Testing between two competing linear models with different lagged independent variables

Question

What would be an appropriate statistical test to determine the best model for two competing linear models?

Both models use the same independent variables (IVs); however, some independent variables are measured at different times. In the first model, the IVs lag the dependent variable (DV) by one period of time, and in the second model, the IVs lag the DV by a longer period of time. I'm interested in determining which lagged period is more appropriate to use. The DV for both models is measured at the same time period. In symbols:

$$ y_{i,t} = x_{i,t-1} + y_{i,t-1} + z_{i,t-1} + \epsilon_i $$

versus

$$ y_{i,t} = x_{i,t-2} + y_{i,t-2} + z_{i,t-2} + \epsilon_i $$

Are AIC, BIC, etc. appropriate for this?

Answer 1

You could use AIC, BIC to decide which model is "better"; it should have lower AIC, BIC values.

You could also form a nesting model

$$y_{i,t} = \beta_0 + \beta_{11} y_{i,t-1} + \beta_{21} x_{i,t-1} + \beta_{31} z_{i,t-1} + \beta_{12} y_{i,t-2} + \beta_{22} x_{i,t-2} + \beta_{32} z_{i,t-2} + \varepsilon_i $$

and test a joint hypothesis $H_{0,1}$: $\beta_{12}=\beta_{22}=\beta_{32}=0$.
If you cannot reject it, you would choose model 1 over model 2.

You could then also test a joint hypothesis $H_{0,2}$: $\beta_{11}=\beta_{21}=\beta_{31}=0$.
If you cannot reject it, you would choose model 2 over model 1.

It could happen that you reject both or reject none of the two, which would not help you much (the result would generally be informative but would not help you select one model over the other). But you could at least try and hope that you are able to reject one hypohesis but not the other.