Question 1 (Total marks = 46 for ETF2100, 60 for ETF5910)
Consider the problem of estimating a demand function that expresses the relationship between
the quantity of tomatoes purchased and the price of tomatoes, and assume that you have the
data given in Table1.1 below.
You are to do all the parts of this question using only a calculator and report results using 3
decimal points.
Table 1.1
Quantities of tomatoes, y
(kg/day)
Price, x
(cents/kg)
20
28
30
5
10
17
12
10
26
20
Part A
(a) Complete the entries in the table. Put the sums in the last row. What are the sample
means x and y ? (10 marks)
x y ( x x − ) ( )
2
x x − ( y y − ) ( xxyy − − )( )
17 20
12 28
10 30
26 5
20 10
i ∑x = i ∑ y = ( ) i ∑ x x − = ( )
2
i ∑ x x − = ( ) i ∑ y y − = ( )( ) i i ∑ x xy y − −=
(b) Assume that the data can be described by the simple linear regression model
i ii =β +β + 1 2 y xe and that all the assumptions hold. Use the least squares principle to
obtain 1 b and 2 b . (3 marks)
(c) Give an economic interpretation to the estimated parameters. (3 marks)
(d) Compute
5 5
2
1 1
, . i ii
i i
x xy
= =
∑ ∑ Using these numerical values, and show that
( )
2 2 2
i i ∑ ∑ x x x Nx −= − and ( )( ) i i i i ∑ ∑ x x y y xy Nxy − −= −
No proof is needed. (2 marks)
(e) Use the least squares estimates from part (b) to compute the fitted values of y, and
complete the remainder of the table below. Put the sums in the last row. (5 marks)
x y ˆi y ˆ
i e 2
ˆ
i e
17 20
12 28
10 30
26 5
20 10
ˆi ∑ y = ˆ
i ∑e = 2
ˆ
i ∑e =
(f) On the graph paper, plot the data points and sketch the fitted regression line
1 2 ˆ . i i y b bx = + (4 marks)
(g) On the sketch in part (f), locate the point of the means ( x y, .) Does your fitted line
pass through that point? (2 marks)
PART B
(h) Show that for these numerical values 1 2 y b bx = + . (2 mark)
(i) Compute 2 σˆ . (2marks)
(j) Compute ( ) 2 var b and ( ) 2 se b . (3 marks)
(k) Find a 95% confidence interval for 2 β . Interpret the results. (5 marks)
(l) Test the hypothesis that 2 β = 0 and interpret. Make sure to justify your choice of
alternative hypothesis and show all steps used to conduct your hypothesis. (5 marks)
For ETF5910 students only
(m) Predict the quantity of tomatoes purchased when the price of tomatoes is 20 cents/kg.
(2 marks)
(n) Find the standard error for the prediction in (m) and construct a prediction interval.
Interpret the results. (5 marks)
(o) Construct a 95% confidence interval for (β −β 2 1 ). (7 marks)
Question 2 (Total marks = 60 for ETF2100, 66 for ETF5910)
Use EViews to obtain the results for this question.
Data on the beginning salary ( ) y and education in years ( ) x for 93 employees of Harris Bank
Chicago in 1977 can be found in the file salary.xlsx. These data come from Schafer, D.W.
(1987), “Measurement-Error Diagnostics and the Sex Discrimination Problem”, Journal of
Business and Economic Statistics 5, 529-537, and were utilised by Dielman T.E. (1991),
Applied Regression Analysis for Business and Economics, Boston: PWS-Kent.
PART A
(a) Consider the relationship between salary and education to be 1 2 . i ii y xe =β +β + What
sign would you expect for β2 . Why? Answer this question before looking at the data.
(2 marks)
(b) Obtain a table of descriptive statistics for salary and education, and a scatterplot for
education against salary. Describe the relationship (if any) you observe from your
scatterplot (make sure that your scatterplot is well labelled and presented). (6 marks)
(c) Estimate the linear relationship between salary and education:
1 2 . i ii y xe =β +β + (1)
Report results the usual way and interpret the slope coefficient. Does it match your
expectations? What interpretation (if any) can you place on the intercept? (6 marks)
(d) Compute the estimated variances and covariance of the least squares estimators 1 b and
2 b . (6 marks)
(e) Test the null hypothesis that years of education has no effect on beginning salary at the
5% level of significance. Be sure to justify your choice of alternative hypothesis and
show all steps used to conduct your hypothesis. (6 marks)
(f) Predict the starting salary for a person with 16 years of education (i.e. a university
graduate). (3 marks)
(g) Calculate a 95% prediction interval around your point estimate in part (f). Explain
whether or not you think that the estimated equation is a good tool for predicting starting
salary. (5 marks)
Part B
(h) Re-estimate the relationship between education and salary using the following
functional forms:
a. ( ) 1 2 ln . i i i y xe =α +α + (2)
b. 1 2
1 . i i
i
y e
x
=γ +γ + (3)
Report results for both equations. (6 marks)
Use all three functional forms in equations (1), (2) and (3) to answer parts (i) to (l) below.
Consider reporting results in a table format when comparing them.
(i) Which functional form is best in explaining the variation in beginning salary? Does
your scatterplot support this? (4 marks)
(j) Use each functional form to predict beginning salary for an individual with 12 years of
education (i.e. a high-school graduate). (6 marks)
(k) Find the elasticity of beginning salary with respect to years of education for high-school
graduates that is implied by the three functional forms. (6 marks)
(l) Are the predictions in part (j) and elasticities in part (k) sensitive to the specification of
functional form? Comment. (4 marks)
For ETF5910 students only
(m) Explain what the functional forms in equations (2) and (3) imply about the relationship