Improved Testing Procedure for Kernel Methods with Time Series Data

This paper develops a robust testing procedure for the nonparametric kernel method in the presence of temporal dependence of unknown forms. We …rst propose a new variance estimator that corrects the …nite sample bias caused by temporal dependence. The new variance estimator is novel in the sense that it is not only robust to temporal dependence in …nite samples but also consistent in large samples, as the e¤ect of the correction vanishes with the sample size. Second, we propose a new reference distribution for hypothesis testing. The conventional reference distribution is based on normal approximation that treats the variance estimate as if it is the true value. We employ the …xed smoothing asymptotics to re‡ect the randomness of the variance estimator. The …xed smoothing asymptotic distribution is well approximated by a student’s t-distribution. Finally, we suggest a simulation-based calibration approach to choose smoothing parameters based on a testing oriented criterion. Simulation shows that the proposed procedure works very well. Keywords: HAR estimator, calibration, …xed-smoothing asymptotics, kernel estimator, robust inference, t-approximation, testing optimal smoothing parameters choice, temporal dependence JEL Classi…cation Number : C12, C14, C22 Email: [email protected] and [email protected]. Any comments and suggestions are appreciated. 1 Introduction This paper proposes a new robust testing procedure for the nonparametric kernel method in the presence of temporal dependence of unknown forms. An important issue in hypothesis testing with time series data is how to take unknown forms of dependence into account to calculate the standard error. Dependence is typical in time series and an estimator with positively dependent data tends to have larger variation than the one with iid data. Therefore, if it 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 it is now standard practice to use the heteroskedasticity and autocorrelation robust (HAR) standard error in empirical studies. Andrews (1991) establishes its asymptotic properties rigorously. No such a procedure has been proposed for nonparametric kernel methods. This stems from the fact that the distribution of a kernel estimator with dependent data is asymptotically equivalent to the distribution with iid data. See Robinson (1983) for details. This is in sharp contrast to the parametric case. The asymptotic equivalence implies that the usual standard error formulae with iid data are still valid for time series data in the “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 causes the usual asymptotic test to over-reject in …nite samples. Conley, Hansen and Liu (1997) and Pritsker (1998) …nd this problem in kernel density estimation using short term interest rate models. Our testing procedure improves upon the conventional asymptotic tests in several aspects. First, based on the “pre-asymptotic" variance of a kernel estimator, we construct a kernel based HAR variance estimator that corrects the …nite sample bias caused by temporal dependence. Using the “pre-asymptotic" variance is considered by Chen, Liao and Sun (2014) for sieve inference on time series models. The proposed HAR estimator is novel in the sense that it is not only robust to temporal dependence in …nite samples but also consistent in large samples, as the e¤ect of the correction vanishes with the sample size. We study its asymptotic properties rigorously. We …nd that the asymptotic bias and variance of the HAR estimator are determined by the two smoothing parameters, the truncation parameter of the HAR estimator ST and the bandwidth parameter of the original kernel estimator hT . In the parametric models, the trade-o¤ between the bias and variance is based on choice of ST . Secondly, we adopt the …xed-smoothing asymptotics to characterize the limiting behavior of the HAR estimator and the associated t-statistic. The conventional reference distribution is based on normal approximation that treats the variance estimate as if it is the true value. Under the …xed-smoothing asymptotics, the ratio of truncation parameter ST to the sample size T , b is assumed to be …xed with the sample size. As the degree of smoothing is …xed with the sample size, the HAR estimator converges in distribution to a random variable which is proportional to the true variance. As a result, the t-statistic is asymptotically equivalent to a fraction of standard normal variable to the square root of the random weighting variable. The randomness of the HAR estimator is embedded in this random weighting variable. The asymptotically equivalent distribution is nonstandard, but easy to simulate because it is a function of T iid standard normal variables. We also extend Sun (2014) to approximate the …xed smoothing asymptotic distribution by a student’s t-distribution and establish its validity. Since Kiefer, Vogelsang and Bunzel (2000) and Kiefer and Vogelsang (2002a, 2002b, 2005),