EMPIRICAL PROJECTObjective to help students put in practice w.docx

EMPIRICAL PROJECT
Objective: * to help students put in practice what they have
learned in Econometrics I
* to teach students how to write an “economic
paper”.
Steps
a) Selecting a topic
Topic areas: Macroeconomics: consumption function,
investment function, demand
function, the Phillips curve…
Microeconomics: estimating production, cost, supply and
demand. Data
are hard to obtain here.
Urban and Regional Economics: demand for housing,
transportation…
International Economics: estimating import and export
functions,
estimating purchasing power parity,
estimating capital mobility…
Development Economics: measuring the determinants of per-
capita
income, testing the per-capita output
convergence among nations…
Labor Economics: testing theories of unionization, estimating
labor force
participation, estimating wage differential
among women, minorities…
Resource and Environmental Economics: estimating water
pollution,
estimating the determinants of toxic
emissions…
The resource journal is JEL (Journal of Economic
Literature) + Internet EconLit .
b) Statement of the Problem

State clearly the problem that you are interested in (what
are you trying
to achieve)
c) Review of literature
Point out (critically) what others have done concerning the
topic of interest.
d) Formulation of a general model
The final model can be derived in several ways: utility
maximization,
profit maximization, cost minimization, etc. The review of
literature is
generally helpful to accomplish this task. In the course of
deriving the model,
one must sort out clearly the dependent variable and the
independent
variables. After transforming the economic model in
econometric model, one
writes up the hypotheses to be tested: expected signs of the
parameters and
magnitudes. To elaborate a bit, let use the following
demand for some good:
Q
P
P
Y
u
be
be
o
=
+
+
+
+

a
b
g
d
where
Q
P
P
Y
and
u
be
be
o
,
,
,
represent the quantity of good of interest, the price
of that good, the price of another good (pork, etc), income
and the error term,
respectively. Here
b
g
<
<>
0
0
,
depending on the nature of the good: >0

if substitute and <0 if complementary. The size of
b
depends on the nature of
product. Thus if the product is a necessity, price and income
elasticities are
expected to be small.
e) Collecting Data
Sources: international, national, regional
primary or secondary.
Notes.
f) Empirical Analysis
Data analysis: outliers, level of variation…
Model estimation and hypothesis testing
g) Writing a Report
Statement of the problem: describe the problem you have
studied,
the questions you have asked, and the broad
hypothesis
you have tested.
Review of Literature: summarize the relevant literature.
Formulation of a general model: describe the initial model
you formulated and
point out the difference between your model
and those reviewed
in the RL. (If a model done by previous
research is used, state so.)
Data Sources and Description: Present a table of variable
names and their
definitions. Specify the source of data and
attach a copy of raw
data.
Model Estimation and Hypothesis Testing. Present the
regression results in tables

with relevant statistics. (for eg
heteroscedasticity, causality, VAR, OLS, unit root,…)
Interpretation of the Results and Conclusion. Present some
concluding remarks
regarding the study and put it in perspective
with other studies.
Limitation of the Study and Possible Extension. Recognize
the limitations of the
study. Limitations might be due to
inadequate data, lack of
appropriate software package, etc.
Acknowledgments
ReferencesUNIVERSITY OF THE WEST INDIES
CAVE HILL CAMPUS
FACULTY OF SOCIAL SCIENCES
DEPARTMENT OF ECONOMICS
ECON 3049 ECONOMETRIC I
ASSIGNMENT # 2
A. CONCEPTS
1. Explain the sample properties of estimators.
2. Provide (and explain) advantages and disadvantages of the
following
data settings: (a) time-series data; (b) cross-sectional data;
(c) panel data
3. Explain the relationship between the normal distribution,
chi-square, Student’s t,

and Fisher’s F distributions.
4. Compare (and contrast) the following methods of
estimation: method of moments,
least squares method and maximum likelihood.
B. PROOFS AND EMPIRICAL QUESTIONS
5. Table 2.1 below presents the data for money supply and
retail price index for
Barbados from 1972 to 2001. Let
t
y
be the retail price index and
t
x
the money
supply E "money supply" .
(a) Fitting the data to the following model
30
,...,
3
,

