Adequacy of Lagrange Multiplier Test

This paper examines the distribution of the Lagrange multiplier test, LM test, and focuses on what factors affect the distribution of the LM test estimator. It is worth noting that due to Chi-square distribution properties, the degree of freedom depends not only on the lagged-number of autocorrelation, but also on the number of independent variables whatever the sample sizes, that is, degree of freedom is the lagged-number of autocorrelation plus the number of independent variables. The result also indicates that the LM test estimator is not necessary to become Chi-square distribution because the different effect of the sample size and the number of independent variables, nevertheless, the law of large number, sample size is larger than 1000, leads the LM test estimator toward to Chi-square distribution.