How do I use BIC (Bayesian Information Criterion) to estimated model AR (auto regressive) lag?

Question

In financial research papers, I have seen several times that the lag length in an ARMA model has been determined using BIC. Do the researchers estimate the lag length before considering other variables?

Would you compare the BIC values of the just the dependent variable and its lags or would you compare the BIC of the full model with the other exogenous variables.

Should I use method 1)

BIC1 Y(t) = c + Y(t-1)

BIC2 Y(t) = c + Y(t-1) + Y(t-2)

...

or should I use method 2)

BIC1 Y(t) = c + Y(t-1) + x1 + x2 + x3

BIC2 Y(t) = c + Y(t-1) + Y(t-2) + x1 + x2 + x3

...

Answer 1

Generally it depends on the model you need to compare; anyway, if I understood your question, you have to estimate the model (2) and in this case it is convenient to compare the Bayesian Information Criterion only for the model (2), since it is exactly what you want to run in the end.

Said that, I suggest you to look at the different criteria that exists in literature (AIC, BIC, DIC, FIC, R2,...) in order to be able to choose the one that is more suitable for your needs.

For instance, in my humble opinion, it is not convenient to use the BIC because of the fact it is neither a good estimator of the Kullback-Leibler's divergence (on which it is based) and asymptotically efficient, as suggested by Burnham & Anderson (2002).

I suggest you to use the AIC in the place of the BIC, that, at least, it is more precise from a theoretical point of view.

Of course, choose the model with the lowest statistic score (in both the case).

Answer 2

To identify the best lags , try to fit models with different lag choices.Plotting of ACF and PACF could give a sense of different lags choices. Further to this BIC could be used to determine for which AR and MA lag BIC is getting minimised.