Selecting best ARIMA model with regressors and dummy variable I have data on GDP, employment rate, inflation and production on two countries and I like to make some ARIMA models. I have done this before, but not with including regressors. Also, the time period includes the European crisis, so I created a dummy for this period. I know that I can select the best ARIMA model by checking the AIC and BIC values. But I have to play around with the regressors, maybe it is better not to include them all. Also, I have to check if the dummy is necessary to include or not (Chow test). How do I start?
I thought


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*First, I select the appropriate ARIMA model (using AIC and BIC), using all regressors, but not the dummy.

*Then, when looking at the significance of the regressors, select the one to use and which not (how to do?). 

*Then compare this model with the same model including the dummy and perform a Chow test on whether to include it or not. 


But I feel like there are many more combinations of regressors and the dummy and so on, like I did not test all possible models. What is the right approach?
 A: One idea would be to first compute an ARIMA model without any regressors (using the usual AIC, BIC etc) and then check, if the residuals of this ARIMA model are correlated with your regressors. You could then select those regressors with "significant" correlation (or those whose correlation is higher than a pre-defined threshold) and re-compute the ARIMA model, this time including the regressors.
One drawback of this procedure is that you are, in effect, applying a kind of "hillclimbing" method. So, you are excluding regressors, which are correlated only in (non-linear) combination with another regressor. Nonetheless, the procedure might give you a good starting point for further investigation.
Finally, of course, you should check, if the AIC and BIC values have improved by including the regressors.
A: The drawback with your approach is that endogeneity among variables and lagged effects are not properly accounted for.
Since these macroeconomic variables are likely to be mutually endogenous and influence each other with some time lags, a VAR or VARMA model would make sense. VAR would be the simpler and more popular alternative; it is often used for modelling the macroeconomic indicators you have. In a VAR model you would select the lag order using information criteria or residual diagnostics. In R the relevant package would be "vars" and function VARselect. 
Vector error correction model (VECM; it is a special case of VAR) could be relevant if your variables are cointegrated. After selecting the lag order for a VAR model in levels, you would proceed with cointegration analysis (function ca.jo from package "urca") and VECM modelling.
Regarding inclusion/exclusion of the crisis dummy, you could compare values of information criteria and do residual diagnostics, similarly to when doing lag order selection.
Some relevant references:


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*Pfaff "Analysis of Integrated and Cointegrated Time Series with R"

*"vars" vignette "VAR, SVAR and SVEC Models: Implementation Within R Package vars"
