marks are allocated to Section A.
40 marks are allocated to Section B.
40 marks are allocated to Section C.
Justify all of your answers.
Show the basis of all calculations.
If you do not wish part of an answer to be marked, delete this part clearly.
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1 of 5
Section A. [20 marks]
Consider the regression output below, where the outcome variable lnwage is the natural
logarithm of hourly wage; belavg is a dummy variable equal to 1 if an individual has below
average looks and zero otherwise; abvavg is a dummy variable equal to 1 if an individual
has above average looks and zero otherwise; female is a dummy variable equal to 1 if
an individual is female and zero otherwise; abvavg_female is an interaction term between
abvavg and female; educ measures the total number of years spent in school; exper mea sures the total number of years spent working; exper2 is the square of exper; bigcity is a
dummy variable equal to 1 if an individual lives in a big city and zero otherwise; smallcity
is a dummy variable equal to 1 if an individual lives in a small city and zero otherwise.
1. Interpret the parameter estimate on belvavg both in terms of its statistical significance
and its magnitude. [4 marks]
2. Discuss whether there is any evidence that women with above average looks earn
more than men with above average looks. [4 marks]
3. Discuss whether there is any evidence of a difference in hourly wages across cities
of different sizes. [4 marks]
4. Bob is a man with below average looks, 5 years of experience, living in a big city, and
with 12 years of education. Donald is a man with below average looks, 5 years of
experience, living in a big city, and with 14 years of education. What is the predicted
difference between Donald’s and Bob’s hourly wages Explain whether you can
conclude if this difference is statistically significant. [4 marks]
5. How would you change the regression if you suspect that the effect of experience on
hourly wage was different for men and women Explain your answer. [4 marks]
2 of 5 TURN OVER
Section B. [40 marks]
Given the random sample {yi, x1i, x2i}N
i=1, each observation satisfies
yi = β0 + β1x1i + β2x2i + ui,
where yi is a dependent variable, x1i and x2i are independent variables, ui is an error term,
and β0, β1, and β2 are parameters. Assume that E(ui|x1i, x2i)=0 and Var(ui|x1i, x2i) = σ2.
1. Derive (but do not solve) the first order conditions for the ordinary least squares (OLS)
estimators β 0, β 1, and β 2. [5 marks]
2. Suppose (for this part only) that x1i and x2i are dummy variables, i.e., equal to 0 or
1, with x2i = 1 x1i. Do the first order conditions for the OLS estimators in Part [1]
guarantee the existence of a unique minimum Explain your answer. [5 marks]
The OLS estimators for β0, β1, and β2 are given by
β 0 = ˉy β 1 ˉx1 β 2 ˉx2,
β 1 =
PN
i=1 x2
2i
PN
i=1 x1i yi PN
i=1 x1i x2i
PN
i=1 x2i yi
PN
i=1 x2
1i
PN
i=1 x2
2i
PN
i=1 x1i x2i
2 ,
β 2 =
PN
i=1 x2
1i
PN
i=1 x2i yi PN
i=1 x1i x2i
PN
i=1 x1i yi
PN
i=1 x2
1i
PN
i=1 x2
2i
PN
i=1 x1i x2i
2 ,
where yˉ = 1
N
PN
i=1 yi, ˉx1 = 1
N
PN
i=1 x1i, and ˉx2 = 1
N
PN
i=1 x2i are sample means, and
where y i = yi yˉ, x1i = x1i ˉx1, and x2i = x2i ˉx2 are deviations from sample means.
3. Prove that β 1 is an unbiased estimator for β1. [10 marks]
The conditional variances of β 1 and β 2 are given by
Var(β 1|{x1i, x2i}N
i=1) = σ2
PN
i=1 x2
1i(1 2)
,
Var(β 2|{x1i, x2i}N
i=1) = σ2
PN
i=1 x2
2i(1 2)
,
where 2 is given by
2 =
PN
i=1 x1i x2i
2
PN
i=1 x2
1i
PN
i=1 x2
2i
4. Interpret and comment on its relation to the precision of β 1 and β 2. [10 marks]
Suppose that you omit x2i from the specification above and instead estimate the model
given by yi = δ0 + δ1x1i + vi, where yi is a dependent variable, x1i is an independent
variable, vi is an error term, and δ0 and δ1 are parameters.
5. Show under what further conditions δ
