计量经济学导论CH10习题答案

79 CHAPTER 10

TEACHING NOTES

Because of its realism and its care in stating assumptions, this chapter puts a somewhat heavier

burden on the instructor and student than traditional treatments of time series regressions, but I

think it is worth it. It is important that students learn that there are potential pitfalls inherent in

using regression with time series data that are not present for cross-sectional applications.

Trends, seasonality, and high persistence are ubiquitous in time series data. By this time,

students should have a firm grasp of multiple regression mechanics and inference, and so you

can focus on those features that make time series applications different from cross-sectional ones.

I think it is useful to discuss static and finite distributed lag models at the same time, as these at

least have a shot at satisfying the Gauss-Markov assumptions. Many interesting examples have

distributed lag dynamics. In discussing the time series versions of the CLM assumptions, I rely

mostly on intuition. The notion of strict exogeneity is easy to discuss in terms of feedback. It is

also pretty apparent that, in many applications, there are likely to be some explanatory variables

that are not strictly exogenous. What the student should know is that, to conclude that OLS is

unbiased – as opposed to consistent – we need to assume a very strong form of exogeneity of the

regressors. Chapter 11 shows that only contemporaneous exogeneity is needed for consistency.

Although the text is careful in stating the assumptions, in class, after discussing strict exogeneity,

I leave the conditioning on X implicit, especially when I discuss the no serial correlation

assumption. As this is a new assumption I spend some time on it. (I also discuss why we did not

need it for random sampling.)

Once the unbiasedness of OLS, the Gauss-Markov theorem, and the sampling distributions under

the classical linear model assumptions have been covered – which can be done rather quickly – I

focus on applications. Fortunately, the students already know about logarithms and dummy

variables. I treat index numbers in this chapter because they arise in many time series examples.

A novel feature of the text is the discussion of how to compute goodness-of-fit measures with a

trending or seasonal dependent variable. While detrending or deseasonalizing y is hardly perfect

(and does not work with integrated processes), it is better than simply reporting the very high R-squareds that often come with time series regressions with trending variables. 80 SOLUTIONS TO PROBLEMS

10.1 (i) Disagree. Most time series processes are correlated over time, and many of them

strongly correlated. This means they cannot be independent across observations, which simply

represent different time periods. Even series that do appear to be roughly uncorrelated – such as

stock returns – do not appear to be independently distributed, as you will see in Chapter 12 under

dynamic forms of heteroskedasticity.

(ii) Agree. This follows immediately from Theorem 10.1. In particular, we do not need the

homoskedasticity and no serial correlation assumptions.

(iii) Disagree. Trending variables are used all the time as dependent variables in a regression

model. We do need to be careful in interpreting the results because we may simply find a

spurious association between yt and trending explanatory variables. Including a trend in the

regression is a good idea with trending dependent or independent variables. As discussed in

Section 10.5, the usual R-squared can be misleading when the dependent variable is trending.

(iv) Agree. With annual data, each time period represents a year and is not associated with

any season.

10.2 We follow the hint and write

gGDPt-1 = 0 + 0intt-1 + 1intt-2 + ut-1,

and plug this into the right-hand-side of the intt equation:

intt = 0 + 1(0 + 0intt-1 + 1intt-2 + ut-1 – 3) + vt

= (0 + 10 – 31) + 10intt-1 + 11intt-2 + 1ut-1 + vt.

Now by assumption, ut-1 has zero mean and is uncorrelated with all right-hand-side variables in

the previous equation, except itself of course. So

Cov(int,ut-1) = E(inttut-1) = 1E(21tu) > 0

because 1 > 0. If 2u= E(2tu) for all t then Cov(int,ut-1) = 12u. This violates the strict

exogeneity assumption, TS.2. While ut is uncorrelated with intt, intt-1, and so on, ut is correlated

with intt+1.

10.3 Write

y* = 0 + (0 + 1 + 2)z* = 0 + LRPz*,

and take the change: y* = LRPz*.