Solved – Estimate the best ARMAX model with one lagged independent variable (time series)

arimaarmaxstationaritytime series

I have two time series to work with, let's say $X_1$ and $X_2$.

First I have to estimate the best pure ARMA model for $X_1$; which is no problem. For that I perform the following steps:

  1. Stationarize (if needed) the time series by differencing
  2. Determine whether AR/MA terms are needed to correct any
    autocorrelation in the differenced series -> tentatively identify the maximum number of AR and/or MA terms using ACF and PACF plots
  3. Then estimate different models, store BIC values, construct a matrix of BIC values, and select the ARMA model with the lowest BIC value.

Now comes the part that confuses me. I have to estimate an ARMAX model for $X_1$, where I need to put $X_2$ as a lagged explanatory variable in the ARMAX model.

  • I don't think that including the exogenous variable will change the number of AR and MA terms for the best model relative to model estimated before. If e.g. ARMA(2,1) has the lowest BIC relative to the other ARMA models, ARMAX(2,1) will also have the lowest BIC relative to the other ARMAX models. Is that true?
  • What is meant by including the 'lagged' explanatory variable? Does that mean the one-period lagged values of $X_2$? Or do I have to find the optimum number of lags for $X_2$ using a statistical technique?

Best Answer

I don't think that including the exogenous variable will change the number of AR and MA terms for the best model relative to model estimated before.

I highly doubt that. I do not remember any result regarding under which conditions this would hold. This would be a special case.

What is meant by including the 'lagged' explanatory variable? Does that mean the one-period lagged values of X2? Or do I have to find the optimum number of lags for X2 using a statistical technique?

It is difficult to know what someone meant without seeing the context, but generally you would either rely on subject-matter knowledge or use statistics (or both).

For a reference, you may find Rob J. Hyndman's blog post about ARIMAX and related models useful. It highlights the differences between the different versions of the model (or should I say, the different but related models), mentions software implementations and has a take on their interpretations, too.

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