Assignment 3 of ETF3200
IMPORTANT: This assignment is assessed. The deadline for submission of work
for this assignment is 5:00pm, Friday October 22nd. Your solutions must be typed
before you may submit them through Assignment Box on the Moodle website.
Late submission of assignments Work submitted late will receive a mark of zero, unless
the Lecturer has been advised as soon as possible of any extenuating circumstances with
supporting evidence (e.g. medical certificate).
Question 1 (10 marks)
Consider a simple time series model of the form
Xt = a+ b t+ cXt−1 + et, t = 1, 2, . . . , T, (1)
where et ∼ N(0, 1) is a sequence of independent and normally distributed (i.n.d.) random
variables. Let X0 = 0, and a, b and c be known constants.
Model (1) covers the following commonly used scenarios.
• Linear trending model:
Xt = a+ b t+ et, t = 1, 2, . . . , T. (2)
What is the probability distribution of Xt ? Provide the necessary detail [3 marks]
• Random walk model without intercept:
Xt = Xt−1 + et, t = 1, 2, . . . , T. (3)
Do you know the probability distribution of Xt ? Provide the necessary detail [3
marks]
• Random walk model with intercept:
Xt = a+Xt−1 + et, t = 1, 2, . . . , T. (4)
Is the probability distribution of Xt different from those of (2) and (3) ? Give your
reasoning [4 marks].
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Question 2 (20 marks)
Macroeconomists are interested in factors that explain economic growth. An aggregate
production function specification was studied by Duffy and Papageorgiou. The data are
in the data file ces being attached below. They consist of cross-sectional data on 82
countries for 28 years, 1960–1987.
• (a) Estimate a Cobb–Douglas production function
LYit = β1 + β2LKit + β3LLit + eit,
where LY is the log of GDP, LK is the log of capital, and LL is the log of labor.
Interpret the coefficients on LK and LL. Test the hypothesis that there are constant
returns to scale, β2 + β3 = 1. (3 marks)
• (b) Add a time trend variable t = 1, 2, …, 28, to the specification in (a). Interpret
the coefficient of this variable. Test its significance at the 5% level. What effect
does this addition have on the estimates of β2 and β3 ? (3 marks)
• (c) Assume β2 + β3 = 1. Solve for β3 and substitute this expression into the model
in (b). Show that the resulting model is
LY Lit = β1 + β2LKLit + λt+ eit,
where LYL is the log of the output–labor ratio, and LKL is the log of the capi-
tal–labor ratio. Estimate this restricted, constant returns to scale, version of the
Cobb–Douglas production function. Compare the estimate of β2 from this specifi-
cation to that in part (b). (3 marks)
• (d) Estimate the model in (b) using a fixed effects estimator. Test the hypothesis
that there are no cross-country differences. Compare the estimates to those in part
(b). (3 marks)
• (e) Using the results in (d), test the hypothesis that β2 + β3 = 1. What do you
conclude about constant returns to scale ? (2 marks)
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• (f) Estimate the restricted version of the Cobb–Douglas model in (c) using the fixed
effects estimator. Compare the results to those in part (c). Which specification do
you prefer ? Explain your choice. (4 marks)
• (g) Using the specification in (b), replace the time trend variable with dummy
variables D2–D28. What is the effect of using this dummy variable specification
rather than the single time trend variable ? (2 marks)
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