Determination of the Number of Common Stochastic Trends under Conditional Heteroskedasticity

Permanent-transitory decompositions and the analysis of the time series properties of economic variables at the business cycle frequencies strongly rely on the correct detection of the number of common stochastic trends (co-integration). Standard techniques for the determination of the number of common trends, such as the well-known sequential procedure proposed in Johansen (1996), are based on the assumption that shocks are homoskedastic. This contrasts with empirical evidence which documents that many of the key macro-economic and financial variables are driven by heteroskedastic shocks. In a recent paper, Cavaliere et al., (2010, Econometric Theory) demonstrate that Johansen's (LR) trace statistic for co-integration rank and both its i.i.d. and wild bootstrap analogues are asymptotically valid in non-stationary systems driven by heteroskedastic (martingale difference) innovations, but that the wild bootstrap performs substantially better than the other two tests in finite samples. In this paper we analyse the behaviour of sequential procedures to determine the number of common stochastic trends present based on these tests. Numerical evidence suggests that the procedure based on the wild bootstrap tests performs best in small samples under a variety of heteroskedastic innovation processes.

Asymptotic Versus Bootstrap Inference for Inequality Indices of the Cumulative Distribution Function

We examine the performance of asymptotic inference as well as bootstrap tests for the Alphabeta and Kobus–Miłoś family of inequality indices for ordered response data. We use Monte Carlo experiments to compare the empirical size and statistical power of asymptotic inference and the Studentized bootstrap test. In a broad variety of settings, both tests are found to have similar rejection probabilities of true null hypotheses, and similar power. Nonetheless, the asymptotic test remains correctly sized in the presence of certain types of severe class imbalances exhibiting very low or very high levels of inequality, whereas the bootstrap test becomes somewhat oversized in these extreme settings.

Fast and wild: Bootstrap inference in Stata using boottest

The wild bootstrap was originally developed for regression models with heteroskedasticity of unknown form. Over the past 30 years, it has been extended to models estimated by instrumental variables and maximum likelihood and to ones where the error terms are (perhaps multiway) clustered. Like bootstrap methods in general, the wild bootstrap is especially useful when conventional inference methods are unreliable because large-sample assumptions do not hold. For example, there may be few clusters, few treated clusters, or weak instruments. The package boottest can perform a wide variety of wild bootstrap tests, often at remarkable speed. It can also invert these tests to construct confidence sets. As a postestimation command, boottest works after linear estimation commands, including regress, cnsreg, ivregress, ivreg2, areg, and reghdfe, as well as many estimation commands based on maximum likelihood. Although it is designed to perform the wild cluster bootstrap, boottest can also perform the ordinary (nonclustered) version. Wrappers offer classical Wald, score/Lagrange multiplier, and Anderson–Rubin tests, optionally with (multiway) clustering. We review the main ideas of the wild cluster bootstrap, offer tips for use, explain why it is particularly amenable to computational optimization, state the syntax of boottest, artest, scoretest, and waldtest, and present several empirical examples.