ETF2100 Introductory Econometrics Assignment 1 Q1 and Q3

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Question 1 (40 marks;)
Life expectancy is an indicator of how long a person can expect to live on average given the
environment s/he is in. It is a standard measure of population wellbeing in general. It is
commonly used by the government for policy planning related to future population ageing in a
country. It is also commonly used in calculating life insurance policy.
There are a number of studies done on the relationship between life expectancy and income. It
is believed that people with higher incomes tend to live longer than people with lower incomes
and the relationship can be linear or nonlinear.
Consider the data on life expectancy and income for the year 2015 from 194 countries obtained
from and reported in LifeExp2015.xls. The variable LE represents
life expectancy at birth in years. The variable GDP represents the country’s Gross Domestic
Product (GDP) per capita after adjusting for purchasing power parity so that GDP from
different countries are comparable. The unit of measurement for GDP is in 1000 international
Consider three functional forms for the relationship between life expectancy and income.
i 1 2 i i LE    GDP  e (1.1)
  1 2ln i i i LE    GDP  v (1.2)
i 1 2 i i LE     GDP   (1.3)
(a) Explain what each of the above functional forms imply in terms of the relationship
between life expectancy and GDP? (6 marks)
(b) Using the data for 194 countries, estimate the three equations and report the results the
usual way. (6 marks)
(c) Plot the fitted equations onto a scatterplot of the data. Which equation seems to best
match your observations? Explain why? (6 marks)
(d) Obtain and comment on residual plots against the explanatory variable for each
equation. Which model seems best specified? (6 marks)
(e) Obtain histograms of your residuals for each model and discuss if they resemble a
normal distribution. Perform a test of normality on each set of residuals. (6 marks)
(f) Interpret and compare the 2 R for each of your three equations, and explain why it is
possible to make this comparison. (4 marks)
(g) Which is your preferred model? Use your answers in parts b) to f) above to justify your
choice. (6 marks)
Question 2 (25 marks for ETF5910 students only)
This question continues from Question 1. It may be of interest to compare life expectancy
between two levels of incomes. Some previous studies have shown that a big proportion of
population around the world receive income in 2015 around 2,000 international dollars. At the
same time, some people at the high end of the global income distribution receive income more
than 30,000 international dollars. Using the preferred model in Question 1 (g):
(a) Predict life expectancy values for the two levels of GDPs. Obtain a 99% prediction
interval for both predicted life expectancy values. Compare and comment on the
finding. (10 marks)
(b) Find the estimates of the slopes
 
 
d LE
at the points where GDP  2,000 and
GDP  30,000. (2 marks)
(c) Find estimates of the elasticities
 
 
 at the point where GDP  2,000 and
GDP  30,000. (3 marks)
(d) Based on the finding from parts (a) to (c), write a short report (not more than half a
page) in terms of life expectancy between low and high income populations.
(10 marks)
QUESTION 3 (45 marks)
Consider the following total cost function where i TC represents total cost for the ith firm and
i Q represents quantity of output.
2 3
i 1 2 i 3 i 4 i i TC     Q   Q   Q  e (3.1)
Two other costs that may be of interest are average cost per quantity of output and marginal
cost for an extra output. Average cost is obtained by dividing total cost by quantity of output.
Using total cost as defined in equation (3.1) average cost function can be defined as:
2 1 3 4
i i i i
AC Q Q v
         (3.2)
where the new error term is i i i v  e Q . Note that the parameters 1 2 3 4  , , and  are the same
parameters as those in equation (3.1).
Marginal cost is obtained from taking derivative of total cost with respect to output. From the
total cost function defined in (3.1) marginal cost function can be derived as:
2 3 4 2 3 i i i MC     Q   Q (3.3)
Data on a sample of 28 firms in the clothing industry are in the file clothes.wf1.
(a) Consider the total cost function (3.1). What sign would you expect for 2 3  , and 4  ?
Explain why. Answer this before looking at the data. (5 marks)
(b) Estimate the total cost function (3.1) and report results. Are the signs for the estimated
coefficients turn out as expected? (5 marks)
(c) Using   0.01 test individually the individual significance of 2 3 4  , and  .(6 marks)
(d) Using   0.01 test the significance of the model. (3 marks)
(e) Test whether the data suggest that a linear total cost function will suffice. (5 marks)
(f) What parameter restrictions imply a linear average cost function? Test these restrictions.
(4 marks)
(g) Calculate the predicted total, average and marginal cost for output 1,2, ,10 i Q  using
the parameter estimates from (b). (3 marks)
(h) Graph the predicted total, average and marginal cost for output 1,2, ,10 i Q  calculated
in (g). (3 marks)
(i) It is believed that it is profitable for firms to produce when price exceeds average cost
and firms will decide to shut down if price is less than average cost. Also, average cost
is at minimum when i i AC  MC . From the graph in (h) roughly find the output level i Q
where i i AC  MC . Calculate the minimum average cost implied by the estimated
parameters in (b). At what price should the firm make a decision to shut down?
(5 marks)
(k) Find estimates of 1 2 3 4  , , and  by applying least squares to the average cost function
defined in (3.2). Report the results. Is it possible to make any decision about which set
of estimates might be “best”? (6 marks)

nt of Econometrics and Business Statistics
ETF2100/5910 Introductory Econometrics
Assignment 1, Semester 1, 2018
Worth 10% of Final Mark
Due 4pm Friday April 20
(Students must submit a soft copy to Moodle
and a hard copy into your tutor’s mailbox in Building H Level 5)
 Notation used in the assignment needs to be written correctly and properly. It is
recommended that students submit hand-written assignments if they cannot type
 Your assignment must have the assignment cover sheet. It can be found in Assignment
section of the Moodle.
 This assignment comprises three questions. ETF2100 students do Questions 1 and 3.
ETF5910 students do all three questions.
 Mark allocations are given for each question. Marks are also awarded for
 Total marks for ETF2100 and ETF5910 students are 85 and 110, respectively.
 Marks will be deducted for late submission on the following basis:
10% for each day late, up to a maximum of 3 days.

Sample Solution