2
,
1
=
+
+
=
t
t
u
t
x
t
y
b
a
obtain the estimated coefficients, RSSE "RSS" (residual sum of
squares), standard errors, t and F statistics. (State the method
you use to obtain the statistics and any assumptions of interest).
Use any appropriate computer software package
(b) By how much does the retail price index increase as a result
of a one unit increase in the money supply?
(c) Compute and interpret the corresponding elasticity from
(b).
(d) Suppose that the above linear model is replaced by the
following log-linear
model:
30

,...,
3
,
2
,
1
=
+
+
=
t
t
u
t
x
Ln
t
y
Ln
b
a
Compute the slope and the elasticity. Comment on the
difference between the latter elasticity and that obtained in (c).
(e) Compute and interpret the slope and the elasticity in each of
the following:
Log-lin
:
t
u

t
x
t
y
Ln
+
+
=
b
a
Lin-log
:
t
u
t
x
Ln
t
y
+
+
=
b
a
Reciprocal:
t

u
t
x
t
y
+
+
=
)
/
1
(
b
a
Table 2.1: Money Supply and Price Index in Barbados, 1972-
2001.
Year
Money Supply
(BDS $000)
Retail Price Index
(1994=100)
Year
Money Supply
(BDS $000)
Retail Price Index
(1994=100)

1972
79,709
16.2
1987
477,672
77.3
1973
76,123
18.9
1988
574,608
80.9
1974
90,507
26.3
1989
514,646
86.0
1975
105,158
31.6
1990
624,738
88.6
1976
118,009
33.2
1991
549,915
94.2
1977
134,873
35.9
1992
610,797
99.9

1978
164,182
39.4
1993
624,738
101.1
1979
229,216
44.5
1994
795,638
101.1
1980
255,440
51.0
1995
899,305
103.0
1981
269,679
58.4
1996
1,036,746
105.5
1982
266,099
64.4
1997
1,263,776
113.6
1983
309,108
67.8
1998
1,317,556
112.2

1984
301,392
71.0
1999
1,399,669
113.9
1985
345,208
73.8
2000
1,519,773
116.7
1986
403,039
74.7
2001
1,584,835
119.7
Source: Central Bank of Barbados, Annual Statistical Digest
2002.
(f) Compute and interpret the following
(i) instantaneous rate of growth of money supply;
(ii) compound rate of growth of money supply;
(iii) absolute rate of increase of money supply.
Mamingi(2005,8)
6. Consider the following saving model
t
t
t

u
Y
S
+
+
=
b
a
where
S
is saving,
Y
stands for income,
u
is the error term and
t
is the time index. Fitting data to the above saving model using
OLS gives the following results:
Table 2.2: Saving (S) and Income (Y) Relationship: OLS
Results, 1970-2000
Variable
Coefficient
Std. Error
t-Statistic
Prob.

Constant
-648.1236
118.1625
-5.485018
0.0000
Y
0.084665
0.004882
17.34164
0.0000
R-squared
0.912050
Mean dependent var
1250.323
Adjusted R-squared
0.909017
F-statistic
300.7324
RSS
1778203.
Prob(F-statistic)
0.000000
Note: Variables are in 1,000 US dollar.
(a) Interpret the results assuming all the assumptions of CLM
hold.
(b) Derive the consumption function results. Which assumption
do you use if any?
(c) Show that the residual sum of squares (RSS) remains the
same.
(d) Show that the corresponding standard errors of estimators
are the same in both models.

(e) Compare the values of
2
R
from the two models.
Mamingi(2005, 8)
7. Consider the following four data sets
Data Set:
Variable
1-3
X
1
Y
2
y
3
Y
4
x
4
Y
Observation
1
10.0

8.04
9.14
7.46
8.0
6.58
2
8.0
6.95
8.14
6.77
8.0
5.76
3
13.0
7.58
8.74
12.74
8.0
7.71
4
9.0
8.81
8.77
7.11
8.0
8.84
5
11.0
8.33
9.26
7.81
8.0

