UNIVERSITY OF ZIMBABWE
2023 July Examination
Faculty: Business Management Sciences
and Economics
Department: Economics and Development
Paper code and Title: ECD 303/ECON313, Applied
Econometrics
Duration: 2 hours
Examiner: Mr. R. Makoto
Authorized Materials: Calculator
INSTRUCTIONS:
1. This paper contains 4 Questions.
2. Answer ANY THREE questions.
3. Start each question on a new page.
4. This question paper comprises 10 printed pages.
5. Statistical tables are attached at the end.
NB: DO NOT TURN OVER THE QUESTION PAPER OR COMMENCE WRITING
UNTIL INSTRUCTED TO DO SO.
Page | 1
Question One [25 marks]
The Kariba Fish Market has operated for over 50 years. The prices for fish are determined daily
by the forces of supply and demand. A researcher collected daily data on the price of Kariba bream
(a common type of fish), quantities sold, and weather conditions (rainy, cold and stormy) during
the period December 2, 2020, to May 8, 2021. The researcher opted for the following demand
equation for this market:
where ln is the logarithm operator, QUANt is
the quantity sold, in kilograms, and PRICEt is the average daily price per kilogram. The variable
STORMYt is an indicator variable indicating stormy weather during the previous three days.
a) How appropriate is the simultaneous equation modelling in the above price and quantity
problem? [2 marks]
b) What reasons are behind the inclusion of the STORMY variable in the model? [2marks]
c) The supply equation (1) was estimated using both Ordinary Least Squares (OLS) and Two-
Stage Least Squares (2SLS). The estimated regression results are reported in table 1 and 2
respectively.
Table 1: OLS estimates
Dependent Variable: LnQUAN
Method: Least Squares
Date: 04/03/23 Time: 16:51
Sample: 12/02/1991 5/08/1992
Included observations: 111
Variable Coefficient Std. Error t-Statistic Prob.
LPRICE -10.11663 1.328711 -7.613872 0.0000
STORMY 8.719686 1.055939 8.257757 0.0000
R-squared -63.068881 Mean dependent var 8.523430
Adjusted R-squared -63.656669 S.D. dependent var 0.741672
S.E. of regression 5.963738 Akaike info criterion 6.427126
Sum squared resid 3876.713 Schwarz criterion 6.475946
Log likelihood -354.7055 Hannan-Quinn criter. 6.446931
Durbin-Watson stat 0.559233
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Table 2: TWO-STAGE LEAST SQUARES estimates
Dependent Variable: LnQUAN
Method: Two-Stage Least Squares
Date: 04/03/23 Time: 16:46
Sample: 12/02/2020 5/08/2021
Included observations: 111
Instrument specification: C MON TUE WED THU RAINY COLD STORMY
Variable Coefficient Std. Error t-Statistic Prob.
LPRICE -27.80033 3.620898 -7.677746 0.0000
STORMY 9.514197 1.715817 5.544996 0.0000
R-squared -167.182166 Mean dependent var 8.523430
Adjusted R-squared -168.725121 S.D. dependent var 0.741672
S.E. of regression 9.662399 Sum squared resid 10176.45
Durbin-Watson stat 0.620913 Second-Stage SSR 435.0416
J-statistic 4.138431 Instrument rank 8
Prob(J-statistic) 0.657949
i) Make a comparison of the OLS and 2SLS results. Which set of results would you
consider to be more appropriate for the analysis and why? [6 marks]
ii) What is your estimate of the elasticity of supply? [2 marks]
iii) Comment on the J-statistic. [3 marks]
d) Post-estimation tests for the equation are presented in tables 3 and 4.
Table 3
Breusch-Godfrey Serial Correlation LM Test:
F-statistic 72.37168 Prob. F(2,107) 0.0000
Obs*R-squared 22.03312 Prob. Chi-Square(2) 0.0000
Test Equation:
Dependent Variable: RESID
Method: Least Squares
Date: 04/03/23 Time: 16:52
Sample: 12/02/1991 5/08/1992
Included observations: 111
Presample missing value lagged residuals set to zero.
Variable Coefficient Std. Error t-Statistic Prob.
LPRICE 2.007700 0.903805 2.221386 0.0284
STORMY -2.689186 0.746736 -3.601256 0.0005
RESID(-1) 0.651155 0.090568 7.189705 0.0000
RESID(-2) 0.233855 0.097051 2.409607 0.0177
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R-squared 0.198497 Mean dependent var 4.050245
Adjusted R-squared 0.176025 S.D. dependent var 4.323104
S.E. of regression 3.924215 Akaike info criterion 5.607581
Sum squared resid 1647.743 Schwarz criterion 5.705221
Log likelihood -307.2207 Hannan-Quinn criter. 5.647190
Durbin-Watson stat 1.559489
Table 4
Heteroskedasticity Test: Breusch-Pagan-Godfrey
F-statistic 58.57031 Prob. F(2,108) 0.0000
Obs*R-squared 57.75328 Prob. Chi-Square(2) 0.0000
Scaled explained SS 35.72806 Prob. Chi-Square(2) 0.0000
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 04/03/23 Time: 16:52
Sample: 12/02/1991 5/08/1992
Included observations: 111
Variable Coefficient Std. Error t-Statistic Prob.
