Consider a case where the X variable causes Y one period in the future. Consider a country where the Income in year "Z" effects or is useful in predicting the consumption one year in the future. X is referred to as a lag variable. The question is how can this lag be detected and subsequently harnessed in order to better describe the behavior of Y ? Initially, if we run a simple multiple regression, we get the following results.

If we examine the autocorrelation of the residuals we find that they do not appear to have any structure. Most analysts would stop here.

However for a causal model to be sufficient one needs to check as to whether or not any remaining or auxiliary information exists in lags of the series of interest. We find a relatively high value of .65 indicating a need to change the model.

Estimating the new model where the additional lag is found to be significant we get...

Finally the proof of sufficiency ... both the ACF and the CCF are approximately zero indicating that there is no need for any more changes.

The data.

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