Question 1. [8 marks] Consider the true population model, which satisfies the assumptions for simple linear regression m

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Question 1. [8 marks] Consider the true population model, which
satisfies the assumptions for simple linear regression model, as y
= βo + β1x + u The OLS estimators can be derived as follows: βˆ 1 =
Pn i=1(xi − x¯)(yi − y¯) Pn i=1(xi − x¯) 2 and βˆ 0 = ¯y − βˆ 1x¯
Show that OLS estimators are unbiased estimators. Question 2. [12
marks] (a) Consider the true population model, which satisfies MLR
assumptions, as follows: wage = β0 + β1edu + β2IQ + u 2 ECON 2123
Where edu = education. Now consider two estimators of β1. The
estimator βˆ 1 comes from the multiple regression wage ˆ = βˆ 0 +
βˆ 1edu + βˆ 2IQ and the estimator β˜ 1 comes from the simple
regression wage ˜ = β˜ 0 + β˜ 1edu Assume that β2 6= 0 and
corr(edu, IQ) > 0. (i) Between βˆ 1 and β˜ 1, which estimator is
unbiased? [1] (ii) Between βˆ 1 and β˜ 1, which estimator has a
lower variance? [1] (iii) Between βˆ 1 and β˜ 1, which estimator
would you prefer? Explain. [3] (b) Suppose that wage is determined
by the following true population model that satisfies the
assumptions for the MLR model. wage = βo + β1edu + β2abil + u where
abil = ability, β2 > 0 and abil = γ0 + γ1edu + w, γ1 > 0. As
abil is not observed, the following model is estimated instead wage
= αo + α1edu + v where u, v, and w are error terms. Show that E[
ˆα1] > β1. [3] (c) Consider the model: log(wage) = β0 + β1female
+ β2exper + β3female ∗ exper + u where exper is the years of work
experience, and female is a dummy variable (1 if the person is
female, and 0 otherwise). How would you measure the difference in
the return of experience between males and females? [2] (d)
Consider the model: log(wage) = β0 + β1female + β2graduate +
β3female ∗ graduate + u 3 ECON 2123 where graduate is a dummy
variable dummy variable (1 if the person has graduated from
college, and female is a dummy variable (1 if the person is female,
and 0 otherwise). How would you measure the return of graduating
from college for men? [2] Question 3. [24 marks] Consider the
following estimated equation that can be used to study the effects
of education, experience, tenure, and number of dependents on
average hourly wage (standard errors are in parentheses): wage ˆ =
−3.347 (0.768) + 0.549 (0.053) educ + 0.189 (0.039) exper + 0.245
(0.051) tenure − 0.023 (0.114) numdep −0.004 (0.001) exper2 − 0.003
(0.002) tenure2 n = 526, SSR = 4676.76, SST = 7160.41 Education,
experience and tenure are measured by the number of years. (a)
Discuss the sign and magnitude of the estimated coefficients on
educ. [2] (b) Calculate an estimate of the error variance. [2] (c)
Compute the values of R2 and adjusted R2 . Interpret them. [4] (d)
Using the standard normal approximation, find the 95% confidence
interval for βtenure and βnumdep. Show your work. [3] (e) Test the
overall level of significance of above model at the 5% significance
level. Show your work. [3] (f) Does the variable exper have the
expected estimated effect on wages. At what point does another year
of experience reduce the wage. Show your work. [3] (g) Is either
exper2 or tenure2 individually significant at the 5% level against
4 ECON 2123 the two-sided alternative? Show your work. [4] (h)
Dropping exper2 and tenure2 from the equation gives (standard
errors are in parentheses) wage ˆ = −3.355 (0.788) + 0.620 (0.053)
educ + 0.025 (0.012) exper + 0.168 (0.022) tenure + 0.177 (0.110)
numdep n = 526, SSR = 4, 941.90, SST = 7, 160.41 Are exper2 and
tenure2 jointly significant in the original equation at the 5%
significance level? Show your work. [3] Question 4. [6 marks] The
following four equations were estimated using the 88 observations
in Wage data (standard errors are in parentheses): Model 1: wage ˆ
= −7.423 (2.387) + 0.917 (0.167) educ + 0.162 (0.036) exper (1) R2
= 0.324, R¯2 = 0.308 Model 2: wage ˆ = −7.969 (2.361) + 0.854
(0.167) educ + 0.385 (0.116) exper − 0.006 (0.003) exper2 (2) R2 =
0.356, R¯2 = 0.333 Model 3: wage ˆ = −3.816 (3.650) + 0.630 (0.276)
educ − 0.038 (0.157) exper − 0.017 (0.013) educ ∗ exper (3) R2 =
0.338, R¯2 = 0.314 Model 4: wage ˆ = −5.958 (3.836) + 0.707 (0.277)
educ + 0.250 (0.233) exper + 0.009 (0.013) educ ∗ exper − 0.005
(0.003) exper2 (4) 5 ECON 2123 R2 = 0.359, R¯2 = 0.328 (a) Which of
these four models would you recommend to use? Why? [2] (b) The
average years of education and experience in the sample are 13
years and 15 years, respectively and wage is measured in
dollar/hour. Calculate the estimated effect of exper on wage at the
mean level of education and experience in model 3 and model 4. [4]
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