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Unformatted text preview: On TestingOptimal Smoothing Parameter Choice in Robust Multivariate Trend Inference Yixiao Sun & Department of Economics, University of California, San Diego July 15, 2009 & Email: [email protected] The author gratefully acknowledges partial research support from NSF under Grant No. SES0752443. Correspondence to: Department of Economics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 920930508. ABSTRACT 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 Fdistribution while when K grows with the sample size, the Wald statistic converges to a Chisquare 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 plugin procedure for implementing the new rule is suggested and simulations show that the new plugin procedure works remarkably well in &nite samples. JEL Classi&cation: C13; C14; C32; C51 Keywords: Asymptotic expansion, Fdistribution, Hotelling T 2 , longrun variance, multiple window estimator, robust standard error, testoptimal 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 equationbyequation 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...
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 Fall '08
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 Economics, Normal Distribution, Big O notation, Estimation theory, Bias of an estimator, Asymptotic analysis

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