Assignment Details

Subject: Economics / General Economics

Question

Econ 326 (004)

Winter Session, Term II, March 2017

Problem Set 3

Due: March 21.

1. Consider the population regression function

log yi =

(a) If 3 = 2 + 3; 0 + 1 xi1 + 2 log xi2 + log xi3 + ui 3 rewrite the regression equation including (b) Explain how one might go about testing the restriction 3.

2 = 3. 2. Consider the multiple regression model with three independent variables (under the

classical assumptions).

y= 0 + 1 x1 + 2 x2 + 3 x3 You are interested in testing the null hypothesis H0 : +u 1 3 2 = 1. (a) Let ^ 1 and ^ 2 denote the OLS estimators of 1 and 2 . Write an expression for

var( ^ 1 3 ^ 2 ) in terms of var( ^ 1 ); var( ^ 2 ) and cov( ^ 1 ; ^ 2 ).

(b) Write out the t-statistic for testing H0 : 1 3 2 = 1.

(c) De…ne 1 = 1 3 2 and ^1 = ^ 1 3 ^ 2 . Write a regression equation involving

^

0 ; 1 ; 2 and 3 that allows you to directly obtain 1 and its standard error.

(d) Explain how you can use your work in (c) to carry out the hypothesis test.

3. If y is regressed on 3 explanatory variables and a constant, the result is:

parameter estimate

standard error

3638:25

112:28

0

0:148

0:017

1

11:13

5:88

2

2:20

1:45

3

n = 706

R2 = :113

(a) Test if either 2 or 3 is individually signi…cant at the 5% level against a two-sided

alternative. Be sure to write out all null and alternative hypotheses.

(b) If dropping x2 and x3 from the equation gives estimation of y = ^ + ^ x1

0 1 parameter estimate

standard error

3586:38

38:91

0

0:151

0:017

1

2

n = 706

R = :103

Test if the parameters

cance. 2 and 3 are jointly signi…cant at the 5% level of signi…- 1 4. A researcher obtains data to estimate the PRF

y= + 0 1 x1 + 2 x2 yi xi1

3 2

5 5

2 3

6 7

7 5

9 8

6 5

11 8 xi2

7

5

3

5

6

2

4

2 + 3 x3 +u xi3

3

4

1

6

4

8

12

11 (a) Obtain the o.l.s. estimates for ^ 0 ; ^ 1 ; ^ 2 and ^ 3 . (A spreadsheet will ease the

computation burden). Find the residuals u^i and the SSR for the regression.

(b) Test the hypothesis that 1; (c) Test the hypothesis that 2 2 and and 3 (d) Test the multiple hypotheses that 3 are jointly equal to zero. can be excluded from the model.

1 = 0 and 2 = 3. 5. If ^ 0 ; ^ 1 ; : : : ; ^ k are the OLS estimates from the regression of yi on xi1 ; xi2 ; : : : ; xik; i =

1; 2; : : : ; n, show that for non-zero c0 ; c1 ; c2 ; : : : ; ck , a regression of c0 yi on c1 xi1 ; c2 xi2 ; : : : ; ck xik ,

i = 1; 2; : : : ; n; gives estimates ~ 0 = c0 ^ 0 ; ~ 1 = cc10 ^ 1 ; : : : ; ~ k = cck0 ^ k .

6. To test the e¤ectiveness of a job training program on the subsequent wages of workers

the following model is estimated:

ln(wage) = 0 1 train + + 2 educ + 3 exper +u Where

train = 1 if worker participated in program

0

otherwise An unobserved factor that in‡uences the wage of a worker is worker ability.

(a) What is the interpretation of the parameter estimates ^ 1 ; ^ 2 and ^ 3 . How would

the interpretation of the estimates change if the dependent variable had been

wage instead of ln(wage)?

(b) If the error term u contains unobserved worker ability, and less able workers have

a greater chance of being selected for the program, will there be bias in the OLS

estimator of 1 ? Is it possible to predict the direction of bias if it exists?

7. In the speci…cation

yi = 0 + 1 xi1 + 1 di1 + ui The variable x1 is a quantitative variable and the variable d1 is a dummy (binary)

variable. Sample size is n. There are n0 observations with di1 = 0 and n1 observations

with di1 = 1. For ease, let the …rst n0 observations correspond to the subgroup with

di1 = 0.

2 (a) Consider calculating the OLS estimators ^ 1 and ^1 by performing simple regressions. Write out the expressions for coe¢ cient estimates for the ancillary regressions. 3