8.47
6
14.0
9.96
8.10
8.84
8.0
7.04
7
6.0
7.24
6.13
6.08
8.0
5.25
8
4.0
4.26
3.10
5.39
19.0
12.50
9
12.0
10.84
9.13
8.16
8.0
5.56
10
7.0

4.82
7.26
6.42
8.0
7.91
11
5.0
5.68
4.74
5.73
8.0
6.89
a. Plot the four data sets and run the four regressions of y on x.
b. Comment on the results by paying special attention to the
issue of outliers (use statistics to detect outliers: i.e.,
studentized residuals)
c. Run the regression for data set 4 without observation # 8.
Comment on the results.
(Maddala, 1992,89-90)
8. Consider the following table related to gross national
income per capita and gross
domestic investment per capita for Gabon in the period
1973 – 1993.
Table 2.3: Per Capita Gross National Income and Per Capita
Gross Investment for Gabon in US dollar 1973-1993
Year
Gni
Gdi
Year
Gni
Gdi

1973
4180.00
1350.00
1984
6480.00
2330.00
1974
6820.00
3630.00
1985
6040.00
2510.00
1975
7130.00
4290.00
1986
4350.00
2070.00
1976
8900.00
5850.00
1987
3430.00
950.00
1977
7550.00
4360.00
1988
3830.00
1820.00
1978
5590.00
1580.00
1989
3950.00
1440.00

1979
5730.00
1860.00
1990
4190.00
1210.00
1980
6740.00
1820.00
1991
4160.00
1300.00
1981
7200.00
2620.00
1992
3820.00
980.00
1982
6420.00
2400.00
1993
3740.00
1020.00
1983
6470.00
2250.00
Source: World Bank: World’s Table 95, Washington, D.C.
Note:
Gni: Gross national income per capita for Gabon in 1987 US
dollars.

Gdi: Gross domestic investment per capita in 1987 US dollars.
(a)
Fit the following two regressions to the above data:
EMBED Equation.3
t
t
t
u
Gdi
b
c
Gni
+
+
=
t
t
t
e
Gdi
L
LGni
+
+
=
d
g

where variables in the first regression are defined as in Table
2.3. and in the second, variables are defined in logarithm forms;
u and e represent the error terms.
(b). Assuming that the assumptions of the classical linear
model are fulfilled, compute
and interpret the elasticities from the two models. Which
elasticity do you prefer?
9. Suppose that a researcher, using data on class size (CS) and
average test scores from
100 third-grade classes estimates the OLS regression,
)
21
.
2
(
)
4
.
20
(
5
.
11
08
.
0
2
82
.
5
4
.

520
^
=
=
-
=
SER
R
CS
TS
a. A classroom has 22 students. What is the regression’s
prediction for that classroom’s average test score?
b. Last year a classroom had 19 students, and this year it has 23
students. What is the regression’s prediction for the change in
the classroom average test score?
c. Construct a 95% confidence interval for
1
b
, the regression slope coefficient.
d. Test
.
0
1
:
0
=
b
H

Do you reject the null hypothesis at the 5% level of
significance? At the 1% level?
e. The sample average class size across the 100 classrooms is
21.4. what is the sample average of the test scores across the
100 classrooms?
Stock and Watson (2003, 132-133).
10. Table 2.4. contains the ACT and the GPA (grade point
average) for eight
college students. GPA is based on a four-point scale and has
been rounded to one digit after the decimal.
Table 2.4. GPA and ACT results
Student
GPA
ACT
1
2.8
21
2
3.4
24
3
3.0
26
4
3.5
27
5
3.6
29
6
3.0

25
7
2.7
25
8
3.7
30
(a) Estimate the relationship between GPA and ACT using
OLS; that is, obtain:
ACT
GPA
1
ˆ
0
ˆ
^
b
b
+
=
Comment on the direction of the relationship. Does the
intercept have a useful
interpretation. How much higher is the GPA predicted to be if
the ACT is increased by five points?
(e) What is the predicted value of GPA when ACT=20?
(f) How much of the variation in GPA for these eight students is
explained by ACT? Explain.
(Woolridge, 2006, 67).

