In the linear regression model, certain type of mis-specification have only mild implications for our inferences. For example, although heteroskedasticity renders the usual estimated covariance matrix for the OLS parameter estimator inconsistent, the parameter estimates themselves only lose efficiency - they are still unbiased and consistent. Similarly, although the omission of relevant regressors from the standard linear regression model generally biases the OLS parameter estimates, this bias vanishes if the included and excluded regressors are orthogonal (uncorrelated in the sample).
These results change if the model is non-linear in the parameters - a fact that is well known (e.g., Maddala & Nelson, 1975; Hurd, 1979; Arabmazar & Schmidt, 1981; White, 1981; Lee, 1982; Ruud, 1983; Kiefer & Skoog, 1984; Yatchew & Griliches, 1985; Blundell, 1987), but largely ignored in most empirical studies. More specifically, these results change (for the worse) in the context of such non-linear models as Logit, Probit, Tobit, and the various extensions of these models. For example, in the presence of heteroskedasticity, the MLE's of the parameters of these models are inconsistent. So, worrying about reporting "robust" standard errors is of second-order importance. There are much more serious matters to be concerned about! Similarly, the omission of relevant covariates (even if they are uncorrelated with the included ones) also renders the MLE's of the parameters themselves inconsistent.
It should be noted that these issues apply not only to maximum likelihood estimation of non-linear models, but to any estimators based on the optimization of a criterion function (see Kiefer & Skoog, 1984). Various Lagrange Multiplier (LM) tests for these types of mis-specification in Logit, Probit, and related models have long been available (e.g., Davidson & MacKinnon, 1984; Smith, 1989; Pagan & Vella, 1989; Murphy, 1994, 1996), but seem to be largely ignored in most empirical research. While this may reflect a lack of familiarity with the theoretical literature, it may also reflect the lack of easy access to the code needed to implement these specification tests.
Listed below are EViews workfiles and program files that can be used to perform various specification tests on binary choice models. Each workfile contains a READ_ME text object.
The files were created using EViews 6 (Quantitative Micro Software, 2007).
The Logit model will be mis-specified, and the MLE's of the parameters will be inconsistent, if the underlying distribution is asymmetric. The Logistic distribution is nested within the Burr Type II family of distributions, whose p.d.f.'s may be symmetric or skewed to the left or to the right, depending on the value of the shape parameter (K > 0). The Logistic distribution is nested within that family, and arises when K = 1. This LM test is for the null hypothesis, H_{0}: K = 1, against H_{A}: K ≠ 1. Asymptotically, the LM test statistic is Chi-Square with 1 degree of freedom, under the null, so we reject H_{0} in favour of H_{A} if the LM test statistic exceeds the appropriate χ^{2}_{(1)}(α) critical value.
Logit and Probit models will be mis-specified, and the MLE's of the parameters will be inconsistent, if the disturbances are heteroskedastic, or if the underlying distribution is mis-specified. We consider two specification tests See Davidson & MacKinnon (2004, pp.464-465) for further details.
In the LM tests for homoskedasticity the form of the skedastic function is taken to be exp(2Z_{t}γ) where Z_{t} is the vector giving the t^{th} observation on the r variables that are believed to give rise to the heteroskedasticity. These variables must not include an intercept.
These LM tests are for the null hypothesis, H_{0}: γ = 0, against H_{A}: γ ≠ 0.
Asymptotically, the LM test statistic is Chi-Square with r degrees of freedom, under the null, so we reject H_{0} in favour of H_{A}
if the LM test statistic exceeds the appropriate χ^{2}_{(r)}(α) critical value.
The LM tests for functional form are similar to the familiar "RESET test" for the standard linear OLS model. Under the alternative hypothesis, the functional form is taken to be F[τ(δX_{t}β) / δ], where F is the c.d.f. for either the Logistic distribution or the Standard Normal distribution; δ is a scalar parameter; and τ( . ) is a scalar monotonic increasing function satisfying:
τ(0) = 0, τ'(0) = 1, & τ"(0) ≠ 0.
These LM tests are for the null hypothesis, H_{0}: δ = 0, against H_{A}: δ ≠ 0. Asymptotically, the LM test statistic is standard normal under the null, so we reject H_{0} in favour of H_{A}
if the absolute value of the test statistic exceeds the appropriate Z(α/2) critical value. So, a 2-sided p-value is reported.
(a) Logit Model
A procedure for the SHAZAM econometrics package is available at http://shazam.econ.ubc.ca/intro/logit3.htm
When a Logit or Probit model is estimated in EViews, various tests are available through the "VIEW" tab in the menu bar for the Equation window. Specifically, EViews incorporates the Likelihood Ratio Test for wrongly omitted/included covariates.
See Bera et al. (1984) and Wilde (2008). EViews code is currently being prepared - Check back soon!
A bivariate probit model is a 2-equation system in which each equation is a probit model. Apart from estimating the system, in the hope of increasing the asymptotic efficiency of our estimator over single-equation probit estimation, we will also be interested in testing the hypothesis that the errors in the two equations are uncorrelated.
Packages such as Stata and LIMDEP/NLOGIT provide routines for the estimation of Bivariate Probit models, and an "add-in" for estimating Bivariate Probit models is now available for EViews 7.1.
It is also easy to create a "LOGL" object in EViews to estimate a Bivariate Probit model if you are using an earlier version of EViews - see the following test workfiles, and the associated documentation.
Last Update: 7 October, 2011
Contact: David Giles; email: dgiles@uvic.ca; Department of Economics, University of Victoria, P.O. Box 1700 STN CSC, Victoria, B.C., CANADA V8W 2Y2; Tel.: +1-250-721-8540