Christopher Dougherty
EC220 - Introduction to econometrics: past
examinations and marking schemes
2011 exam
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Summer 2011 examination
EC220
Introduction to Econometrics
2010/2011 syllabus only. Not for resit candidates.
Instructions to candidates
Time allowed: 3 hours + 15 minutes reading time
This paper contains EIGHT questions. Answer any FOUR questions. All questions
will be given equal weight (25%).
You are supplied with: Graph paper
Statistical tables
Calculators are NOT allowed in this examination.
 LSE 2011/EC220
Page 1 of 14
Abbreviations used in this examination
iid
IV
OLS
s.e.
independently and identically distributed
instrumental variables
ordinary least squares
standard error
Results that may be assumed
For the purposes of this examination, you may assume without proof the following results (both from
the Weak Law of Large Numbers). Suppose that a random variable X is distributed with population
mean X and finite variance  X2 . Then the mean X of a sample of n independent observations of X is
a consistent estimator of X:
plim X   X
and the mean square deviation is a consistent estimator of the variance:
plim
1
n
 X
n

 X   var X    X2  E  X i   X 
2
i
2

i 1
You may also assume without proof the following associated result (Continuous Mapping Theorem).
Suppose that random variables X and Y are distributed with population means X and Y and finite
variances. Then
plim
 LSE 2011/EC220
1
n
 X
n
i
 X Yi  Y   cov X , Y   E X i   X Yi   Y 
i 1
Page 2 of 14
1. Some practitioners of econometrics advocate ‘standardizing’ each variable in a regression by
subtracting its sample mean and dividing by its sample standard deviation. Thus, if the original
regression specification is
Yi   1   2 X i  u i
the revised specification is
Yi*  1*   2* X i*  v i
where
Yi* 
Yi  Y
X X
and X i*  i
,
sY
sX
Y and X are the sample means of Y and X, sY and sX are the sample standard deviations of Y and
X, defined as
sY 
1
n
2
 Yi  Y 
n
and s X 
i 1
1
n
 X
n
 X ,
2
i
i 1
and n is the number of observations in the sample. Let the fitted models for the two specifications
be written
Yˆ  b  b X
i
1
2
i
and
Yˆi*  b1*  b2* X i*
(a) [3 marks] Taking account of the definitions of Y * and X * , provide an interpretation of b2*
(b) [3 marks] Show that b1*  0 . [Note: To do this, you must use the regression expression for
b1* .]
(c) [3 marks] Write down the regression expressions for b2* , first assuming the regression
specification should include an intercept, second assuming that it should not
include an intercept, and show that the expressions are equivalent.
s
(d) [3 marks] Show that b2*  X b2 . [Note: To do this, you must use the regression expressions
sY
for b2* and b2 .]


1 ˆ
(e) [3 marks] Hence show that Yˆi* 
Yi  Y .
sY
(f) [2 marks] Hence show that ei* 
1
ei .
sY
 
