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ECON3330 Project: Assignment and brief

report.

October 14, 2022

Answer all questions and submit your solution as a single pdf file on black-

board. Marks as shown. Marks will be awarded for correct answers and for

clarity of written expression. You can submit your report with the answers as

a handwritten or typed document.

Question 1 (4 marks)

You are interested in estimating a wage equation using quarterly data. To this

purpose you consider the following two regressions of the log of wage wt against

the log of education et:

w = α1e+ Sα2 + u (1)

where S = [s1 s2 s3 s4] are seasonal dummy variables;

w = β1e+Dβ2 + u (2)

where D = [1n s2 s3 s3] includes a constant and three dummy variables for

quarters 2, 3, 4.

1. What is the relationship between the coefficients of regression (1) and

regression (2)?

2. Interpret the coefficients of the two regressions.

3. Suppose that wages show significant seasonality. Explain what happens if

you omit S from regression (1) and D from regression (2).

Question 2 (4 marks)

You are estimating a consumption function by regressing the log of consumption

ct against the log of income yt and a set of control variables Zt (where Zt is a

1× (K ? 1) vector):

ct = β0 + β1yt + Ztβ2 + ut

1

You suspect that age may affect the marginal rate of consumption (assume

age is observed in the dummy variable gt, with gt = 1 for people below 40 years

old and gt = 0 for people above 40 years old).

1. Explain how you would test for the hypothesis that the marginal rate of

consumption is not affected by age.

2. What test statistic is appropriate in this context?

Question 3 (4 marks)

Consider the following production function estimation, where the log of output

yt is regressed against a costant, the log of labour Lt and the log of capital Kt:

yt = β0 + β1Lt + β2Kt + ut

Suppose we want to test the hypothesis of constant returns to scale (β1+β2 =

1) against the alternative of either increasing or decreasing returns to scale.

1. Write the restriction in standard form Rβ = r.

2. Transform the model in order to have the previous restriction expressed

as a single parameter restriction (γ = 0).

3. Explain how you would test for constant returns to scale and list the

assumptions you would be using.

Question 4 (3 marks)

You are estimating a consumption function by regressing the log of consumption

ct against the log of income yt and a number of control variables Zt (with Zt

being a 1×K vector):

ct = β1yt + Ztβ2 + ut

Suppose that these control variables have no effect on consumption (β2 = 0).

1. Explain what happens to the estimate of β1 if you include these variables

in the regression.

2. Explain how you would test for β2 = 0 and list the assumptions you need

in order to do so.

Question 5 (4 marks)

Consider the error component model (panel data)

yit = Xβ + vi + it

Assume that the regressors display no within group variation (i.e. Xit = Xis)

and the data are balanced, with m groups and T observations per group. Show

that the GLS and OLS estimators are identical in this special case.

2

Question 7 (3 marks)

Consider the following model:

y = β1n + u

and the following three estimators for the parameter β:

Determine if they are unbiased and consistent estimators of the parameter of

interest. If you are forced to use one of these three estimators (instead of the

OLS), which one would you use?

Question 8 (3 marks)

Production data for 22 firms in a certain industry produce the following (where

y = log (output) and x = log (labor)):

1. Compute the least squares estimates of α, β in the model: yt = α+βxt+ut.

2. Test the hypothesis that β = 1 (assume that ut is i.i.d. and normally

distributed).

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