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 The former approach can be considered outdated.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.
• AIC does not rely on normality; the likelihood can be arbitrary. Many of the implementations of ARIMA use normal likelihood, though. The important thing is, which one does better in forecasting. I place my bet on intelligent AIC-based selection routines such as auto.arima which has a proven track record and good performance on benchmark datasets. (Meanwhile, ACF and PACF are fine for pedagogical purposes when learning about ARIMA models.) Since this has been discussed and debated extensively before on Cross Validated, I will stop here. Feb 19 '20 at 7:35