Using the information in Table 2.5 for Jamaica, do the
following:
(a) Estimate the linear regression equation
t
u
t
IM
t
CPI
+
+
=
b
a
where CPI is the consumer price index E "consumer price
index" , IM is the import price index and
u
is the error term.
(b) Test the significance of ( and (.
© Compute r2 using the results derived in (b) and test for
its significance.
(d) Comment on the overall relevance of the regression.
(e) Test whether there is a structural change in the year
1985.
Table 2.5. Price Statistics: Jamaica, 1972-1997

Year
IM
CPI
Year
IM
CPI
1972
0.8
1.0
1985
9.0
10.2
1973
1.0
1.2
1986
8.7
11.7
1974
1.5
1.5
1987
10.1
12.5
1975
1.7
1.7
1988
10.9
13.5
1976
1.8
1.9
1989
12.2
15.5

1977
2.1
2.1
1990
16.7
18.9
1978
2.1
2.9
1991
27.2
28.5
1979
2.4
3.7
1992
54.8
50.6
1980
3.0
4.7
1993
59.6
61.7
1981
3.3
5.3
1994
84.6
83.4
1982
3.1
5.7
1995
100.0
100.0

1983
3.7
6.3
1996
119.8
126.4
1984
6.8
8.1
1997
115.9
138.6
Source: Statistical Institute of Jamaica XE "Jamaica" ,
Statistical Abstract (various issues).
Note: IM: Import Price Index (1995=100); CPI: Consumer
Price Index (1995=100).
UNIVERSITY OF THE WEST INDIES
CAVE HILL CAMPUS
ECON 3049 ECONOMETRICS I
ASSIGNMENT #3
A. CONCEPTS
1. Discuss the following statement: multiple regression is more
useful than simple regression in quantifying economic
relationships.
2. Write a careful essay on
stat

F
R
R
-
the
and
2
,
2
. Include the issue of modeling in your essay.
3. The zero-one values attributed to a dummy variable in a
model are meaningless per se. Comment on the statement.(If
necessary, use a model to back up your arguments.)
B. PROOFS AND RELATED QUESTIONS
4. Suppose we have the following regression
u
X
Y
+
=
b
where Y is nx1, X is an nxk
matrix, ( is a kx1 vector and u is an nx1 vector of errors.
(a) Prove that R2 is the square of the single correlation
between Y and
.

(b) Show that while the OLS estimator of ( attains the
Cramer-Rao MVB, the
estimator of (2 does not.
(c) Prove that the OLS estimator
)
/(
ˆ
'
ˆ
2
ˆ
k
n
u
u
-
=
s
and the ML estimator of (2
are both consistent.
5. Consider the following regression model
U
X
Y

+
=
b
where Y is an n x 1 vector of observations on the dependent
variable, X is an n x k matrix of explanatory variables
including the vector column of ones,
b
is a k x 1 vector of parameters and U is an n x 1 vector of
disturbances. Moreover, the number of observations, n, is equal
to the number of parameters, k.
(a) What is the implication of
k
n
=
in terms of
X
?
(b) Derive
b
ˆ
, the OLS estimator of
.

b
(c) Derive
,
ˆ
Y
the predicted values of Y. Comment on the result.
(d) Derive
.
2
R
Comment on the result.
(e) Derive the F-statistic. Comment on the result.
(f) What major issue does this regression imply?
6. The LR, W, and LM are the F test equivalent in large
samples. Show why these
tests follow a chi-square distribution with the number of
restrictions as the number of degrees of freedom, rather than an
F distribution.
7. You are given the following information:
12
ˆ

'
ˆ
120
88
420
5
1
2
2
2
4
2
1
20
1
=
=
=
=
÷
÷
ø
ö
ç
ç
è
æ
=
¢
÷
÷
ø
ö
ç
ç
è

æ
=
¢
å
å
å
=
u
u
X
X
Y
Y
X
X
X
t
t
where the components of the matrices X’X and X’Y are
measured in deviation form,
and
u
ˆ
is the vector of OLS residuals.
(a) Find the least squares vector
, given that

t
u
t
X
t
X
t
Y
+
+
+
=
2
2
1
1
0
b
b
b
(b) Are (1 and (2 statistically different from zero?
© Compute R2 and
and explain the results carefully.
8. In their unpublished paper entitled “The monetary approach
to the balance of
payments: An application to Barbados”, Howard and Mamingi
used, among others, the following version of the standard
reserve flow equation to see whether or not the monetary

approach to the BOP holds for Barbados:
t
t
t
t
t
t
t
t
t
t
t
t
t
u
D
R
D
m
m
i
i
Y
Y
P
P
D
R
R
+
+
D
+

