Asymptotic distribution theory for nonparametric entropy measures of serial dependence.

*(English)*Zbl 1152.91729Summary: Entropy is a classical statistical concept with appealing properties. Establishing asymptotic distribution theory for smoothed nonparametric entropy measures of dependence has so far proved challenging. In this paper, we develop an asymptotic theory for a class of kernel-based smoothed nonparametric entropy measures of serial dependence in a time-series context. We use this theory to derive the limiting distribution of C. Granger and J. L. Lin [J. Time Ser. Anal. 15, No. 4, 371–384 (1994; Zbl 0807.62067)] normalized entropy measure of serial dependence, which was previously not available in the literature. We also apply our theory to construct a new entropy-based test for serial dependence, providing an alternative to P. M. Robinson’s [Rev. Econ. Stud. 58, No.3, 437–453 (1991; Zbl 0719.62055)] approach. To obtain accurate inferences, we propose and justify a consistent smoothed bootstrap procedure. The naive bootstrap is not consistent for our test. Our test is useful in, for example, testing the random walk hypothesis, evaluating density forecasts, and identifying important lags of a time series. It is asymptotically locally more powerful than Robinson’s test, as is confirmed in our Simulation. An application to the daily S&P 500 stock price index illustrates our approach.

##### MSC:

91B84 | Economic time series analysis |

62E20 | Asymptotic distribution theory in statistics |

62G07 | Density estimation |

62G09 | Nonparametric statistical resampling methods |

62G20 | Asymptotic properties of nonparametric inference |

62M07 | Non-Markovian processes: hypothesis testing |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |