta}_{ols}$ and $\hat{\beta}_{2sls}$ denote the leading $d_{y_{1}}\times1$ subvectors of $\hat{\theta}_{ols}$ and $\hat {\theta}_{2sls}$ respectively. The Hausman test statistic can be characterized by
where $\hat{Y}_{1}\equiv P_{Z}Y_{1}$, $\hat{\sigma}_{1}^{2}(\hat{Y}_{1}^{\top }M_{Z_{1}}\hat{Y}_{1})^{-1}$ and $\hat{\sigma}_{2}^{2}(Y_{1}^{\top}M_{Z_{1} }Y_{1})^{-1}$ are estimators of the asymptotic variances of $\hat{\beta }_{2sls}\ $and $\hat{\beta}_{ols}$ respectively, and $\hat{\sigma}_{1}^{2}$ and $\hat{\sigma}_{2}^{2}$ are (possibly different) estimates of the variance $\sigma_{\varepsilon}^{2}$ of the error term $\varepsilon$ in the structural equation ((ref)).
In practice, $\sigma_{\varepsilon}^{2}$ can be estimated by the sample variance of the fitted residual in the OLS estimation: \[ \hat{\sigma}_{ols}^{2}\equiv n^{-1}(Y_{2}-X\hat{\theta}_{ols})^{\top} (Y_{2}-X\hat{\theta}_{ols}), \] or through its counterpart in the 2SLS estimation: \[ \hat{\sigma}_{2sls}^{2}\equiv n^{-1}(Y_{2}-X\hat{\theta}_{2sls})^{\top} (Y_{2}-X\hat{\theta}_{2sls}), \] resulting in three popular versions of the Hausman test with test statistics $t_{H_{1}}\equiv t_{H,n}(\hat{\sigma}_{ols}^{2},\hat{\sigma}_{ols}^{2})$, $t_{H_{2}}\equiv t_{H,n}(\hat{\sigma}_{2sls}^{2},\hat{\sigma}_{2sls}^{2})$, and $t_{H_{3}}\equiv t_{H,n}(\hat{\sigma}_{2sls}^{2},\hat{\sigma}_{ols}^{2})$ respectively ( wooldridge2010 ; Baum2003 ).
Alternatively, we can apply the CF approach and obtain the estimators of $\theta$ and $\rho$ in ((ref)) as
where $\hat{V}\equiv M_{Z}Y_{1}$, and then test the endogeneity of \ $y_{1}$ using the Wald test with the test statistic
where $\widehat{Asv}(\hat{\rho}_{cf})$ denotes an estimator of the asymptotic variance of $\hat{\rho}_{cf}$. Applying the partitioned regression formula (which is also known as the Frisch-Waugh-Lovell Theorem, see, e.g., davidson1993 ) to ((ref)) yields
where\ $U\equiv(u_{1},\ldots,u_{n})^{\top}$, $V\equiv(v_{1},\ldots ,v_{n})^{\top}$, and the second equality follows by ((ref)). Although the regressor $v$ is estimated and hence the estimation error $\hat{V}-V$ should be taken into account when calculating $\widehat{Asv}(\hat{\rho}_{cf})$, the expansion in\ ((ref)) shows that this estimation error can be ignored under the null\ that $\rho=0$. Therefore, in the rest of the paper we use the estimator
where $\hat{\sigma}_{u}^{2}$ is the sample variance of the fitted residual in the OLS regression of ((ref))
As we shall see in the next section, choosing this estimator makes the Wald statistic $t_{CF}$ comparable to\ $t_{H,n}(\hat{\sigma}_{1}^{2},\hat{\sigma }_{2}^{2})$.
Throughout this paper, we assume that $X^{\top}X$, $X^{\top}P_{Z}X$, and $(X,\hat{V})^{\top}(X,\hat{V})$ are non-singular so that the estimators $\hat{\theta}_{ols}$, $\hat{\theta}_{2sls}$, $\hat{\theta}_{cf}$ and $\hat{\rho}_{cf}$ are well defined. Under these primitive conditions, we have $Y_{1}^{\top}M_{Z_{1}}Y_{1}>0$, $\hat{Y}_{1}^{\top}M_{Z_{1}}\hat{Y}_{1}>0$ and \[ Y_{1}^{\top}M_{Z_{1}}Y_{1}-\hat{Y}_{1}^{\top}M_{Z_{1}}\hat{Y}_{1}=Y_{1}^{\top }M_{Z}Y_{1}>0, \] which further implies that
In view of ((ref)) and the definitions of $t_{H_{j}}$ ($j=1,2,3$) and $t_{CF}$, we also assume that $\hat{\sigma}_{ols}^{2}$, $\hat{\sigma} _{2sls}^{2}$ and $\hat{\sigma}_{u}^{2}$ are strictly positive so that these test statistics are well defined.