D
+
D
+
D
+
D
+
=
+
D
)
(
)
(
6
5
4
3
2
1
g
g
g
g
g
g
where R=International reserves in Barbados $million;
i=nominal interest in %;
P= consumer price index; Y = real income in Barbados
$million;
M=money multiplier; D=domestic credit in Barbados $
million;

D
= first difference operator; u = error term.
Using data for Barbados in the period 1973 to 1998 and OLS,
they obtained the following results
181067
.
0
)
(
007
.
1
685
.
0
152
.
0
097
.
1
158
.
0
058
.
0

)
(
=
+
+
D
-
D
-
D
-
D
+
D
+
=
+
D
RSS
u
D
R
D
m
m
i
i
Y
Y
P
P
D
R
R
t
t

t
t
t
t
t
t
t
t
t
t
t
ú
ú
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ê
ê
ë
é
-
-

-
-
-
-
-
012637
.
0
011609
.
0
048476
.
0
5
55
.
6
005216
.
0
008131
.
0
006431
.
0
001509
.
0
015812
.
0
350372
.
0

002804
.
0
009637
.
0
001820
.
0
022712
.
0
026738
.
0
000535
.
0
000954
.
0
000552
.
0
00742
.
0
001967
.
0
000657
.
0
E
where RSS = residual sum of squares and […] = variance-

covariance matrix of estimators.
(a) test the significance of each slope coefficient.
(b) The monetary approach predicts that
.
1
6
-
=
g
Test the latter hypothesis.
(c) Test the hypothesis that
.
6
5
g
g
=
(d) Two further regressions , based on the original specification
were computed for the subperiods 1973 to 1985 and 1986 to
1998 yielding residuals sum of squares of 0.100101 and
0.050142, respectively. Test the hypothesis that the coefficients
are identical in the two periods.
(e) Write a concise report on the results of the exercise.
9.
A production function is specified as

Yi = (1 + (2 X2i + (3 X3i + (i
where Yi = log output, X2i = log labour input and X3i = log
capital input. The data refer to a sample of 23 firms, and the
observations are measured as deviations from the sample means
(a) Estimate (2, (3, their standard errors, and R2.
(b) What does the hypothesis H0: (2 + (3 = 1 mean? Test it.
(c) Suppose now that you wish to impose a prior restriction that
(2 + (3 = 1. What is the least squares estimates of (2 and its
standard error? What is the value of R2 in this case? Compare
these results with those obtained in (a) and comment.
10. Consider the following CLM
t
t
t
t
e
Z
X
Y
+
+
+
=
2
1
0

a
a
a
where Y is the dependent variable, X and Z are the
independent variables, and
e is a well behaved error term.
(a) Test the assumption
2
1
a
a
-
=
using two variants of the t-test.
(b) Test the above hypothesis using the F-test.
11. Gasoline sales in a regional market are modeled by the
following regression
equation, estimated with quarterly data:
where Q is sales, P is price, Y is disposable income, and the
are quarterly dummy

variables. The expected paths of P and Y for the next year are
as follows:
Quarter
1
2
3
4
P
110
116
122
114
Y
100
102
104
103
(a) Carefully interpret the results of the model.
(b) Calculate the sales of gasoline to be expected in each
quarter of the year.
(c) Suppose another researcher proposes to use the same data
to estimate an equation of the same form, except she wishes to
employ dummy variables
Write down the equation that will come from her calculation.
(d) Yet another investigator proposes to suppress the intercept
and use all four seasonal
dummies. Write down the results of his estimation.

