Suppose I have a time series $y_t$. I use ARIMA model. I do grid search to select the lag order $(p_0, d_0, q_0)$ that minimizes the AIC. Then I find another time series $x_t$ as an exogenous variable for $y_t$ and turn to an ARIMAX model.
Should I stick to $(p_0, d_0, q_0)$ or should I use grid search for the new model with the exogenous variable included to arrive at $(p_1, d_1, q_1)$? Why?