Department of Economics
The University of Melbourne
ECOM30001/ECOM90001: Basic Econometrics
Semester 1, 2022
Assignment 1
Introduction
Consider the following econometric model for manufacturing firms operating in a small
developing country:
ln Yit = β0 + βK ln Kit + βL ln Lit + βM ln Mit
+ βKK [ln Kit]
2 + βLL [ln Lit]
2 + +βMM [ln Mit]
2
+ βKL [ln Kit ∗ ln Lit] + βKM [ln Kit ∗ ln Mit] + βLM [ln Lit ∗ ln Mit] + εit (1)
where:
Yit = Real value of output (1991 firm specific output prices) for firm i in period t
Kit = Real replacement value of plant and machinery (1991 prices) for firm i in period t
Lit = Total number of workers in firm i in period t
Mit = Real total cost of raw materials (1991 firm specific output prices) for firm i in period t
and lnX denotes the natural logarithm of variable X.
Note that all of the variables in the econometric model (1) are measured in natural
logarithms. Assume that the random error εi satisfies all of the assumptions required for
the Gauss-Markov theorem to hold.
The data assign1.csv, accessible from the Canvas page, contains a sample for 1,867
firms in the manufacturing industry.
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The data file contains the following variables:
firm = Unique identifier for firm
year = Year of observation: 1991 to 2002
lnroutput = Natural logarithm of real firm output
lnrcapital = Natural logarithm of real value of the capital stock
lnlabour = Natural logarithm of the labour input
lnrmaterials = Natural logarithm of the real value of the materials input
exports = 1 if firm exports any products, 0 otherwise
age = firm age, in years
sectorj = 1 if firm operates in sector j , 0 otherwise, j = 1, 2, . . . 8
You will need to install following packages to complete this assignment:
stargazer : for easily generating output in R
car : for easily conducting hypothesis tests in R
lmtest : for easily conducting the Ramsey RESET test in R
These can be installed directly in RStudio from the packages tab or by using the com-
mand install.packages() and inserting the name of the package in the brackets.
Question 1(d) and Question 2(c) Ramsey RESET Test [Week 4, Lecture 1]:
In order to apply a Ramsey RESET test you will need to use the lmtest package. First,
estimate your model using the lm command. For the RESET test with squared fitted
values:
resettest(model name, power=2, type="fitted")
where model name is the name of your saved regression results from the lm command.
For the RESET test with squared and cubed fitted values:
resettest(model name, power=2:3, type="fitted")
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Sample code for Question 2(a): Suppose you wish to find the sample mean for
the variables lnrcapital, lnlabour, and lnrmaterials (in the data file assign1.csv).
Suppose you have read the raw data into R and saved it as n R data file named assign1:
assign1 <- read.csv("assign1.csv")
meanlnrcapital <- mean(assign1$lnrcapital)
meanlnlabour <- mean(assign1$lnlabour)
meanlnrmatrerials <- mean(assign1$lnrmaterials)
print(meanlnrcapital)
print(meanlnlabour)
print(meanlnrmatrerials)
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Question 1 [Total 36 marks]
Consider the following restricted version of model (1):
ln Yit = β0 + βK ln Kit + βL ln Lit + βM ln Mit + εit (2)
Economic theory provides the following restrictions on the population parameters:
βK > 0, βL > 0, βM > 0
a) [6 marks] What is the interpretation of the population parameters βK , βL, and
βM?
b) [2 marks] Estimate the econometric model (2) in R and report the results. Do the
signs of your estimated coefficients turn out as you expect? Why or why not?
c) Constant returns to scale implies that a doubling of all factor inputs (K, L,and M)
exactly doubles production. In terms of the model (2), this imposes the restriction
(βK + βL + βM) = 1
i) [6 marks] Using the car package in R, test the hypothesis of constant returns
to scale, for model (2), at a 5% level of significance using an F-test.
ii) [6 marks] Without using the car package in R, test the hypothesis of constant
returns to scale, for model (2), at a 5% level of significance using an F-test.
