Improvement of the Likelihood Ratio Test Statistic in ARMA Models

In this paper, we develop a Bartlett correction for the likelihood ratio statistic used to test hypotheses about parameters of a Gaussian stationary and invertible model belonging to the ARMA (autoregressive moving average) family. Alternative hypotheses with and without disturbance parameters are considered. The correction formulae are written in matrix form with the advantage of being easily implemented with the aid of some symbolic or numerical matrix language. Some simulation results are also presented.

[1]  谷口 正信.  Higher order asymptotic theory for time series analysis , 1991 .

[2]  D. Cox,et al.  Parameter Orthogonality and Approximate Conditional Inference , 1987 .

[3]  M. Bartlett.  Properties of Sufficiency and Statistical Tests , 1992 .

[4]  L. Chambers.  Practical methods of optimization (2nd edn) , by R. Fletcher. Pp. 436. £34.95. 2000. ISBN 0 471 49463 1 (Wiley). , 2001, The Mathematical Gazette.

[5]  P. Harris,et al.  A note on Bartlett adjustments to likelihood ratio tests , 1986 .

[6]  Kees Jan,et al.  Exact Geometry of Autoregressive Models , 1999 .

[7]  Jurgen A. Doornik,et al.  Object-orientd matrix programming using OX , 1996 .

[8]  J. Magnus.  The moments of products of quadratic forms in normal variables , 1978 .

[9]  P. Young,et al.  Time series analysis, forecasting and control , 1972, IEEE Transactions on Automatic Control.

[10]  Richard A. Davis,et al.  Time Series: Theory and Methods (2Nd Edn) , 1993 .

[11]  Andrew Chesher,et al.  Bartlett corrections to likelihood ratio tests , 1995 .

[12]  Gauss M. Cordeiro,et al.  On the corrections to the likelihood ratio statistics , 1987 .

[13]  D. Lawley.  A GENERAL METHOD FOR APPROXIMATING TO THE DISTRIBUTION OF LIKELIHOOD RATIO CRITERIA , 1956 .

[14]  Francisco Cribari-Neto,et al.  On bartlett and bartlett-type corrections francisco cribari-neto , 1996 .

[15]  James P. Braselton,et al.  The Maple V handbook , 1994 .

[16]  T. Hayakawa,et al.  The likelihood ratio criterion and the asymptotic expansion of its distribution , 1977 .

[17]  J. R. Koehler,et al.  Modern Applied Statistics with S-Plus. , 1996 .

[18]  Francisco Cribari-Neto,et al.  Econometric Programming Environments: GAUSS, Ox and S-PLUS , 1997 .

[19]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[20]  B. Efron.  Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency) , 1975 .

[21]  Ulrich Küsters,et al.  Matrix Programming Languages for Statistical Computing : a Detailed Comparison of GAUSS, MATLAB and Ox , 1996 .

[22]  Masanobu Taniguchi.  Asymptotic expansions of the distributions of some test statistics for Gaussian ARMA processes , 1988 .

[23]  A. Markushevich.  Analytic Function Theory , 1996 .

[24]  Nalini Ravishanker,et al.  DIFFERENTIAL GEOMETRY OF ARMA MODELS , 1990 .

[25]  G. Box,et al.  A general distribution theory for a class of likelihood criteria. , 1949, Biometrika.

[26]  Masanobu Taniguchi,et al.  Third-order asymptotic properties of a class of test statistics under a local alternative , 1991 .

[27]  E. Mammen.  The Bootstrap and Edgeworth Expansion , 1997 .