Problem Set for ECON 3150/4150 Spring 05
In order to get familiar with the textbook and the problems we should be able to solve in this
introductory econometric course, the first problems for the seminars are taken from the
textbook “Undergraduate Econometrics” by R. C. Hill, W.E. Griffiths and G. G. Judge.
Problem set 1. These problems are taken from chapter 2, and are intended to refresh and
revive your memory of well known facts you have learnt in previous courses in statistics.
Solve the exercises: 2.17, 2.18, 2.22, 2.26
Problem set 2. The problems here are meant to illustrate what type of information we get from
linear regression analysis. In addition they are exercises in using regression programs.
Solve the exercises: 3.9, you will find the data in the file capm.xls, 3.15, 3.16 using the data in
the file br-1.xls.
Problem set 3. The problems here summarize properties of the least square method (OLS). In
particular they remind you that OLS estimators and estimators in general are random
variables.
Solve the exercises: 4.5, 4.9, 4.12, 4.16
Problem set 4. The point with the exercises here is to become familiar with procedures for
testing hypotheses on regression coefficients and constructing confidence intervals.
Solve the exercises: 5.13 using data in the file insur.xls, 5.14 using data in the learn.xls, 5.15,
5.18.
Problem set 5. The step from simple regression to multiple regression is in many ways
immediate. The problems listed below is meant to give you some training in using multiple
regression.
Solve the exercises: 7.9 using the data in meat.xls, 7.10 using the data in clothes.xls.
Problem set 6. Testing simultaneous hypotheses on the regression coefficients - F tests.
Solve the exercises: 8.9 using the file codd.xls, 8.13 using the data in the file cars.xls.
Problem set 7. This is an exercise in demand analysis. You will find the data in the file table
8-3.xls
Demand analysis:
We are concerned with analysing the demand for beer in a sample of households. Our sample
reports observations on 5 variables: the quantity of beer demanded (QB ) , the price of beer
(PB) , the price of other liquor ( PL) , a price index of the remaining goods and services on the
households’ budgets (PR) , and finally the households income (INC ). We are not certain of
what will be the appropriate specification of this demand so we will try different forms. We
start with the ln-ln functional form:
(1) ln(QBi )  0  1 ln( PBi )  2 ln( PLi )  3 ln( PRi )  4 ln( INCi )   i
where  i denotes the random disturbances.
Question A. How will you interpret the regression parameters in this demand function.
Use PcGive and the attached data file table 8-3.xls to estimate this model.
Question B. Do you think the signs of the estimates are reasonable? Substantiate your
assertions. Explain how the numbers in the column P  t are calculated.
Standard consumer demand theory tells us that if prices and income increase by the same
proportion we should expect no change in the quantity demanded. The consumers are said to
have no money illusion.
Question C. Show that the assumption of no money illusion applied to the demand function
(1) implies the restriction:
(2) 1   2  3   4  0
Question D. Use results and information from your regression to test the hypothesis:
H 0 : 1   2  3   4  0
against
with significance level   0.05 .
H a : 1   2  3   4  0
Now we are told that the ln-ln functional form (1) has certain theoretical drawbacks. In
addition to (1) we thus wish to analyse two forms which are linear in the relative price and the
real income.
(3) QBi  1  2 ( PLi PBi )  3 ( PRi / PBi )  4 ( INCi PBi )  ui
(4) QBi  0 (1 PBi )  1  2 ( PLi / PBi )  3 ( PRi / PBi )   4 ( INCi / PBi )  vi
where u i and vi denote random disturbances.
Use PcGive to estimate the regression equations (3) and (4).
Question E. Which of the two equations (3) or (4) do you think is appropriate for testing the
assumption of no money illusion? State the reason for your choice and explain how you
would test this hypothesis.
Question F. Assume that the number of household members ( Ni ) have been wrongly
excluded from the regressions (3) and (4). Choose one of these regressions for further study,
and explain formally how this misspecification effects the OLS estimates of the  ' s when:
(i) N i is uncorrelated with the explanatory variables already used in the equation.
(ii) N i is correlated with the explanatory variables already used in the equation.
Problem set 8. The next exercise illustrates the use of dummy variables in regression analysis.
Problem set 9. Households’ expenditures on food.
Problem set 10. Exercises with dummy variables.
Solve the exercises: 9.6 using the data file tuna.xls, 9.8
Problem set 11. This is an exercise with non-linear models.
