Lagrance Multiplier Tests of Autoregressive Models

Abstract

The Lagrange multiplier test procedure is applied to hypotheses concerning autoregressive time series models. One reasonable method of testing the specification of a statistical model is by overfitting: the null hypothesis that the model is correct is tested against a suitable alternative hypothesis of which the null hypothesis is a special case. Considering that the white noise process is distributed as t with v degrees of freedom and an extreme-value (Weibull) distribution with shape parameter, c, Lagrange Multiplier test statistics have been derived under the null hypothesis. The estimates of the parameters under the null hypothesis have been calculated by solving the system of nonlinear equations with the help of IMSL, NAG and FORTRAN 77 programming. Simulation results are presented to assess the performance of the tests. Consider the modelling of time series data with a lower limit of detection L, i.e. all the observations below L are censored, the score test statistics have been derived under the null hypothesis to test the autodependence in the data. Some simulation experiments are performed to assess the closeness and validity of the test under the null hypothesis. To illustrate the methodology in a real situation, data from the Niagara River containing on Total Lead Concentration (mg∕L) for the period 1986-1992 have been used for illustration.