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An application of the empirical likelihood method to non-Gaussian locally stationary processes is presented. Based on the central limit theorem for locally stationary processes, we give the asymptotic distributions of the maximum empirical likelihood estimator and the empirical likelihood ratio statistics, respectively. It is shown that the empirical likelihood method enables us to make inferences on various important indices in a time series analysis. Furthermore, we give a numerical study and investigate a finite sample property.

The empirical likelihood is one of the nonparametric methods for a statistical inference proposed by Owen [

In this paper we extend the empirical likelihood method to non-Gaussian locally stationary processes with time-varying spectra. First, We derive the asymptotic normality of the maximum empirical likelihood estimator based on the central limit theorem for locally stationary processes, which is stated in Dahlhaus [

As an application of this method, we can estimate an extended autocorrelation for locally stationary processes. Besides we can consider the Whittle estimation which is stated in Dahlhaus [

This paper is organized as follows. Section

The stationary process is a fundamental setting in a time series analysis. If the process

Consider the following AR

As an extension of the stationary process, Dahlhaus [

A sequence of stochastic processes

There exists a constant

The time-varying spectral density is defined by

Consider an inference on a parameter

Let us set

Consider the problem of fitting a parametric spectral model to the true spectral density by minimizing the disparity between them. For the stationary process, this problem is considered in Hosoya and Taniguchi [

Now, we set

We can also use the following alternative estimating function:

To show the asymptotic properties of

The functions

The parameters

The data taper

For

Assumption

Now we give the following theorem.

Suppose that Assumption

In addition, we give the following theorem on the asymptotic property of the empirical likelihood ratio

Suppose that Assumption

Denote the eigenvalues of

If the process is stationary, that is, the time-varying spectral density is independent of the time parameter

In our setting, the number of the estimating equations and that of the parameters are equal. In that case, the empirical likelihood ratio at the maximum empirical likelihood estimator,

In this section, we present simulation results of the estimation of the autocorrelation in locally stationary processes which is stated in Example

We set a confidence level as

90% confidence intervals of the autocorrelation with lag

Lower bound | Upper bound | Interval length | Successful rate | |
---|---|---|---|---|

0.057 | 0.439 | 0.382 | 0.854 | |

0.172 | 0.382 | 0.210 | 0.866 | |

0.203 | 0.332 | 0.129 | 0.578 | |

0.203 | 0.356 | 0.154 | 0.826 | |

0.225 | 0.308 | 0.084 | 0.444 | |

−0.087 | 0.225 | 0.312 | 0.890 | |

0.001 | 0.169 | 0.168 | 0.910 | |

0.028 | 0.104 | 0.076 | 0.515 | |

0.023 | 0.139 | 0.116 | 0.922 | |

0.047 | 0.087 | 0.040 | 0.384 | |

0.060 | 0.449 | 0.388 | 0.841 | |

0.176 | 0.393 | 0.216 | 0.871 | |

0.201 | 0.332 | 0.131 | 0.586 | |

0.203 | 0.359 | 0.156 | 0.827 | |

0.226 | 0.310 | 0.083 | 0.467 |

In this subsection we give the three lemmas to prove Theorems

Suppose (

Let

Suppose (

We set

Next, we calculate the covariance of

The

Suppose (

First we calculate the mean of

Next we calculate the second-order cumulant:

This is equal to

Using the lemmas in Section

Using the lemmas in Section

The author is grateful to Professor M. Taniguchi, J. Hirukawa, and H. Shiraishi for their instructive advice and helpful comments. Thanks are also extended to the two referees whose comments are useful. This work was supported by Grant-in-Aid for Young Scientists (B) (22700291).