Hint: You will need to estimate a restricted version of (2), imposing the
restriction of constant returns to scale, to obtain the restricted residual sum
of squares. For example, using βK = (1 − βL − βM) we have the following
restricted model:
[lnYit − lnKit] = β0 + βL [lnLit − lnKit] + βM [lnMit − lnKit] + εit
iii) [8 marks] Without using the car package in R, test the hypothesis of constant
returns to scale, for model (2), at a 5% level of significance using a t-test.
Hint: You will only need the parameter estimates and the estimated covari-
ance matrix of the OLS estimators that you obtained in Question 1(b).
You may find the following result useful. If X, Y , and Z are random variables:
VAR(aX + bY + cZ) = a2 VAR(X) + b2 VAR(Y ) + c2 VAR(Z)
+ 2 a b COV(X, Y ) + 2 b c COV(Y, Z) + 2 a c COV(X,Z)
d) [6 marks] Test the hypothesis that model (2) is correctly specified using the Ram-
sey RESET test, at the 5% level, using both the squares and the cubes of the fitted
values. What is your conclusion?
e) [2 marks] An equivalent version of the model (2) may be obtained by subtracting
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(ln Li) from both sides:
[lnYit − lnLit] = β0 + (βL − 1) lnLit + βK lnKit + βM lnMit + εit (3)
This alternative model has been estimated using the data in assign1.csv and a
value of R2 = 0.8723 was obtained. Consider the following statement:
“Given the sample of data, the econometric model (2) is preferred to the
alternative econometric model (3) because it has a larger R2.”
Do you agree with this statement? Why or why not?
Assignment continues on the next page
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Question 2 [Total 40 marks]
Consider the following extended econometric model:
ln Yit = β0 + βK ln Kit + βL ln Lit + βM ln Mit
+ βKK [ln Kit]
2 + βLL [ln Lit]
2 + +βMM [ln Mit]
2
+ βKL [ln Kit ∗ ln Lit] + βKM [ln Kit ∗ ln Mit] + βLM [ln Lit ∗ ln Mit]
+
8∑
i=2
βSj sectorijt + βexp exportsit + βage ageit + βt yeart + εit (4)
a) [2 marks] Report the sample means for all three of the factor input variables
(lnrcapital, lnlabour, and lnrmaterials) in the data file assign1.csv. Esti-
mate the econometric model (4) in R and report the results.
b) [7 marks] Using your estimates for model (4), calculate the following estimated
elasticities, evaluated at the sample means:
i) Elasticity of sales with respect to capital
ii) Elasticity of sales with respect to labour
iii) Elasticity of sales with respect to materials
Compare and contrast your answers to those reported for the restricted model (2)
in Question 1(b).
Hint: You will need to use the sample means for the data, calculated in Question
2(a) to answer this part.
c) [7 marks] What is the interpretation of the parameter βt in model (4). At the 5%
level of significance, test the hypothesis that, on average, the real level of output is
increasing over time.
d) [6 marks] Test the hypothesis that the model (4) is correctly specified using the
Ramsey RESET test, at the 5% level, using both the squares and the cubes of the
fitted values. What is your conclusion?
e) [6 marks] Using the car package in R, test whether the data are consistent with
the restricted model (2) at a 5% level of significance, using an F-test.
f) [6 marks] Constant returns to scale implies that a doubling of all factor inputs
(K, L,and M) exactly doubles production. In terms of the model (4), this imposes
the restrictions:
βK + βL + βM = 1
2 βKK + βKM + βKL = 0
2 βLL + βKL + βLM = 0
2 βMM + βKM + βLM = 0
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Using the car package in R, test the hypothesis of constant returns to scale, for
model (4), at a 5% level of significance using an F-test.
g) [6 marks] Using the car package in R, at a 5% level of significance test the hy-
pothesis that (industry) sector has no effect on the value of output.
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