This article presents a theoretical case for replacement of conventional heteroskedasticity-consistent and cluster-robust variance estimators with jackknife variance estimators, in the context of linear regression with heteroskedastic and/or cluster-dependent observations. We examine the bias of variance estimation and the coverage probabilities of confidence intervals. Concerning bias, we show that conventional variance estimators have full downward worst-case bias, while our jackknife variance estimator is never downward biased. Concerning confidence intervals, we show that intervals based on conventional standard errors have worst-case coverage equalling zero, while the jackknife-based confidence interval has coverage probability bounded by the Cauchy distribution, under the auxiliary assumption of normal errors. We also extend the Bell and McCaffrey (2002) student t approximation to our jackknife t-ratio, resulting in confidence intervals with improved coverage probabilities. Our theory holds under broad assumptions, allowing arbitrary cluster sizes, regressor leverage, within-cluster correlation, heteroskedasticity, regression with a single treated cluster, fixed effects, and delete-cluster invertibility failures. Our theoretical findings are consistent with the extensive simulation literature investigating heteroskedasticity-consistent and cluster-robust variance estimation.

This paper studies the asymptotic properties of instrumental variable (IV) estimates of multivariate cointegrating regressions and allows for deterministic and stochastic regressors as well as quite general deterministic processes in the data-generating mechanism. It is found that IV regressions are consistent even when the instruments are stochastically independent of the regressors. This phenomenon, which contrasts with traditional theory for stationary time series, is a beneficial artifact of spurious regression theory whereby stochastic trends in the instruments ensure their relevance asymptotically. Problems of inference are also addressed and some promising new theoretical results are reported. These involve a class of Wald tests which are modified by semiparametric corrections for serial correlation and for endogeneity. The resulting test statistics which we term fully-modified Wald tests have limiting X2 distributions, thereby removing the obstacles to inference in cointegrated systems that were presented by the nuisance parameter dependencies in earlier work. Some simulation results are reported which seek to explore the sampling behaviour of our suggested procedures. These simulations compare our fully modified (semiparametric) methods with the parametric error-correction methodology that has been extensively used in recent empirical research and with conventional least squares regression. Both the fully-modified and errorcorrection methods work well in finite samples and the sampling performance of each procedure confirms the relevance of asymptotic distribution theory, as distinct from super-consistency results, in discriminating between statistical methods.