Introductory Econometrics: A Modern Approach 6th Edition by Jeffrey M Wooldridge

Exercise 12

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.)

Step 1 of 3

For the finite distributed lag model:

It has been discovered that the equation form of this model turns out to be

For such kind of models, the estimate of ,and are discovered to be imprecise and individually statistically insignificant, hence, the model is said to be given with the multicollinearity problem.

To overcome this problem, the lag coefficients are assumed to follow quadratic in lag

Such that

Consider for the purpose of concreteness, the finite distributed lag model is given with four lags and no other explanatory variables, and is represented by:

(i)

On plugging the formula for each into the distributed lag model, the result is:

The result is:

Hence, on re-writing the finite distributed lag model with four lags in terms of the parameters for h = 0, 1, 2, the result obtained is:

The finite distributed model, thus become polynomial distributed model.