[Solved] Assignment 219488

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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

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