12. Suppose that a researcher, using wage data on 250 randomly
selected male workers and 280 female workers, estimates the
OLS regression,
)
84
.
0
(
)
18
.
0
(
10
.
3
,
06
.
0
2
,
79
.
2
68
.
12
^
=
=
+
=

SER
R
Male
Wage
where wage is measured in $/hour and Male is a binary variable
that is equal to o ne if the person is male and 0 if the person is a
female. Define the wage gender gap as the difference in mean
earnings between men and women.
a. What is the estimated gender gap?
b. Is the estimated gender gap significantly different from
zero?
c. in the sample, what is the mean wage of women? Of men?
d. Another researcher uses the same data, but regress Wages on
Female (1 if female and 0 if not). What are the regression
estimated calculated from this regression?
L
L
L
L
=
=
+
=
SER
R
Female
Wage
,
2
,

^
(Woolridge, 2006)
UNIVERSITY OF THE WEST INDIES
CAVE HILL CAMPUS
ECON 3049 ECONOMETRICS I
ASSIGNMENT #5
A. ESSAYS
1. Write a concise note on each of the following:
a. Heteroscedasticity
b. Autocorrelation
c. Multicollinearity
6. Although the DW test statistic is the most popular test for
autocorrelation, it has
several limitations. Discuss these limitations and
provide some alternatives.
7. Outline the Hendry’s approach to model selection (one page
maximum).
4. Write a short note on tests of nonnested hypothesis.
5. Write a concise note on the identification problem in

simultaneous equations models.
B. PROOFS AND EMPIRICAL QUESTIONS
2. From the regression
where Y is an n x 1 vector of observations, X is an n x k
nonstochastic matrix of exploratory variables including the
vector of ones,
is an n x 1 vector of disturbances and
I
uu
E
2
)
'
(
s
=
.
a. Provide and Explain four factors that affect the accuracy of
.
ˆ
b
b. Provide and explain three possible causes of wrong signs
associated occasionally with some components of

.
ˆ
b
c. Suppose the existence of a competing model
e
Z
Y
+
=
g
where variables and parameters are defined similarly as above
and
.
X
Z
¹
Suppose in both cases, the errors are well-behaved.
(i) Explain whether the J test proposed by Davidson and
MacKinnon can be applied to discriminate between the two
models.
(ii) If yes, explain its implementation and provide its
limitations.
3. Applying OLS to data on expenditure on clothing (Y), total
expenditure (
) and the

price of clothing (
) yields the following results:
where variables are measured in arbitrary units and (.)
are standard errors.
a. Evaluate the results on a priori economic criteria
b. Evaluate the statistical significance of the results (
c. Test for the first order autocorrelation using the Durbin-
Watson d statistic. Does a significant d necessarily mean
presence of autocorrelation ? (Be as explicit as possible in your
answer).
d. A regression of
on
yields
and

Conduct a test for omitted variables (you must provide the
name of the test).
e. A regression of
on
where
represents the residuals from the original regression, yields
and
Conduct a test for heteroscedasticity (you must provide the
name of the test).
4. An investigator has specified two models and proposes to use
them in some empirical work with macroeconometric time
series data
Model 1.
where
are jointly dependent variables and

and
are predetermined variables.
Model 2.
where
and
are jointly dependent variables and
and
are predetermined variables
a. Assess the identifiability of the parameters that appear as
coefficients with the above models (treating the models
separately).
b. Obtain the reduced form equation for

in model 1 and the reduced-form equation for
in model 2.
c. Assess the identifiability of the two-equation model
comprising the reduced-form equation for
and the reduced-form equation
in model 2 (using the rank condition).
5. Consider the simple Keynesian model of income
determination:
Consider the following demand and supply model for money:
Demand for money:
t
t
t
t
d
t
u
P
R
Y
M
1

3
2
1
0
+
+
+
+
=
b
b
b
b
Supply of Money:
t
t
s
t
u
Y
M
2
1
0
+
+
=
a
a
Equilibrium condition:
M
M

M
s
t
d
t
=
=
where M=money, Y=income, R=rate of interest, P=price, u:
stochastic disturbance term.
Assume that R and P are predetermined,
0
)
(
=
u
E
with
I
u
u
E
ij
j
i
s
=
)'
(
(the errors in the different equations are contemporaneously
correlated but are independent over time).

