Introductory Econometrics 4th Edition by Jeffrey Wooldridge

Exercise 15

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 0 , 1 , and 2. 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 h , for h = 0,1,2.
(ii) Explain the regression you would run to estimate the h.
(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.)