A Fixed-bandwidth View of the Pre-asymptotic Inference for Kernel Smoothing with Time Series Data

This paper develops robust testing procedures for nonparametric kernel methods in the presence of temporal dependence of unknown forms. Based on the …xed-bandwidth asymptotic variance and the pre-asymptotic variance, we propose a heteroskedasticity and autocorrelation robust (HAR) variance estimator that achieves double robustness — it is asymptotically valid regardless of whether the temporal dependence is present or not, and whether the kernel smoothing bandwidth is held constant or allowed to decay with the sample size. Using the HAR variance estimator, we construct the studentized test statistic and examine its asymptotic properties under both the …xed-smoothing and increasing-smoothing asymptotics. The …xed-smoothing approximation and the associated convenient t-approximation achieve extra robustness — it is asymptotically valid regardless of whether the truncation lag parameter governing the covariance weighting grows at the same rate as or a slower rate than the sample size. Finally, we suggest a simulation-based calibration approach to choose smoothing parameters that optimize testing oriented criteria. Simulation shows that the proposed procedures work very well in …nite samples. Keywords: heteroskedasticity and autocorrelation robust variance, calibration, …xed-smoothing asymptotics, …xed-bandwidth asymptotics, kernel density estimator, local polynomial estimator, t-approximation, testing-optimal smoothing-parameters choice, temporal dependence JEL Classi…cation Number : C12, C14, C22 Email: [email protected], [email protected], and [email protected]. For helpful comments, we thank seminar and conference participants at OSU, U of Guelph, CESG, KAEA-KEA, and 2015 ESWC. Kim gratefully acknowledges research support from SSHRC (430-2015-00527). Sun gratefully acknowledges partial research support from NSF under Grant No. SES-1530592. Address correspondence to Yixiao Sun, Department of Economics, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0508, USA. 1 Introduction This paper proposes new robust testing procedures for nonparametric kernel methods in the presence of temporal dependence of unknown forms. An important issue in hypothesis testing with time series data is how to take nonparametric dependence into account in calculating the standard error. Dependence is typical in time series, and an estimator with positively dependent data tends to have larger variation than that with iid data. Therefore, if temporal dependence is not properly considered, we may have an over-rejection problem. For parametric models this has been a well-researched problem since Newey and West (1987) and Andrews (1991). It is now standard practice to use the heteroskedasticity and autocorrelation robust (HAR) standard error in empirical studies. No such procedure for nonparametric kernel methods has been proposed in the literature. The fact that the distribution of a kernel estimator with dependent data is asymptotically equivalent to the distribution with iid data may have masked the need for more robust nonparametric kernel methods in …nite samples. See Robinson (1983) for detail on the asymptotic equivalence. This is in sharp contrast to the parametric case. The asymptotic equivalence implies that the usual standard error formula with iid data is still valid for time series data in an asymptotic sense. However, in …nite samples, temporal dependence does a¤ect the sampling distribution of a kernel estimator. In particular, when a process is highly persistent and/or the sample size is not large enough, the asymptotic variance tends to understate the true …nite sample variation of a kernel estimator, and this understated variation causes the usual asymptotic test to over-reject in …nite samples. See, for example, Conley, Hansen and Liu (1997) and Pritsker (1998), who discuss this problem in kernel density estimation based on short term interest rates. In developing new and more accurate testing procedures, the paper makes several contributions. Firstly, based on the …xed-bandwidth asymptotic variance and the “pre-asymptotic” variance of a kernel estimator, we construct a kernel based HAR variance estimator that captures temporal dependence. The “pre-asymptotic” approach has also been used in Chen, Liao and Sun (2014) in sieve inference on time series models. Here the “pre-asymptotic”variance is further justi…ed using the new …xed-bandwidth asymptotics where the kernel-smoothing bandwidth h is held …xed. The proposed HAR variance estimator achieves double robustness — it is asymptotically valid regardless of whether the temporal dependence is present or not, and whether the kernel smoothing bandwidth is held constant or allowed to decay with the sample size. Secondly, we consider the asymptotic properties of the HAR variance estimator and the associated test statistics under various speci…cations of the smoothing parameters. There are two smoothing parameters in our testing procedures. The …rst is the kernel smoothing bandwidth parameter h, and the second is the truncation lag parameter b for covariance weighting; which is parametrized as the ratio of the truncation lag ST to the sample size T: Both h and b can be …xed or small in our asymptotic speci…cations. Under the small-h speci…cation, h ! 0 but hT ! 1: Similarly, under the small-b speci…cation, b! 0 but bT !1: Under the …xed-h and small-b speci…cation, the asymptotic properties of the HAR variance estimator and the associated test statistic resemble what one would obtain in a parametric setting. Under the small-h and small-b speci…cation, the asymptotic bias and variance of the HAR variance estimator are determined jointly by h and b: This is in contrast with the parametric setting where the bias and variance trade-o¤ is dictated by b only. Regardless of whether h is …xed or small, the …xed-b asymptotics delivers new limiting dis-