# Clarifying lag number selection in AR,VAR, VECM etc. models

When it comes to optimum lag length selection, we are supposed to comply with certain information criteria such as Akaike, Schwarz etc. As far as I know, either of them suggest the proper lag number that we should embed in our model, which they are implicitly supposed to be in serial order.

For instance, if we have a VAR(3) of $$Y_t$$ and $$X_t$$ variables, we are supposed to embed in the model $$Y_{t-1}$$, $$Y_{t-2}$$, $$Y_{t-3}$$ and $$X_{t-1}$$, $$X_{t-2}$$, $$X_{t-3}$$ respectively. And here is my question:

Why should the lagged values always be consecutive? Is it prohibitive (according to economic and statistic theory) for a model to include non serial lagged values such as that below?

E.g.,

$$Y_t = a + bY_{t-1} + cY_{t-3} + dY_{t-5} + eX_{t-1} + fX_{t-3} +gX_{t-5} + e_t$$

• Precise and explicit answer. Thank you very much Christoph! Commented Dec 13, 2018 at 19:07
• You should post this under the answer, otherwise Christoph will not get notified of your comment. Commented Dec 14, 2018 at 8:53
• Thank you for the guidance Richard, since I'm a new member of this community. I still have so much to learn! Commented Dec 14, 2018 at 9:12
• You are very welcome here! Commented Dec 14, 2018 at 9:28

My interpretation of the tradition to nevertheless proceed in this fashion is that, even for a simple AR model with maximum lag $$p_{\max}$$, one would need to compare $$2^{p_{\max}}$$ models when combinations of intermediate lags need not enter the model, while we only need to compare $$p_{\max}$$ models when the search is only performed over the maximal lag.
Suppose you entertain lags at "business cycle frequencies" of a few years in quarterly data, so something like 20 lags. Then, $$2^{20}=1048576$$ need to be compared. While the computational cost of doing so may be less prohibitive nowadays, it still somewhat of a burden, and the benefits to finding models with missing intermediate lags may be limited.