Should I change the lag order after including an exogenous regressor in ARIMA?

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?

In general, there is no reason to believe the same lag order should be optimal after inclusion of a new exogenous variable as before the inclusion. Hence, you should reconsider the lag order after the inclusion.

Since you are finding the lag order $$k$$ by:
$$\text{argmin}_{k} AIC := \text{argmin}_k \{2k -ln(\hat{L})\}$$
where k is the number of parameters and $$\hat{L}$$ is the maximum likelihood function, adding an exogenous variable $$x_t$$ (which may also be serially correlated) would increase the AIC for all values of $$k$$ previously considered in the grid-search.

Perhaps you could instead identify the order of the variables by the Autocorrelation and Partial Autocorrelation functions above a certain confidence level.

• Adam, welcome to CV! Adding $x_t$ would not only increase the values of $k$, but also change the value of the likelihood. The formula of AIC uses log-likelihood, not likelihood. Why use ACF and PACF instead of AIC? The former approach can be considered outdated. AIC is sounder from information-theoretic perspective and model selection literature. Feb 18 '20 at 19:15
• Interesting - thanks for pointing that out! Do you have any references/further reading on the advantages of using AIC instead of ACF/PACF for order selection? I had meant to point out that the log-likelihood will change as well, but the point was that the optimal $k$ before adding $x_t$ is not necessarily optimal afterwards.
• After digging around the literature, it seems that selecting the order of an ARIMA-class model is not an exact science but something that has to be done empirically with some subjectivity. The original method proposed by Box et al. (1994) is essentially: 1. Identification of significant lags via ACF/PACF 2. Estimation of parameters 3. Diagnostics (checking residuals are white noise, comparing different candidate orders via AIC/BIC). Consensus in literature seems that using the AIC by itself tends to over-estimate number of parameters even as $N\to\infty$, and relies heavily on normality.