(Score Test for Serial Correlation) In the dynamic regression (20.8) with autoregressive disturbances

(20.1), if there is no autocorrelation in {$\epsilon_t$} ($\phi_1 = 0$), then the OLS estimator remains consistent and asymptotically efficient. Testing for autocorrelation has more importance than when $x_t$ contains no lagged values of $y_{t-1}$, because the OLS estimator is inconsistent when autocorrelation is present.

Again following Breusch (1978) and Godfrey (1978a, 1978b), the score test method still works for the null hypothesis $\phi_1 = 0$, but the test itself is no longer based simply on the OLS regression of the OLS fitted residual $\hat{\epsilon}_t$ on its lagged value $\hat{\epsilon}_{t-1}$, as described in Section 19.4.1." Show that the score test augments the explanatory variables of the auxiliary regression of $\hat{\epsilon}_t$ on $\hat{\epsilon}_{t-1}$ with all of the explanatory variables $x_t$ in the conditional mean of $y_t$. We then test the statistical significance of the coefficient of $\hat{\epsilon}_{t-1}$. Comment on the need for these additional explanatory variables.