Under the null that $y_{1}$ is exogenous, along with other regularity conditions, one can establish that the asymptotic distributions of $t_{H_{j}}$ ($j=1,2,3$) and $t_{CF}$ are Chi-square with $k_{1}$ degrees of freedom (denoted as $\chi^{2}(k_{1})$). Consequently, the Hausman tests and the Wald test reject the null hypothesis if the corresponding test statistic exceeds the $1-\alpha$ quantile of $\chi^{2}(k_{1})$, where $\alpha$ denotes the significance level. As we shall see in the next section, the test statistics $t_{H_{j}}$ ($j=1,2,3$) and $t_{CF}$ can be numerically ordered, indicating their relative rejection properties in finite samples.
Our first objective is to establish a numerical relationship between the OLS and 2SLS estimators and the estimators in the CF approach. This result serves as a foundation for further investigating the numerical order among $t_{H_{j} }$ ($j=1,2,3$) and $t_{CF}$ in finite samples.
The lemma above carries two important implications. First, $\hat{\theta}_{cf}$ is numerically equivalent to $\hat{\theta}_{2sls}$, implying that $\hat {\theta}_{cf}$ shares the same standard error as $\hat{\theta}_{2sls}$. This finding has been well recognized in the literature since at least hausman1978 (see also davidson1993 and wooldridge2010 ). Second, $\hat{\rho}_{cf}$ is a linear transformation of $\hat{\beta} _{ols}-\hat{\beta}_{2sls}$, establishing a connection between the Wald test based on $t_{CF}$ and the Hausman tests based on\ $t_{H,n}(\hat{\sigma} _{1}^{2},\hat{\sigma}_{2}^{2})$. To the best of our knowledge, this numerical relationship is a novel contribution to the literature. The expression of $t_{CF}$ in ((ref)) further suggests that $t_{CF}$ is a special case of $t_{H,n}(\hat{\sigma}_{1}^{2},\hat{\sigma}_{2}^{2})$ with $\hat{\sigma} _{1}^{2}=\hat{\sigma}_{2}^{2}=\hat{\sigma}_{u}^{2}$.
Since $t_{H_{j}}$ ($j=1,2,3$) and $t_{CF}$ differ only in how the variance estimators $\hat{\sigma}_{1}^{2}\ $and $\hat{\sigma}_{2}^{2}$ in ((ref)) are calculated, their relative performances are determined by the differences of these variance estimators. Intuitively,\ $\hat{\sigma} _{u}^{2}$ should not be larger than $\hat{\sigma}_{ols}^{2}$ since, compared with the OLS estimation of ((ref)), the CF approach includes an extra regressor $v$ in ((ref)), and the resulting $R^{2}$ should not be smaller. Moreover, we have $\hat{\sigma}_{ols}^{2}\leq \hat{\sigma}_{2sls}^{2}$ by the definitions of OLS and 2SLS estimation. Therefore, we expect a weak order among these variance estimators: $\hat{\sigma}_{u}^{2}\leq \hat{\sigma }_{ols}^{2}\leq \hat{\sigma}_{2sls}^{2}$, which together with the definitions of $t_{H_{j}}$ ($j=1,2,3$) and $t_{CF}$ implies a weak numerical order among the test statistics:\ $t_{CF}\geq t_{H_{1}}\geq t_{H_{2}}\geq t_{H_{3}}$. Our next lemma establishes the exact relationships among the variance estimators $\hat{\sigma}_{u}^{2}$, $\hat{\sigma}_{ols}^{2}$ and $\hat{\sigma}_{2sls}^{2} $, enabling us to obtain a strong numerical order among them as well as among the test statistics $t_{H_{j}}$ ($j=1,2,3$) and $t_{CF}$.
Since $t_{H_{j}}$ ($j=1,2,3$) and $H_{n}$ are non-negative, from ((ref)) we immediately obtain $\hat{\sigma}_{2sls}^{2}\geq \hat{\sigma}_{ols}^{2} \geq \hat{\sigma}_{u}^{2}$, which implies that $t_{CF}\geq t_{H_{1}}\geq t_{H_{2}}\geq t_{H_{3}}$. Moreover, when $\hat{\beta}_{ols}-\hat{\beta} _{2sls}\neq0$, $t_{H_{j}}$ and $H_{n}$ are strictly positive. In this case, we can deduce from ((ref)) that $\hat{\sigma}_{2sls}^{2}>\hat{\sigma} _{ols}^{2}>\hat{\sigma}_{u}^{2}$, and a strong numerical order among $t_{H_{j}}$ ($j=1,2,3$) and $t_{CF}$, which is summarized in the lemma below.
Lemma (ref) demonstrates that in finite samples, the endogeneity test based on $t_{CF}$ has the largest rejection probability compared with $t_{H_{j}}$\ ($j=1,2,3$), although these four tests are asymptotically equivalent under the null hypothesis and local alternatives where $\hat{\beta }_{ols}-\hat{\beta}_{2sls}=o_{p}(1)$.
This paper explores the endogeneity test using the CF approach in linear IV models. We find that the OLS estimator of the coefficients of the generated CF is a linear transformation of the difference between the OLS and 2SLS estimators of the coefficients of endogenous regressors. This finding allows us to demonstrate that the endogeneity test using the CF approach is a variant of the Hausman test. In addition, we establish a numerical order among the test statistics employed in the Hausman tests and the endogeneity test using the CF approach. This numerical order highlights that the endogeneity test using the CF approach exhibits the highest rejection probability in finite samples.