# Modeling different lag structures

I know there are various information criteria that can be used to compare model specifications, including those with different lag structures. I can easily compare the Akaike Information Criterion (AIC) of two models.

My question is how to develop models of different lag structures for comparison?

Example problem:

Dependent variable: consumer price index (CPI)

Independent variables: money supply (M), interest rates (r,) exchange rate (XR)

My initial specification is: CPIt=a0+a1Mt+a2rt+a3XRt.

How would a one period lag structure be specified? And how would a two period lag structure be specified? Is the specification in the following link correct? It was unanswered.

Testing between two competing linear models with different lagged independent variables

• I have answered the linked question now. – Richard Hardy Apr 23 '15 at 18:18
• I guess an error term is missing in your initial specification. As it stands now, your model is a deterministic equation, some kind of definition used in accountancy rather than a regression model. – javlacalle Apr 23 '15 at 20:01

## 1 Answer

This is not really an answer, but I hope it will be helpful.

First, your "initial specification" is rather problematic. For the model to be estimable by OLS you would need the regressors (the right-hand-side variables) to be exogenous with respect to the regressand (the left-hand-side variable). Here you have interest rates that are anything but exogenous with respect to CPI. Central banks normally follow the development of CPI closely and try to affect CPI using the main instrument they have, i.e. the interest rates. That is, the interest rate depends on CPI due to the actions of the central bank. Moreover, exchange rate and money supply could also be argued to be endogenous. Thus there is very little chance that the regressors are uncorrelated with errors. In presence of correlation between regressors and the error term, the OLS estimates will not have their nice properties of unbiasedness and consistency. Therefore you should rethink the model specification. But perhaps you are analyzing a country where the central bank has some other target than inflation stability, that does not have a free-floating currency or is peculiar in some other way. Then my remarks may not apply.

Second, I suppose it depends on the definition of what a "one period lag structure" is. Have you looked carefully into the source you are following (textbook, lecture notes, slides, research paper) to find the definition? Or are you asking for the definition here so that What is a one period lag structure? actually is you question?

• (+1) Interesting interpretation. The Central Bank probably does not react immediately after a change in CPI. There may a lag between the changes in CPI and the decision to change the interest rate, $r_t$. In that case, CPI and $r_t$ are not independent of each other but they are contemporaneously uncorrelated, that is, given a time point $t$, $E(\hbox{error}_t r_t)=0$ (the shocks at time $t$ do no affect the interest rate at time $t$ because the Central Bank has not yet reacted). In this case, OLS will be biased but will at least be consistent and inference will be valid in large samples. – javlacalle Apr 23 '15 at 20:03
• @javlacalle, The central bank may be forward looking rather than backward looking. If the central bank is able to predict CPI, it will not wait for the CPI to realize but will try to affect it before that. Even if the information flow may be messy (CPI $\rightarrow$ interest rate, vice versa or both ways), I suspect endogeneity may be a worry nevertheless. – Richard Hardy Apr 23 '15 at 20:09
• A forward looking Central Bank is a possible scenario. A dynamic factor model or some other multivariate specification that captures the comovement among the variables may be more appropriate. – javlacalle Apr 23 '15 at 20:35