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

Exercise 4

Let gGDPt denote the annual percentage change in gross domestic product and let int t denote a short-term interest rate. Suppose that gGDP tis related to interest rates by

,

where ut is uncorrelated with int,, intt_v and all other past values of interest rates. Suppose that the Federal Reserve follows the policy rule:

where ?1 > 0. (When last year's GDP growth is above 3%, the Fed increases interest rates to prevent an "overheated" economy.) If vt is uncorrelated with all past values of intt and ut, argue that intt must be correlated with ut_1. (Hint: Lag the first equation for one time period and substitute for gGDPt-1 in the second equation.) Which Gauss-Markov assumption does this violate?

Step 1 of 2

Consider as annual percentage change in gross domestic product.

Given is related to short term interest rate by:

,

where is uncorrelated with and all other past values of interest rates.

At one lag time period, such that, at, is related to short term interest rate by

,

where is uncorrelated with and all other past values of interest rates.

That implies,

Now substitute for in, where and is uncorrelated with all the past values of and

The result is:

In the equation,

When is uncorrelated with all the past values of and, is uncorrelated with and, that implies,

Now, in order to observe whether is correlated with, consider the covariance of with

The result is:

Since, holds as given, it is concluded that there is correlation between and , Assume

Hence, it is concluded that is correlated with