There has been increasing concern recently over the use of the simple first order Markov form to model error autocorrelation in regression analysis. The consequence of misspecifying the error model will be especially serious when the regressors include lagged values of the dependent variable. The purpose of this paper is to develop Lagrange multiplier tests of the assumed error model against specified ARMA alternatives. It is shown that all of the tests can be regarded as asymptotic tests of the significance of a coefficient of determination, and a table is provided which gives details of two general tests and several special cases.