Improved Inference under Heteroscedasticity of Unknown form Using a New Class of Bootstrap and Nonparametric Estimators

Abstract

It is well-known that use of ordinary least squares for estimation of linear regression model with heteroscedastic errors, always results into inefficient estimates of the parameters. Additionally, the consequence that attracts the serious attention of the researchers is the inconsistency of the usual covariance matrix estimator that, in turn, results in inaccurate inferences. The test statistics based on such covariance estimates are usually too liberal i.e., they tend to over-reject the true null hypothesis. To overcome such size distortion, White (1980) proposes a heteroscedasticity consistent covariance matrix estimator (HCCME) that is known as HC0 in literature. Then MacKinnon and White (1985) improve this estimator for small samples by presenting three more variants, HC1, HC2 and HC3. Additionally, in the presence of influential observations, Cribari-Neto (2004) presents HC4. An extensive available literature advocates the use of HCCME when the problem of heteroscedasticity of unknown from is faced. Parallel to HCCME, the use of bootstrap estimator, namely wild bootstrap estimator is also common to improve the inferences in the presence of heteroscedasticity of unknown form. The present work addresses the same issue of inference for linear heteroscedastic models using a class of improved consistent covariance estimators, including nonparametric and bootstrap estimators. To draw improved inference, we propose adaptive nonparametric versions of HCCME, bias-corrected versions of nonparametric HCCME, adaptive wild bootstrap estimators and weighted version of HCCME using some adaptive estimator, already available in literature, namely, proposed by Carroll (1982). The performance of all the estimators is evaluated by bias, mean square error (MSE), null rejection rate (NRR) and power of test after conducting extensive Monte Carlo simulations.