C 64.59571 3.819727 16.91108 0.0000
LPRICE 70.84491 7.564108 9.365931 0.0000
STORMY -55.32326 6.349156 -8.713483 0.0000
R-squared 0.520300 Mean dependent var 34.92534
Adjusted R-squared 0.511416 S.D. dependent var 39.74053
S.E. of regression 27.77814 Akaike info criterion 9.513031
Sum squared resid 83335.49 Schwarz criterion 9.586261
Log likelihood -524.9732 Hannan-Quinn criter. 9.542738
F-statistic 58.57031 Durbin-Watson stat 1.396858
Prob(F-statistic) 0.000000
What conclusion can be drawn from these results regarding the estimated coefficients and standard
errors of equation (1)? [5 marks]
e) The following system of equations represent the Keynesian model of income
determination:
Where C = consumption expenditure
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I = investment expenditure
Y = income
G = government expenditure
Determine whether the investment equation (I) is identified by the order condition? [5 marks]
Question Two [25 marks]
Dhillon, Shilling, and Sirmans (‘‘Choosing between Fixed and Adjustable Rate Mortgages,’’
Journal of Money, Credit and Banking, 19(1), 1987, 260–267) estimate a probit model designed
to explain the choice by homebuyers of fixed versus adjustable rate mortgages. They use 78
observations, taken over the period January 1983 to February 1984.ADJUST = 1 if an adjustable
mortgage is chosen. The explanatory variables are FIXRATE = fixed interest rate; MARGIN = the
variable rate less the fixed rate; YIELD = the ten-year Treasury bill rate less the one-year rate;
MATURITY = ratio of maturities on adjustable to fixed rates; POINTS = ratio ofq1 percentage
points paid on an adjustable mortgage to those paid on a fixed rate mortgage; NETWORTH =
borrower’s net worth.
(a) LPM and Probit models were estimated. The least squares estimates of the linear
probability model explaining the choice of an adjustable mortgage, using the explanatory
variables listed above are:
Table 5: LPM estimates
Dependent Variable: ADJUST
Method: Least Squares
Date: 04/03/23 Time: 17:06
Sample: 1 78
Included observations: 78
Variable Coefficient Std. Error t-Statistic Prob.
C -0.070774 1.287665 -0.054963 0.9563
FIXRATE 0.160391 0.082203 1.951162 0.0550
MARGIN -0.131802 0.049831 -2.644984 0.0101
MATURITY -0.034135 0.190766 -0.178938 0.8585
POINTS -0.088710 0.071130 -1.247151 0.2164
NETWORTH 0.028894 0.011787 2.451385 0.0167
YIELD -0.793202 0.323471 -2.452161 0.0167
R-squared 0.315162 Mean dependent var 0.410256
Adjusted R-squared 0.257289 S.D. dependent var 0.495064
S.E. of regression 0.426649 Akaike info criterion 1.219750
Sum squared resid 12.92411 Schwarz criterion 1.431250
Log likelihood -40.57026 Hannan-Quinn criter. 1.304417
F-statistic 5.445703 Durbin-Watson stat 0.594338
Prob(F-statistic) 0.000111
Page | 5
i) Determine the coefficients which are statistically significant. [4 marks]
ii) Are the signs of the coefficients consistent with expectations? [4 marks]
iii) What is the major shortfall of LPM? [2 marks]
b) Table 6 presents the Probit estimates
TABLE 6: Probit estimates
Dependent Variable: ADJUST
Method: ML - Binary Probit (Quadratic hill climbing)
Date: 04/03/23 Time: 17:07
Sample: 1 78
Included observations: 78
Convergence achieved after 6 iterations
Covariance matrix computed using second derivatives
Variable Coefficient Std. Error z-Statistic Prob.
C -1.877265 4.120677 -0.455572 0.6487
FIXRATE 0.498728 0.262476 1.900092 0.0574
MARGIN -0.430951 0.173910 -2.478010 0.0132
MATURITY -0.059185 0.622583 -0.095064 0.9243
POINTS -0.299914 0.241388 -1.242458 0.2141
NETWORTH 0.083829 0.037854 2.214524 0.0268
YIELD -2.383963 1.083047 -2.201163 0.0277
McFadden R-squared 0.257472 Mean dependent var 0.410256
S.D. dependent var 0.495064 S.E. of regression 0.422708
Akaike info criterion 1.184798 Sum squared resid 12.68645
Schwarz criterion 1.396298 Log likelihood -39.20713
Hannan-Quinn criter. 1.269465 Deviance 78.41426
Restr. deviance 105.6045 Restr. log likelihood -52.80224
LR statistic 27.19021 Avg. log likelihood -0.502655
Prob(LR statistic) 0.000133
Obs with Dep=0 46 Total obs 78
Obs with Dep=1 32
i) Comment on the McFadden, LR and Deviance statistics. [5 marks]
ii) How do the Probit estimates differ from the LPM estimates reported in table 5.
