In Example 10.4, 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 (j follow a relatively simple pattern. For concreteness, consider a model with four lags:
yt = a0 + (0zt + (1zt-1 + (2zt-2 + (3zt-3 + (4z t-4 + ut.
Now, let us assume that the 8. follow a quadratic in the lag, j:
(j = (0 + (1j + (2j2,
for parameters (0 + (1, and (2, This is an example of a polynomial distributed log (PDL) model.
(i) Plug the formula for each (j 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? (Think F test.)