Simple Robust Testing of Regression Hypotheses

IN THIS PAPER WE CONSIDER the problem of hypothesis testing in models with errors that have serial correlation or heteroskedasticity of unknown form. This situation is often encountered in regression models applied to economic time series data. It is a classic textbook result that while ordinary least squares (OLS) estimates of regression parameters remain consistent and asymptotically normal when errors are heteroskedastic or autocorrelated (provided usual regularity conditions hold and no lagged dependent variables are in the model), standard tests are no longer valid. If the true form of serial correlation/heteroskedasticity were known, then generalized least squares (GLS) provides efficient estimates and standard inference can be conducted on the GLS transformed model. But, in practice the form of serial correlation/heteroskedasticity is often unknown, and this has led to the development of techniques that permit valid asymptotic inference without having to specify a model of the serial correlation or heteroskedasticity. The most common approach is to estimate the variance-covariance matrix of the OLS estimates nonparametrically using spectral methods (heteroskedasticity and autocorrelation consistent (HAC) estimators) and construct standard tests using the asymptotic normality of the OLS estimates. HAC estimators have been extensively analyzed in the econometrics literature and important contributions are given by Andrews (1991), Andrews and Monahan (1992), Gallant (1987), Hansen (1992), Newey and West (1987), Robinson (1991, 1998), and White (1984) among others. The benefit of HAC estimator tests is asymptotically valid inference that is robust to general forms of serial correlation/heteroskedasticity in the errors. We propose an alternative method of constructing robust test statistics. We apply a nonsingular data dependent stochastic transformation to the OLS estimates. The asymptotic distribution of the transformed estimates does not depend on nuisance parameters. Then, test statistics that are asymptotically invariant to nuisance parameters (asymptotic pivotal statistics) are constructed. The resulting test statistics have nonstandard asymptotic distributions that only depend on the number of restrictions being tested, and critical values are easy to simulate using standard techniques. The main advantage of our approach compared to standard approaches is that estimates of the variance-covariance matrix are not explicitly required to construct the tests. This is potentially important for two reasons. First, with the exception of the estimator proposed by Robinson (1998),2 consistent nonparametric estimates of variance-covariance matrices in models with serial