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Biometrika 1998 85(3):727-740; doi:10.1093/biomet/85.3.727
© 1998 by Biometrika Trust
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Tests for white noise against alternatives with both seasonal and nonseasonal serial correlation

DONALD W. K. ANDREWS, XUEMEI LIU and WERNER PLOBERGER

Cowles Foundation for Research in Economics, Yale University New Haven, Connecticut 06520-8281, U.S.Adandrews{at}minerva.cis.yale.edu
Department of Statistics, Yale University New Haven, Connecticut 06520-8290, U.S.A.liu{at}stat.yale.edu
Department of Economics, University of Rochester Rochester, New York 14627-0156, U.S.A.wplo{at}troi.cc.rochester.edu

This paper considers tests for seasonal and nonseasonal serial correlation in time series and in the errors of regression models. The problem of testing for white noise against multiplicative seasonal ARMA(1, 1) × (1, 1), alternatives is investigated. This testing problem is nonstandard because of nuisance parameters that appear under the alternative but not under the null hypothesis. The likelihood ratio, sup Lagrange multiplier, and average exponential Lagrange multiplier and likelihood ratio tests are considered and are shown to be asymptotically admissible for multiplicative seasonal ARMA (1, 1) × (1, 1), alternatives. In addition, they are shown to be consistent against all weakly stationary strong mixing non-white noise alternatives. Simulation results compare the tests to several tests in the literature. The average exponential test, Exp-LR{infty}, is found to be the best test overall. It performs substantially better than the Box-Pierce, Durbin-Watson and Wallis tests.

Key Words: Autoregressive moving average model • Consistent test • Lagrange multiplier test • Likelihood ratio test • Multiplicative seasonal ARMA model • Nonstandard testing problem • Seasonal serial correlation • Seasonality • Test of white noise


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