a) Show that
.
0
)
,
(
2
¹
t
t
u
Y
Cov
(State clearly any assumption you use).
(b) Show that the OLS estimator of
1
a
is inconsistent.
© Using order and rank conditions, determine whether the
demand equation is identified.
(d) Using order and rank conditions, determine whether the
supply equation is identified.
(e) Which method would you use to estimate the parameters of
the identified equation(s) Why?

f) Suppose we modify the supply function by adding the
following explanatory variables:
1
-
t
Y
and
.
1
-
t
M
What happens to the identification problem?
10. Consider the following model:
t
t
t
t
t
u
X
X
Y
Y
1
2

12
1
11
2
12
1
11
=
+
+
+
g
g
b
b
t
t
t
t
t
u
X
X
Y
Y
2
2
22
1
21
2
22
1

21
=
+
+
+
g
g
b
b
where the Y’s are endogenous variables and the X’s exogenous
variables.
Study the identifiability of each equation using the following
restrictions:
(a)
;
1
11
=
b
;
1
12
11
=
+
g
g
.
1
,

1
21
21
22
=
+
=
g
b
b
(b)
;
1
11
=
b
;
12
11
g
g
-
=
;
1
22
=
b
0
22
=

g
11. Consider the following two-equation model capturing the
labour market in Barbados
for the period 1970-1996:
t
t
t
t
t
t
t
t
t
u
Lbnisee
T
Lbwager
lblabor
u
Lbniscor
Lbgdp
Lbwager
lblabor
2
4
3
2
1
1
4
3

2
1
+
+
+
+
=
+
+
+
+
=
b
b
b
b
a
a
a
a
where variables are logarithms of variables defined as in Table
1 with the exception of the trend T. Use a 10% level of
significance throughout.
The first equation of the system represents labour demand and
the second, labour supply. Lblabor and Lbwager are the
endogenous variables of the model.
(a) In this system the left hand side variables are the same. Why
is this so?
(b) Is the system complete? Why or why not?
(c) What are the expected signs of the parameters of the model?

(d) Study the identification of the model.
Table 1: Some Labour Statistics for Barbados, 1970 - 1996
Year
BLABOR
BWAGE
BWAGER
BGDP
BNISCOR
BNISEE
1970
83.600
30.6
1.159091
627.600
85.71
100.00
1971
83.300
35.3
1.188552
629.500
100.00
100.00
1972
83.400
39.4
1.238994
637.600
100.00
100.00
1973
84.600
43.1
1.158602

654.640
100.00
100.00
1974
86.400
48.3
0.936047
640.000
100.00
100.00
1975
88.500
53.1
0.853698
627.800
100.00
100.00
1976
87.600
58.2
0.892638
655.300
100.00
100.00
1977
88.200
70.0
0.990099
679.400
101.43
100.00
1978
88.800
77.5
1.001292
712.500

101.43
100.00
1979
95.000
83.2
0.919337
768.600
100.00
100.00
1980
100.300
100.0
0.967118
802.300
100.00
100.00
1981
101.900
110.5
0.931703
786.800
141.14
158.30
1982
96.600
123.0
0.940367
748.000
185.71
216.70
1983
95.700
130.1
0.944808
751.800
228.57

266.70
1984
93.100
142.0
0.985427
778.500
228.57
266.70
1985
92.100
148.9
0.992667
786.900
217.86
250.00
1986
96.100
155.2
1.024422
827.000
217.86
250.00
1987
105.200
157.8
1.006378
848.200
257.14
325.00
1988
108.300
169.4
1.031669
877.500
257.14
325.00

1989
111.700
174.0
0.996564
909.100
257.14
325.00
1990
113.300
182.8
1.016120
879.100
257.14
325.00
1991
107.100
184.4
0.964435
844.800
275.00
370.80
1992
101.700
181.1
0.892998
796.200
339.29
416.70
1993
100.400
184.0
0.897212
802.900
307.14
341.70
1994

105.600
181.0
0.881637
834.700
253.57
266.70
1995
110.100
181.0
0.871030
858.900
253.57
266.70
1996
114.400
181.0
0.850964
903.600
253.57
266.70
Sources:
Downes, A. and McLean, W. (1988,115-35) and Central Bank of
Barbados, Annual Statistical Digest, 1998.
Note:
Blabor = Barbados labor employment in thousands; bwage:
Barbados wage index; Bwager: Barbados real wage index
(bwage/consumer price index); Bgdp=Barbados gross domestic
product at constant prices in millions of Barbados dollars;
Bniscor: index of contributions of employers to national
insurance scheme; Bnisee: index of contributions of employees
to national insurance scheme
(e) Obtain the OLS estimates of the parameters. Fully comment
on the results.

