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Relatively new to stats. I use linear regression and get R^2, which is quite low.

MODEL 1

    lmoutar=lm(formula = ts_y ~ ts_y_lag + ts_x)

So switched to arima with external regressor. Using "auto.arima", I formulate arimax model

MODEL 2

    fitarima <- auto.arima(ts_y, xreg=ts_x)
    arimaout<-arima(ts_y,order=c(2,0,5),xreg=ts_x)

How can I compare the explanability of AR model with arima model. From the thread How can I calculate the R-squared of a regression with arima errors using R?, I understand R^2 is not an option for ARIMA.

From the thread Model comparison between an ARIMA model and a regression model, AIC/BIC is not the right criteria and MSE from forcast/predict can be possible criteria for comparison across AR and ARIMA model. Is MSE the best option for model comparison, if so how would I generate MSE for AR and ARIMA?

I tried to compare the above ar and arima model using anova, but I get following error message

anova.lm(lmoutar,arimaout)
   Warning message:
    In anova.lmlist(object, ...) :
            models with response ‘"NULL"’ removed because response differs from model 1

What does this error message mean?

EDIT

Thanks for the response so far and insight that AR is nested within ARIMA. How would one answer this question, if I rephrase as "How to compare AR, ARIMA and General Linear Models?". The first model I listed has AR(1) and independent variable; it is a general linear model. So how would I compare a GLM versus ARIMAX model? Any thing else besides MSE that I could use to judge between GLM and ARIMAX

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  • $\begingroup$ What is your loss function? For instance, how does overly optimistic overshooting compare with pessimistic forecasts? In the restaurant industry, for instance, it might be more OK to have ingredients rot than to run out of them. Is being off by a factor of 4 worse than two times the cost of being off by a factor of 2? $\endgroup$ – Dimitriy V. Masterov Sep 11 '14 at 20:58
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I believe you can use some information criterion since ARIMAX nests AR. Regression is a different beast, hence the objection in the other post. The procedure that auto.arima uses to pick p and q is described here. It used AICc to pick the model.

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Since the two models are nested. The AR model is just an ARIMA model with all the moving average coefficient restricted to 0. You could use a likelihood ratio test (But really any test in the same vein will do)

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  • $\begingroup$ +1 for "The AR model is just an ARIMA model with all the moving average coefficient restricted to 0" $\endgroup$ – kashili kashili Sep 17 '14 at 0:01

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