Publication Date: December 2018
The usual t test, the t test based on heteroskedasticity and autocorrelation consistent (HAC) covariance matrix estimators, and the heteroskedasticity and autocorrelation robust (HAR) test are three statistics that are widely used in applied econometric work. The use of these signiﬁcance tests in trend regression is of particular interest given the potential for spurious relationships in trend formulations. Following a longstanding tradition in the spurious regression literature, this paper investigates the asymptotic and ﬁnite sample properties of these test statistics in several spurious regression contexts, including regression of stochastic trends on time polynomials and regressions among independent random walks. Concordant with existing theory (Phillips, 1986, 1998; Sun, 2004, 2014), the usual t test and HAC standardized test fail to control size as the sample size n→∞ in these spurious formulations, whereas HAR tests converge to well-deﬁned limit distributions in each case and therefore have the capacity to be consistent and control size. However, it is shown that when the number of trend regressors K→∞, all three statistics, including the HAR test, diverge and fail to control size as n→∞. These ﬁndings are relevant to high dimensional nonstationary time series regressions.
Keywords: HAR inference, Karhunen-Loève representation, Spurious regression, t-statistics
JEL Classification Codes: C12, C14, C23