A note on the finite-sample distribution of lagrange multiplier tests for univariate time series models

Abstract

Godfrey's (1979) Lagrange multiplier (LM) test for examining the adequacy of an autoregressive-moving average (ARMAI process of order (p,q) is based on testing restrictions, r, against an alternative of ARMA (p+r,q) or A M (p,q+r). This paper investigates the finite-sample distribution of the LM test for different choices of r. Additionally, the effect of the nature of the data on the empirical performance of the test is examined. Monte Carlo results indicate that (i) the chi-squared approximation to the distribution of the LM test may fail when the value of r is large relative to the sample size, (ii) its empirical variances are consistently less than the theoretical value even when the sample size is as large as 100, and (iii the empirical power of the LM test can be significantly affected by both the choice of r and the nature of the data (seasonal vs. non-seasonal data).