Theory of Multivariate Statistics

2010

The paper develops a novel testing procedure for hypotheses on deterministic trends in a multivariate trend stationary model. The trends are estimated by the OLS estimator and the long run variance (LRV) matrix is estimated by a series type estimator with carefully selected basis functions. Regardless of whether the number of basis functions K is …xed or grows with the sample size, the Wald statistic converges to a standard distribution. It is shown that critical values from the …xed-K asymptotics are second order correct under the large-K asymptotics. A new practical approach is proposed to select K that addresses the central concern of hypothesis testing: the selected smoothing parameter is testing-optimal in that it minimizes the type II error while controlling for the type I error. Simulations indicate that the new test is as accurate in size as the nonstandard test of Vogelsang and Franses (2005) and as powerful as the corresponding Wald test based on the large-K asymptotics. The new test therefore combines the advantages of the nonstandard test and the standard Wald test while avoiding their main disadvantages (power loss and size distortion, respectively).

Robust Trend Inference with Series Variance Estimator and Testing-Optimal Smoothing Parameter

2010

Let's Fix it: Fixed-b Asymptotics Versus Small-b Asymptotics in Heteroscedasticity and Autocorrelation Robust Inference

2010

In the presence of heteroscedasticity and autocorrelation of unknown forms, the covariance matrix of the parameter estimator is often estimated using a nonparametric kernel method that involves a lag truncation parameter. Depending on whether this lag truncation parameter is speci…ed to grow at a slower rate than or the same rate as the sample size, we obtain two types of asymptotic approximations: the small-b asymptotics and the …xed-b asymptotics. Using techniques for probability distribution approximation and high order expansions, this paper shows that the …xed-b asymptotic approximation provides a higher order re…nement to the …rst order small-b asymptotics. This result provides a theoretical justi…cation on the use of the …xed-b asymptotics in empirical applications. On the basis of the …xed-b asymptotics and higher order small-b asymptotics, the paper introduces a new and easy-to-use asymptotic F test that employs a …nite sample corrected Wald statistic and uses an F-distribution as the reference distribution. Finally, the paper develops a bandwidth selection rule that is testing-optimal in that the bandwidth minimizes the type II error of the asymptotic F test while controlling for its type I error. Monte Carlo simulations show that the asymptotic F test with the testing-optimal bandwidth works very well in …nite samples.

A New Asymptotic Theory for Vector Autoregressive Long-Run Variance Estimation and Autocorrelation Robust Testing

Kaplan

2010

We develop a new asymptotic theory for autocorrelation robust tests using a vector autoregressive (VAR) covariance matrix estimator. In contrast to the conventional asymptotics where the VAR order goes to in…nity but at a slower rate than the sample size, we have the VAR order grow at the same rate, as a …xed fraction of the sample size. Under this …xed-smoothing asymptotic speci…cation, the associated Wald statistic remains asymptotically pivotal. On the basis of this asymptotics, we introduce a new and easy-to-use F test that employs a …nite sample corrected Wald statistic and uses critical values from an F distribution. We also propose an empirical VAR order selection rule that exploits the connection between VAR variance estimation and kernel variance estimation. Simulations show that the new VAR F test with the empirical order selection is much more accurate in size than the conventional chi-square test.