B. Brown

Spring 2008

Midterm Exam II: Answers

Answer three of the following questions. You must answer question 5. The questions are weighted equally. You have 75 minutes. You may use a calculator and one side of an 8.5 × 11 as a cheat sheet. Relevant tables are attached. Please return the exams to my ofﬁce by classtime on Tuesday. Brevity is recommended.

1. Professor I. M. Economist asked his research assistant to estimate the parameters of the regression model Yi = α + βXi + γZi + δWi + ui .

He suspected that multicollinearity might be a problem and suggested an analysis of the possibility. Suppose you are the research assistant. a. Explain exactly what is meant by multicollinearity in this model. Extreme multicollinearity means that one of the RHS variables is perfectly linearly related to the remaining variables. Near extreme means that the correlation between one of the RHS variables and a linear combination of the remaining variables is close to one. b. What would the consequences of collinearity be for estimation and inference? OLS will still be unbiased, consistent and BLUE (BUE under normality). Likewise the usual statistic, such as the t-ratios, will have the usual distribution, such as the t-distribution, under the null hypothesis. Under the alternative hypothesis, however, there will be loss of power since the variance of the estimates on the collinear terms will become highly variable. c. How might you go about detecting the presence and severity of collinearity? If the coefﬁcient estimate of the suspected collinear term is signiﬁcant, then collinearity is not a problem. If the coeffcient is insigniﬁcantly different from zero when considered alone but the possible collinear terms reject a test of the null hypothesis that they all are zero, then collinearity is a problem.

d. Given that collinearity is present and poses serious complications, what is a possible approach to solving or circumventing the problem. (i) Gather more data either through additional observations or independent estimates. (ii) Impose economic theory on one or more of the collinear terms to effectively eliminate one of the problem variables. (iii) Ridge regression to introduce bias as a trade-off against the high variances resulting from the collinearity.

2. For i = 1, 2, . . . , n suppose

(i) (ui , xt ) i.i.d.

(ii) E [ui |xi ] = 0

(iii) E [u2 |xi ] = σ 2

i

i

(iv) E [xi x0i ] = Q, p.d.

(v) E [u2 xi x0i ] = M

i

then establish the asymptotic behavior (convergence in probability or distribution) of P

xi x0i /n. Since xi and hence xi xi0 are i.i.d. and E [xi x0i ] = Q we apply the a. X 0 X/n =

LLN to yield X 0 X/n −→p Q, p.d.

P

b. X 0 u/n = xi ui /n. Since {ui , xi } and hence xi ui are i.i.d. and E [xi ui ] = E [xi E [ui |xi ]] = 0, then LLN yields X 0 u/n −→p 0.

1

ˆ

ˆ

c. β = (X 0 X )−1 X 0 y where y 0 = (y1 , y2 , . . . , yn ) and y = Xβ + u. Rewrite β = 0

−1 0

0

−1 0

0

−1 0

(X X ) X y = β + (X X ) X u = β + (X X/n) X u/n −→p β by (a) and (b). P

√

√

√ˆ

xi ui / n and we know from (b) that xi ui is i.i.d. with

d. n(β − β ). Now X 0 u/ n =

√

zero mean and from (v) has covariance M , so by the CLT X 0 u/ n −→d N (0.M ). And √

√ˆ

n(β − β ) = (X 0 X/n)−1 X 0 u/ n −→d Q−1 · N (0, M ) −→d N (0, Q−1 MQ−1 ). √ˆ

e. n(β i − β 0 )/([Q−1 MQ−1 ]ii )1/2 under H1 : β i = β 1 > β 0 . Adding and subtracting i

i

i

√ˆ

√ˆ

√

we get n(β i − β 0 )/([Q−1 MQ−1 ]ii )1/2 = n(β i − β 1 )/([Q−1 MQ−1 ]ii )1/2 + n(β 1 − i

i

i

1/

β 0 )/([Q−1 MQ−1 ]ii )√2 and the ﬁrst term will be asymptotically N (0, 1) but the second i

shift term will be O( n) in the positive direction.

3. Suppose we are interested in estimating the consumption relationship Csi = α + βYsi + usi

where Csi is consumption and Ysi is disposable income for individual i in state s. The unobservable usi are assumed to have the usual ideal properties. Unfortunately, data are not available on an individual basis but only for statewide aggregates....