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Semester – IV Question Paper 2015
Introductory Econometrics
Duration : 3 Hours
1.
2.
Maximum Marks : 75
State whether the following statements are true or false. Give reasons for your
answer.
(a)
In a two-variable PRF, if the slope coefficient β 2 is zero, the intercept β1
is estimated by the sample mean.
(b)
In regression through origin models the conventionally computed R 2
may not be meaningful.
(c)
In the presence of heteroscedasticity OLS estimators are biased as well
as inefficient.
(d)
The Durbin-Watson d test can also be applied to models that include the
lagged value of the dependent variable as one of the explanatory
variables.
(e)
In a double log model, the slope and elasticity coefficients are the same.
(5 x 3 = 15)
(a)
You are given the following data based on 10 pairs of observation on Y
and X.
X
X
i
 1700,  Yi  1110,  Xi Yi  205, 500
2
i
 322,000,  Yi2  132,100
Suppose the assumptions of the simple two variable CLRM are fulfilled,
obtain
(b)
(i)
OLS estimators, b1 and b 2 .
(ii)
Standard errors of these estimators.
(iii)
What is the value of r 2 ?
(9)
Consider the following model:
GNPt  B1  B2 M t  B3M t 1  B4 M t  M t 1   u t
where
GNPt  GNP at time t. M t  money supply at time t. M t 1  money
supply at time (t – 1) and M t  M t 1  = change in the money supply
between time t and time (t – 1).
The model thus postulates that the level of GNP at time t is a function of
the money supply at time t and time (t – 1) as well as the change in the
money supply between these time periods. Assuming you have data to
estimate the preceding model, can you estimate all the coefficients of
this model? Why or why not? If not, what coefficients can be estimated?
(6)
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3.
(a)
Regression results for Korean savings-income data are presented for the
period 1970-1995,
Ŷt 1.0161  152.4786Dt  0.0803Xt  0.0655 Dt Xt 
t = (0.0504) (4.6090)
(5.5413)
R 2  0.8819
(–4.0963)
where Yt  savings
X t  income
D t  1 for observations in 1982–1995
= 0 otherwise
(b)
4.
(a)
(i)
Interpret the regression results and obtain the regressions for the
two time periods, that is, 1970–1981 and 1982–1995.
(ii)
What do you infer by the statistical significance of the differential
intercept and the differential slope coefficients?
(6)
(i)
What are the practical consequences of in perfect multicollinearity?
(ii)
Outline the White’s
heteroscedasticity.
test
to
detect
for
the
presence
of
(9)
Suppose the true model is :
Yi  β1  β2X2i  u i
But we add an irrelevant variable X 3 to the model and estimate
Yi  A1  A 2 X 2i  A3X3i  vi
(i)
Are the estimates of β1 and β 2 in the first regression unbiased?
(ii)
Does inclusion of “irrelevant” variable X 3 affect the variances of
A 2 and A3 in the second regression?
(iii)
(b)
Would R 2 for the second regression be larger than that for the
first regression?
(9)
To study the rate of growth of population in an economy over the period
1970-1992, the following models were estimated:
Model I: ln pop t = 4.73 + 0.024t
t = (781.25) (54.71)
Model II: ln pop t
t=
= 4.77 + 0.015t - 0.075 D t + 0.011 D t t 
(2477.92) (34.01)
(–17.03)
(25.54)
where pop = population in millions
t = trend variable
D t  1 for 1970–1979
= 0 otherwise
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5.
(a)
(i)
In model I, what is the rate of growth of population over the
sample period?
(ii)
Are the population growth rates statistically different pre and post
1980?
(iii)
If they are different, then what are the growth rates for 1970–79
and 1980–1992?
(6)
Based on the data on annual percentage change in wages (Y) and
percent annual unemployment rate (X) for the years 1950 to 1966,
following regression was obtained:
 1 
Ŷ  1.4282  8.7243 
 Xt 
se  2.0675
R 2  0.3849
(b)
2.8478
F1,15  9.39
(i)
Interpret the above regression
(ii)
What would be the slope of the regression? What would be some
likely shapes of the curve corresponding to the above regression?
(iii)
What is the elasticity of Y with respect to X at mean values of
Y  4.8% and X  1.5% .
(7)
Consider the following two regressions:
Ct  26.19  0.6248GNPt  0.4398D t
se  2.73
0.0060
0.0736
R 2  0.99
Dt
1
 C 
 0.6246  0.4315

  25.92
GNPt
GNPt
 GNP 
se = (2.22)
(0.0068)
(0.0597)
R 2  0.875
where C = aggregate private consumption expenditure
GNP = gross national product
D = national defence expenditure
t = time
6.
(a)
(i)
What might be the reason for transforming the first equation into
the second equation?
(ii)
What assumption has been made about the error variance?
(iii)
According to (i) above, how do you know that the problem has
been corrected in the transformed regression?
(8)
State and prove the Gauss Markov theorem for the slope coefficient in
the classical linear regression model.
(6)
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(b)
Consider the following Cobb Douglas production function estimated for
Taiwan for the period 1955–1974.
lnGDP t  1.6524  0.3397lnL t  0.8460lnK t
t = (–2.725) (1.8295) (9.0625)
R 2  0.9951
RSS UR  0.0136
where GDPt  GDP at time t, L t = labour at time t, K t = capital at time t,
ln = natural logarithms.
(i)
Interpret the coefficients of the regression and comment on their
individual significance.
(ii)
Comment on the returns to scale experienced by the Taianese
economy.
(iii)
By imposing the restriction of constant returns to scale, the
 GDP 
K
  0.4947  1.0153ln  
 L t
 L t
following regression was obtained : ln 
T=
(–4.0612)
(28.1056)
R 2  0.9777 RSSR  0.0166
Interpret the above regression.
(iv)
7.
(a)
Use a test statistic to see whether the economy is characterized by
constant returns to scale.
(9)
For the two variable regression model Yi  B1  B2 Xi  u i , show that.
 y ŷ 

 y  ŷ 
x y

x
2
(b)
(i)
r
(ii)
β̂ 2
2
i
2
i
i
i
2
i
i
2
i
Consider the following model of Indian imports estimated using data for
40 years for the period 1945–1985. (Standard errors are given in
parentheses)
lnY t  1.5495  0.9972lnX 2t  0.3315lnX 3t  0.5284lnY t 1
se = (0.0903) (0.0191) (0.0243)
(0.024)
R 2  0.994
d  1.8
where Y = imports X 2  GDP, X3  CPI
(i)
Does the model suffer from first-order autocorrelation? Which test
statistic do you use and why?
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(ii)
Outline the steps of the test used. Compute the test statistic and
test the hypotheses that the preceding regression does not suffer
from first-order autocorrelation.
(iii)
If the general model is given
Yi  B1  B2 X 2i  B3X3i  u i where
errors follow AR(1) scheme, that is u t  ρu t 1  ε t and where ε t is a
white noise error term. Then how would you transform the model
to correct for the problem of autocorrelation.
(9)
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