In we saw that our estimates of the individual lag coefficients in a distributed lag model...

In we saw that our estimates of the individual lag coefficients in a distributed lag model were very imprecise. One way to alleviate the multicollinearity problem is to assume that the 8. follow a relatively simple pattern. For concreteness, consider a model with four lags:

Now, let us assume that the 8. follow a quadratic in the lag, j:

for parameters β0, β1, and β2. This is an example of a polynomial distributed lag (PDL) model.

(i) Plug the formula for each 8j into the distributed lag model and write the model in terms of the parameters βh, for h = 0,1,2.

(ii) Explain the regression you would run to estimate the βh.

(iii) The polynomial distributed lag model is a restricted version of the general model. How many restrictions are imposed? How would you test these? (Hint: Think F test.)