We are interested in investigating how households’ expenditures on food vary with their
income . We have observations on the households’ expenditures on food (Y) and incomes
( R ) for 50 Norwegian households, both variables are measured in 1000 kroner. In addition do
we have observations on the number of members in the households.
~
For a given income R do we assume that the expected value of Y , denoted Y , is given by
the function:
(1)
R
~
Y 
R
where  and  denote unknown, positive parameters.
(a) Discuss if, in your opinion, the function (1) gives a good description of the relation
between expenditures on food and income.
Since (1) is non-linear in the parameters, we are unable to estimate the parameters  og 
with ordinary OLS regression. We shall therefore approximate (1), first with a linear and then
with quadratic funksjon. The linear approximation Yi is given by:
(2)
Yi   1 Ri   i
where  1 


and  i denotes random disturbances,
i  1,2, ….. ,50
(b) Show that OLS estimator of  1 is given by:
50
(3)
ˆ1 
Y R
i 1
50
i
i
R
i 1
2
i
Out-print 1 shows the results of the OLS applied to regression (2) together with the
histogram of the residuals ˆi .
(c) Give your comments to this regression, calculate the Jarque-Bera observator when
(skewness) S  0.6332 and kurtosis k  4.5288.
~
In order to improve the approximation of Y , we now approximate (1) by a quadratic
function.. The quadratic approximation of Yi is given by:
(4)
Yi   1 Ri   2 Ri   i
2
where  2  

2
and  i are random disturbances,
i  1,2,......50
Outprint 2 shows the results of OLS applied to the regression (4) together with histogram of
the residuals ˆi . The variable RR in the outprint corresponds to R 2 in the regression
equation (4).
(d) Do you think that the results as they appear in this out-print are reasonable?
Substantiate your answer.
(e) Explain how you will use these results to deduce estimates of  og  and calculate
̂ and ˆ .
(f) Use the expenditure function (1) to derive the Engel elasticity. Use your results above
to calculate this elasticity for a household with income equal to 100 000
We suspect that the parameter  1 in equation (4) depends on the number of member in the
household S , but we are uncertain how to specify this dependency. There are two proposals:
(5)  1   0  1 S
eller
(6)  1   0  1 S  
where  denotes the random disturbances satisfying the usual conditions.
Out-print 3 shows the results of the regression:
i  1,2,.......,50
(7) Yi   0 Ri  1 ( RS )i   2 Ri2  ui
where u i denotes the random disturbances in this regression
(g) Give your comments to this out-print.
(h) Explain your opinion about choosing (5) or (6).
Out-print 1
EQ( 1) Modelling Y by OLS-CS (using data.eksoppg.h2003.xls)
The estimation sample is: 1 to 50
R
sigma
Coefficient
0.233338
6.55912
Std.Error
0.01760
RSS
t-value
13.3
t-prob
0.000
2108.08312
no. of observations
50 no. of parameters
1
mean(Y)
12.5025 var(Y)
37.0663
Density
0.45
r:Y
N(0,1)
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
no. of observations
50 no. of parameters
1
mean(Y)
12.5025 var(Y)
37.0663
Out-print 2
Q( 2) Modelling Y by OLS-CS (using data.eksoppg.h2003.xls)
The estimation sample is: 1 to 50
R
RR
Coefficient
Std.Error
0.371463
0.05604
-0.00213237 0.0008260
t-value
6.63
-2.58
t-prob
0.000
0.013
sigma
6.20997 RSS
1851.05693
no. of observations
50 no. of parameters
2
mean(Y)
12.5025 var(Y)
37.0663
4.5
0.5
Density
r:Y
N(0,1)
0.4
0.3
0.2
0.1
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Out-print 3
Q( 3) Modelling Y by OLS-CS (using data.eksoppg.h2003.xls)
The estimation sample is: 1 to 50
R
RS
RR
Coefficient
0.170144
0.0623471
-0.00238358
Std.Error
t-value
0.06098
2.79
0.01248
5.00
0.0006765 -3.52
sigma
5.07192 RSS
1209.04536
no. of observations
50 no. of parameters
3
mean(Y)
12.5025 var(Y)
37.0663
t-prob
0.008
0.000
0.001
Density
r:Y
0.40
N(0,1)
0.35
0.30
0.25
0.20
0.15
0.10
0.05
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0