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Then it may be of interest to test the hypothesis aP 0. Under H0 , ordinary least squares is maximum likelihood and thus the LM approach is attractive for its simplicity. An alternative formula tion of (55) which shows how it fits into Crowder's framework is: e � H0 : a = · · · = = ( 56 ) where 1[;1_ 1 is the past information in both y and x. Thus, again under H0 the regression simplifies to OLS but under the alternative, there are non-linear restrictions. The formulation (56) makes it clear that serial correlation can also be viewed as a restricted model relative to the general dynamic model without the non-linear restrictions.
This is typically approximated by a linear regression without any test of the validity of the approximation. The LM test generally provides a simple test of such restrictions because it uses estimates only under the null hypothesis. While it is ideal for the case where the model is linear under the null and non-linear under the alternative, the procedures also greatly simplify the calculation when the null is non-linear. Three examples will be presented which show the usefulness of this set of procedures.
Evans, G. B. A. and N. E. Savin ( 1 982), " Conflict Among the Criteria Revisited; The W, LR and LM tests", Econometrica, 50:737-748. Ferguson, T. S. ( 1 967), Mathematical Statistics. New York: Academic Press. Godfrey. L. G. ( 1 978), " Testing for Multiplicative Heteroskedasticity", Journal of Econmnetrics, 8 : 227-236. Godfrey. L . G. ( 1 978a), " Testing Against general Autoregressive and Moving Average Error Models When the Regressors Include Lagged Dependent Variables", Econometrica, 46 : 1 293 - 1 302.