econ

Problem Set 3, Econ 120B

Prof. Yixiao Sun

1. Consider the simple linear regression model with one explanatory variable,

Y = β0 + β1X + u.

Suppose we estimate the coeffi cients using OLS on a sample of observations (Xi,Yi), i = 1, · · · ,n. Recall the following definitions of the sample covariance, sample variances, and sample correlation:

ĉov (X,Y ) = 1

n

n∑ i=1

(Xi − X̄)(Yi − Ȳ ),

v̂ar (X) = 1

n

n∑ i=1

(Xi − X̄)2,

v̂ar (Y ) = 1

n

n∑ i=1

(Yi − Ȳ )2,

ρ̂XY = ĉov (Y,Y )√

v̂ar (X) √ v̂ar (Y )

.

(a) Using the facts that β̂0 = Ȳ − β̂1X̄ and β̂1 = ĉov (X,Y ) /v̂ar (X), show that

1

n SSE =

[ĉov (X,Y )] 2

v̂ar (X) .

(b) Show that R2 = ρ̂2XY .

2. Consider the following causal model

y ← bx + v, x ← cy + u,

where b 6= 0,c 6= 0 and bc 6= 1. Suppose that the values of (u,v) are generated from

(U,V ) ∼ N [ 0,

( σuu σuv σuv σvv

)] .

We do not observe (U,V ) but we could observe the equilibrium solution (X,Y ).

1

(a) Consider the special case with σuu = 0 but σvv 6= 0 so that the model becomes

y ← bx + v, x ← cy.

What is best linear prediction of Y given X (i.e., suppose I give you the value of X but withhold the value of Y, what would be your best guess of Y according to the MSE criterion?). What is best linear prediction of X given Y ?

(b) Consider the special case with σvv = 0 but σuu 6= 0 so that the model becomes

y ← bx, x ← cy + u

What is best linear prediction of Y given X? What is best linear prediction of X given Y ?

(c) Now suppose σuu 6= 0 and σvv 6= 0. Under what condition(s) is X useless as a predictor of Y ? Under what condition(s) is Y useless as a predictor of X? Explain.

3. Consider the following model

Yi = Xiβ1 + ui, i = 1, ...,n

withoutan intercept. Thismodel isusedonlyunder theassumption that E(Y ) = E(X) = 0. We maintain this assumption here.

(a) Derive the OLS estimator β̂1 for β1. Show your steps.

(b) Given a dataset, suppose we know that the sample correlation coeffi cient between X and Y is 0.5, the sample variance of X is 4 and that of Y is 60. Compute β̂1.

(c) Now suppose we regress X on Y without an intercept. Given the information in (b), can you find the OLS estimate for the coeffi cient on Y ?

4. Suppose a researcher, using wage data on 250 randomly selected male workers and 280 female workers, obtained the OLS regression estimates (standard errors are given in paren- theses).

ŵage = 12.52 (0.23)

+ 2.12 (0.36)

male, R2 = 0.06. (1)

Here wage is measured in dollars/hour and male is a binary variable which equals 1 if the person is male. Define the gender wage gap as the difference in earnings between males and females.

(a) What is the estimated gender wage gap?

(b) In the sample, what is the mean wage for men and women?

(c) Another researcher uses the same data but regresses wage on female, a variable that equals 1 if the person is female. what are the regression estimates in this new regres- sion? Also compute the R2 for this new regression.

2

5. In your sample, all individuals have received a stimulus check of $1,200/month. Let Y be the monthly income (measured in dollars) and X be the total hours worked in a month. You decide to run a regression of Y net, the monthly income on hours worked, net of the $1,200 stimulus check (Y net = Y − 1200). The estimated coeffi cients are reported below:

Ŷ net = 1256 + 30 X.

(a) What would happen to the estimates if instead you ran the regression using gross income (hence including the $1,200 for all individuals) on hours worked? Explain.

(b) Suppose you construct a new variable, the total hours worked in a week Xweek = X/4. What would be the corresponding slope estimate from a regression of Y net on Xweek?

6. Work on the the Stata Assignment.

3