On Testing-Optimal Smoothing Parameter Choice in Robust Multivariate Trend Inference

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 multiple window estimator with carefully selected data windows. The multiple window estimator is asymptotically invariant to the model parameters and thus does not su¤er from the usual demeaning bias that hurts the performance of conventional kernel LRV estimators. The number of data windows K, the underlying smoothing parameter, plays a key role in determining the asymptotic properties of the long run variance estimator and the associated semiparametric tests. When K is xed, the modi ed Wald statistic converges to an F-distribution while when K grows with the sample size, the Wald statistic converges to a Chi-square distribution. We show that the critical values from the xed-K asymptotics are second order correct under the large-K asymptotics. We propose a novel approach to select K which minimizes the type II error hence maximizes the power of the test while controlling for the type I error. The new selection rule is fundamentally di¤erent from the conventional rule based on the mean square error criterion. A plug-in procedure for implementing the new rule is suggested and simulations show that the new plug-in procedure works remarkably well in nite samples. JEL Classi cation: C13; C14; C32; C51 Keywords: Asymptotic expansion, F-distribution, Hotelling T2, long-run variance, multiple window estimator, robust standard error, test-optimal smoothing parameter choice, trend inference, Type I and Type II errors. 1 Introduction Trend regression is one of simple and important regressions in economic and climatic time series analysis. In this paper, we consider a linear trend regression with multiple dependent variables. For example, the dependent variables may consist of DGPs from a number of countries. Vogelsang and Franses (2005) provide more empirical examples. Estimation of the trends is relatively easy as the equation-by-equation OLS estimator is asymptotically as e¢ cient as the system GLS estimator. Hence, for point estimation, there is no need to take error autocorrelation into account in large samples. However, trend inference is subtle as the variance of the OLS trend estimator depends on the long run variance (LRV) of the error process. Since the LRV is proportional to the spectral density of the error process evaluated at zero, many nonparametric spectral density methods can be used to estimate the LRV. In this paper, we focus on multiple window estimators, a very powerful class of spectral estimators proving to be very valuable in the signal processing literature whenever the spectrum of interest is detailed or varies rapidly with a large dynamic range. See Percival and Walden (1993, ch. 7). The multiple window estimators also produce asymptotically valid tests on trend hypotheses that are very easy to use in practice. The smoothing parameter in the multiple window estimator is the number of data windows or basis functions employed. When the number of basis functions K is xed, the LRV estimator is inconsistent and converges to a scaled Wishart distribution. It is now well known that in cases like robust hypothesis testing, consistent LRV estimates are not needed in order to produce asymptotically valid tests, see for example, Kiefer, Vogelsang and Bunzel (2000), Kiefer and Vogelsang (2002a, 2002b, 2005). Indeed, under the assumption that K is xed, we can show that the Wald statistic converges to a nonstandard distribution. The xed-K asymptotics is in the spirit of the xed-b asymptotics as in Kiefer and Vogelsang (2005). This type of asymptotics captures the randomness of the LRV estimator and tests based on it often have better nite sample size properties than tests based on consistent LRV estimates. The novelty here is that we design a set of basis functions so that the nonstandard limiting distribution becomes the standard F distribution. For these basis functions, the LRV estimator is asymptotically invariant to the intercepts and trend parameters. As a result, the LRV estimator does not su¤er from the bias arising from the estimation uncertainty of model parameters. This is contrast with the conventional kernel LRV estimators where the estimation uncertainty gives rise to a demeaning bias. This type of bias is of the same order of magnitude as the asymptotic variance, see Sun, Phillips and Jin (2008, SPJ hereafter). Under the asymptotic mean-squared error (MSE) criterion, the squared form of the bias is of smaller order than the variance. Hence the demeaning bias is asymptotically negligible under the MSE criterion. However, for testing problems, this type of bias will have the same order e¤ect on the size and power properties as the variance of the LRV estimator. By selecting the basis functions appropriately, we completely remove this type of bias to the order we care about. This is a desirable property as we generally prefer an estimator with fewer bias terms. Another great advantage of using the new LRV estimator is that the critical values from the xed-K asymptotics are readily available from statistical tables and econometrics software programs. The computational burden of simulating critical values from nonstandard distributions is completely removed. While the LRV estimator is inconsistent when K is nite, it becomes consistent when K grows with the sample size at a certain rate. The smoothing parameter K is an important 1 tuning parameter that determines the asymptotic properties of the LRV estimator. Following the conventional approaches (e.g., Andrews, 1991, and Newey and West, 1987, 1994), Phillips (2005) chooses the smoothing parameter K to minimize the asymptotic MSE of the LRV estimator. This approach follows what has long been standard practice in the context of spectral estimation (Grenander and Rosenblatt, 1957; Hannan, 1970) where the focus of attention is the spectrum or the LRV. Such a choice of the smoothing parameter is designed to be optimal in the MSE sense for the point estimation of the spectrum or LRV, but is not necessarily best suited for semiparametric testing. Through its e¤ect on the LRV estimator, the smoothing parameter K a¤ects the type I and type II errors of the associate test. It therefore seems sensible that the choice of K should take these properties into account. To develope an optimal choice of K for semiparametric testing, we rst have to decide on which test to use. There are two choices. One is the traditional Wald test which is based the Wald statistic and uses critical values from a chi-square distribution. The other is the new F test given in this paper, which is based on the modi ed Wald statistic and uses critical values from an F-distribution. One of main contributions of the paper is to show that the critical values from the F-distribution are higher order correct under the conventional large K asymptotics. A direct implication is that the F test has smaller size distortion than the traditional Wald test. On the basis of this theoretical result and the emphasis on the size control in the econometrics literature, we employ and recommend the F test to conduct inference on the trend parameters. Another main contribution of the paper is to develop an optimal procedure for selecting the smoothing parameter K that addresses the central concerns of semiparametric testing. For testing problems, we do not care about the LRV estimator per se. Instead, we are interested in the LRV estimator only to use it to construct the F statistic. The ultimate goal of any testing problem is to minimize the type II error hence maximize the power while controlling for the type I error. It is thus desirable to choose the smoothing parameter to achieve this goal. We propose to choose K to minimize the type II error subject to the constraint that the type I error is bounded. The resulting optimal K is said to be testoptimal for the given bound. The bound is de ned to be ; where is the nominal type I error and > 1 is the parameter that captures the users tolerance on the discrepancy between the nominal and true type I errors. The parameter is allowed to be sample size dependent. For a smaller sample size, we may have a higher tolerance while for the larger sample size we may have a lower tolerance. The introduction of the tolerance parameter into the optimal K selection is a conceptually new idea, which does not seem to appear elsewhere in the literature. The proposed approach to selecting the test-optimal K requires asymptotic measurements of type I and type II errors of the F test. These measurements are provided by means of high order asymptotic expansions of the nite sample distribution of the F statistic under the null and local alternative hypotheses. We show that the type I error depends on the nonparametric bias of the LRV estimator. When the nonparametric bias is negative, the type I error is expected to be larger than the nominal type I error. The result is supported by simulations in Vogelsang and Franses (2005) as well as the present paper. The type II error of the F test depends on the local alternative hypothesis through a noncentrality parameter k~ ck, where ~ c is a vector that characterizes the local departure of the alternative hypothesis from the null. To the rst order, the type II error depends on