(f) Obtain the ILS estimates of the parameters.
(g) Obtain the 2SLS estimates of the parameters. Fully
comment on the results (e.g., compare these results with those
obtained with OLS).
(h) What are the total effect, the direct effect and the indirect
effect of GDP on labour.
12. Consider the following model for joint determination of
unemployment rate, price
inflation and wage inflation:
t
t
t
t
u
T
AD
w
1
3
2
1
0
+
+
+
+
=
b
b

b
b
p
(1)
t
t
t
u
UN
2
t
2
1
0
Prod
+
+
+
=
a
p
a
a
(2)
t
t

t
t
u
UN
w
3
3
2
0
+
+
+
=
p
g
g
g
(3)
where
t
UN
is unemployment rate;
t
p
is inflation rate;

t
w
is wage inflation (rate of change of wages);
Prodt is productivity defined as gdp/total employment;
ADt is rate of change of aggregate demand (gdp growth);
T is time trend;
(a) Tentatively explain the presence of the variable “T” in the
inflation equation.
(b) Study the identifiability of each equation and of the system.
(c) What are the additional restrictions to make the system (i)
triangular; (ii) fully recursive.
Consider the following data for Barbados:
Table 2: Data for Simultaneous Determination of Price
Inflation, Wage Inflation and Unemployment Rate: The Case of
Barbados, 1975-1996
Obs
p
(%)

w
(%)
UN (%)
AD (%)
PROD
1975
18.6833
9.4745
22.40000
-1.9247
7.093785
1976
4.7104
9.1708
16.20000
4.2872
7.480594
1977
8.0986
18.4610
17.00000
3.6117
7.702948
1978
9.0541
10.1783
12.20000
4.7570
8.023649
1979
15.6363
7.0969

13.40000
7.5791
8.090526
1980
13.3255
18.3923
12.90000
4.2912
7.999003
1981
13.7152
9.9845
10.80000
-1.9509
7.721295
1982
9.7913
10.7169
13.70000
-5.0571
7.743271
1983
5.1408
5.6119
15.20000
0.5067
7.855799
1984
4.5430
8.7524
17.20000
3.4899
8.361976
1985
4.0128
4.7448

18.70000
1.0732
8.543974
1986
0.9950
4.1440
18.40000
4.9704
8.605619
1987
3.4385
1.6614
18.00000
2.5312
8.062738
1988
4.6114
7.0934
17.90000
3.3960
8.102493
1989
6.1412
2.6793
17.70000
3.5378
8.138765
1990
2.9903
4.9337
17.60000
-3.3556
7.759047
1991
6.0919
0.8715

17.20000
-3.9799
7.887955
1992
5.8900
-1.8058
23.00000
-5.9250
7.828909
1993
1.1277
1.5985
24.30000
0.8380
7.997012
1994
0.0975
-1.6537
21.70000
3.8842
7.904356
1995
1.2104
0.0000
19.60000
2.8580
7.801090
1996
2.3307
0.0000
15.80000
5.0734
7.898601
.
Source:

Same as Table 1
Note:
w
:
wage inflation computed as 100*[Log (wage) – Log(wage(-1))].
p
:
price inflation computed as 100*[Log (CPI)-Log(CPI(-1))].
UN:
unemployment rate in %.
AD:
GDP growth as 100*[ LOG(GDP)-LOG(GDP(-1))];
PROD:
productivity as the ratio of real GDP to Labor employment;
(d) Estimate the model by all appropriate methods. Comment
on the results.
Use a 10% level of significance throughout.
(e) Test for the endogeneity of inflation rate in the
unemployment rate equation.
Nlandu
Mamingi, Ph.D.
51

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