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

Exercise 8

An interesting economic model that leads to an econometric model with a lagged dependent variable relates yt to the expected value of xt, say, xt*, where the expectation is based on all observed information at time t - 1:

yt = ?0 + a1xx* + ut.

A natural assumption on {ut} is that E(ut|It-1) = 0, where It-1 denotes all information on y and x observed at time t — 1; this means that E(yt|It-1) = ?0 + ?1xt*. To complete this model, we need an assumption about how the expectation x* is formed. We saw a simple example of adaptive expectations in Section 11.2, where x* = xt-1. A more complicated adaptive expectations scheme is

xt* - xt-1* = ?(xt-1 - × t-1*)

where 0<?<1. This equation implies that the change in expectations reacts to whether last period's realized value was above or below its expectation. The assumption 0<?<1 implies that the change in expectations is a fraction of last period's error.

(i) Show that the two equations imply that

yt = ??0 + (1 - ?)yt-1 + ??1xt-1 + ut - (1 - ?)ut-1.

[Hint: Lag equation one period, multiply it by (1 - ?), and subtract this from. Then, use.]

(ii) Under E(ut|It-1) = 0, {ut} is serially uncorrelated. What does this imply about the new errors, vt = ut - (1 - ?)ut-1?

(iii) If we write the equation from part (i) as yt=?0+ ?1yt-1+ ?2xt-1+vt how would you consistently estimate the ?j?

(iv) Given consistent estimators of the ?j., how would you consistently estimate ? and ?1?Equation yt = ?0 + a1xx* + ut.

xt* - xt-1* = ?(xt-1 - × t-1*)

Step 1 of 3

(i)

Here in this problem, the given equation is as follows:

The given economic model leads to an econometric model with a lagged dependent variable relating to the expected value of, say, where the expectation is based on all observed information at time.

When this equation is lagged one time once, multiply it by, and subtract it from the equation to get the new equation as follows:

Now, the given equation in the problem is as follows:

, where

Rewrite the above equation as follows:

Apply the above equation in the first equation to obtain the below equation as follows:

Hence, the above mentioned two equations imply the required equation.