[3 marks]
iii) Given an economic interpretation of MARGIN, NETWORTH and YIELD.
[3 marks]
iv) Estimate the marginal effect of an increase in the variable MARGIN, with all
explanatory variables fixed at their sample means, AND explain the meaning of
this value. [4 marks]
FIXRATE MARGIN MATURITY NETWORTH POINTS YIELD
Mean 13.24936 2.291923 1.058333 3.504013 1.497949 1.606410
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Question Three [25 marks]
a) Consider the following finite distributed lag model:
𝑦𝑡 = 𝛼 + 𝛽1 𝑥𝑡 + 𝛽2 𝑥𝑡−1 + ⋯ + 𝛽𝑛 𝑥𝑡−𝑛 + 𝑒𝑡 𝑡 = 𝑛 + 1, … 𝑇
i) Why are finite distributed lag models usually difficult to estimate
using least squares? [3 marks]
ii) Explain how is the maximum lag length determined? [4 marks]
b) An autoregressive model AR(2) of inflation (INF)dynamics in Zimbabwe has the
following regression output produced using EViews statistical software:
Dependent Variable: INF
Method: Least Squares
Date: 04/03/23 Time: 16:10
Sample (adjusted): 3 104
Included observations: 102 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.171921 0.114853 1.496875 0.1376
INF(-1) 1.415103 0.088591 15.97353 0.0000
INF(-2) -0.457098 0.088075 -5.189898 0.0000
R-squared 0.962666 Mean dependent var 4.994804
Adjusted R-squared 0.961911 S.D. dependent var 3.058878
S.E. of regression 0.596980 Akaike info criterion 1.835103
Sum squared resid 35.28210 Schwarz criterion 1.912308
Log likelihood -90.59027 Hannan-Quinn criter. 1.866366
F-statistic 1276.357 Durbin-Watson stat 2.058526
Prob(F-statistic) 0.000000
i) Write down the fitted regression equation. [2 marks]
ii) Given the following data for inflation, forecast inflation rate for the 2-period ahead 2023
and 2024. [4 marks]
Year Inflation
2020 2.09
2021 2.32
2022 2.59
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iii) Construct a 95% prediction interval for 2023 and comment on the finding, given that the
variance of the forecast error, var(f) = 0.425. [6 marks]
a) Under what properties of time series are the following models recommended in economic
analysis:
i) Vector autoregressive model (VAR) [2marks]
ii) Granger causality [2 marks]
iii) Autoregressive Conditional Heteroscedasticity (ARCH) [2 marks]
Question Four [25 marks]
a) In the following figures, four time series are plotted on line graphs. Explain the time
series property depicted on each graph. [4 marks]
b) Define the following terms:
i) weak stationarity [2 marks]
ii) cointegrated series [2 marks]
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iii) Spurious relationship [2 marks]
c) An AR(1) model of the form is normally employed in the
test for stationarity.
i) State the assumptions of the error term, , for the model to be varied. [2 marks]
ii) State the null and alternative hypotheses in the test for stationarity. [2 marks]
d) The augmented Dickey–Fuller (ADF) tests computed using EViews for two
macroeconomic series of GDP and inflation in levels are presented below:
Null Hypothesis: GDP has a unit root
Exogenous: Constant
Lag Length: 1 (Automatic - based on SIC, maxlag=12)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic 0.892728 0.9951
Test critical values: 1% level -3.495677
5% level -2.890037
10% level -2.582041
*MacKinnon (1996) one-sided p-values.
Null Hypothesis: INF has a unit root
Exogenous: Constant
Lag Length: 8 (Automatic - based on SIC, maxlag=12)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -1.260950 0.6450
Test critical values: 1% level -3.500669
5% level -2.892200
10% level -2.583192
*MacKinnon (1996) one-sided p-values.
i) Determine whether both series are stationary? [4 marks]
ii) Is it possible to estimate a regression equation with both stationary and non-stationary series?
[2 marks]
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e) GDP and inflation were found to be stationary at first difference (I(1)) but not cointegrated.
The following equation was recommended for the analysis:
, where -GDP and -inflation.
Dependent Variable: D(GDP)
Method: Least Squares
Date: 04/03/23 Time: 15:46
Sample (adjusted): 3 104
Included observations: 102 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 44.21401 10.38756 4.256437 0.0000
D(GDP(-1)) 0.571614 0.083526 6.843532 0.0000
D(INF) -4.739218 9.853553 -0.480965 0.6316
D(INF(-1)) 9.215295 1.773586 5.195855 0.0000
R-squared 0.330092 Mean dependent var 101.6765
Adjusted R-squared 0.309584 S.D. dependent var 71.73841
S.E. of regression 59.60836 Akaike info criterion 11.05189
Sum squared resid 348209.3 Schwarz criterion 11.15483
Log likelihood -559.6466 Hannan-Quinn criter. 11.09358
F-statistic 16.09621 Durbin-Watson stat 2.112128
Prob(F-statistic) 0.000000
i) What is the difference between order of integration and cointegration? [2 marks]
ii) Interpret the regression coefficients. [3 marks]
-END-
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