(g) [3 marks] Hence show that s.e. b2* 
sX
s.e. b2  . [Note: s.e. = standard error.]
sY
(h) [2 marks] Hence find the relationship between the t statistic for b2* and the t statistic for b2.
(i) [3 marks] Also, find the relationship between R2 for the original specification and R2 for the
revised specification.
 LSE 2011/EC220
Page 3 of 14
2. A data set contains data on the years of formal training, training qualifications, years of work
experience, and hourly earnings in rupees in 2010 for a sample of 100 refrigeration mechanics in
India. Most refrigeration mechanics acquire their skills through informal apprenticeships and have
no formal training. Some have one year of formal training at a training institute and earn the
Refrigeration Mechanic Certificate if they pass the test at the end. Some take a second year of
training and earn the Refrigeration Mechanic Diploma if they pass the test at the end. Trainees
may continue to the second year of training only if they have passed the test at the end of the first
year. Dummy variables RMC and RMD are defined as follows:
RMC = 1 for those who have had at least one year of formal training and have passed the test for
the Refrigeration Mechanic Certificate, but have not passed the test for the Refrigeration Mechanic
Diploma. RMC = 0 for all others.
RMD = 1 for those who have had two years of formal training and have passed the test for the
Refrigeration Mechanic Diploma. RMD = 0 for all others.
Three regressions are performed:
(1) The logarithm of hourly earnings on EXP, number of years of work experience, and
TRAINING, number of years of formal training at a training institute.
(2) The logarithm of hourly earnings on EXP, RMC, and RMD.
(3) The logarithm of hourly earnings on EXP, TRAINING, RMC, and RMD.
The regression results are shown in the table (standard errors in parentheses; RSS = residual sum of
squares.)
EXP
TRAINING
RMC
RMD
constant
R2
RSS
(1)
0.030
(0.005)
0.200
(0.020)
—
—
2.00
(0.60)
0.35
105.0
(2)
0.028
(0.005)
—
0.18
(0.04)
0.30
(0.06)
2.24
(0.61)
0.40
100.0
(3)
0.029
(0.005)
0.100
(0.045)
0.10
(0.05)
0.20
(0.08)
2.10
(0.70)
0.42
95.0
(a) [4 marks] Provide an interpretation of the coefficient of TRAINING in regression (1),
justifying it mathematically.
(b) [4 marks] Provide an interpretation of the coefficient of RMD in regression (2), justifying it
mathematically.
(c) [3 marks] Provide an interpretation of the coefficient of RMD in regression (3), with a brief
explanation in general terms.
(d) [3 marks] Someone asserts that the earnings of diploma-holders are no higher than those of
certificate-holders. Explain what you would need to do to test this hypothesis.
This question continues on the next page.
 LSE 2011/EC220
Page 4 of 14
(e) [4 marks] Someone else asserts that the extra earnings of a diploma-holder, compared with
those of a certificate-holder, are equal to the extra earnings of a certificate-holder,
compared with the earnings of those with no formal training. Explain what you
would need to do to test this hypothesis.
(f) [4 marks] Someone else asserts that the TRAINING variable is sufficient and that adding the
dummy variables in specification (3) does not improve the explanatory power of
the model in any material way. Explain whether you would agree with this
assertion.
(g) [3 marks] Some of the training institutes were public and others were private. The researcher
said that he thought that the holders of certificates and diplomas from the private
institutes tended to earn less than those who had been trained in public institutes.
Explain how you might test this assertion, given access to the data set.
 LSE 2011/EC220
Page 5 of 14
3. A researcher has a sample of 43 observations on a dependent variable, Y, and two potential
explanatory variables, X and Z. He defines two further variables V and W as the sum of X and Z
and the difference between them:
Vi  X i  Z i
Wi  X i  Z i
He fits the following four regressions
(1) A regression of Y on X and Z
(2) A regression of Y on V and W
(3) A regression of Y on V
(4) A regression of Y on Z and V
The table shows the regression results (standard errors in parentheses; RSS = residual sum of
squares; there was an intercept, not shown, in each regression). Unfortunately, a goat ate part of
the regression output and some of the numbers are missing. These are indicated by letters.
V
(1)
0.60
(0.04)
0.80
(0.04)
—
W
—
R2
RSS
0.60
200
X
Z
(2)
—
(3)
—
(4)
—
—
—
A
(B)
C
(D)
E
F
0.72
(0.02)
—
H
(I)
J
(K)
—
G
220
L
M
Each regression included an intercept (not shown).
(a) [20 marks] Reconstruct each missing number if this is possible, giving a brief explanation.
Detailed mathematical analysis is not required. If the calculation is too
complicated to do without a calculator, you may instead earn full marks by
indicating how the missing value should be calculated. If it is not possible to
reconstruct a number, give a brief explanation.
[1 mark each: A B C D E F H J L M
2 marks: K
4 marks each: G I ]
(b) The correlation between X and Z was high. That between V and W was low. Explain the
implications, if any, for a comparison of the regression results for specifications (1), (2), and
(3)
(i) [2 marks] making no assumption concerning the true values of the coefficients of X and
Z in specification (1)
(ii) [3 marks] assuming that the true coefficients of X and Z are the same.
 LSE 2011/EC220
Page 6 of 14
4. (a) A school has introduced an extra course of reading lessons for children starting school and a
researcher is evaluating the impact of the course on the scores on a literacy test taken at the
age of seven. In the first year of its implementation, those children whose surnames being A–
M are assigned to the extra course, while the rest have the normal curriculum. The researcher
hypothesizes that
Y   1   2 D   3 CA  u
where Y is the score on the literacy test, D is a dummy variable that equal to 1 for those
assigned to the extra course and 0 for the others, and CA is a measure of the cognitive ability
of the child when starting school, and u is an iid (independently and identically distributed)
disturbance term assumed to have a normal distribution. Unfortunately, the researcher has no
data on CA. Using OLS (ordinary least squares), she fits the regression
Yˆ  b1  b2 D
(i) [5 marks] Demonstrate, with a detailed mathematical proof, that b2 is an unbiased
estimator of 2.
(ii) [3 marks] A commentator says that the standard error of b2 will be invalid because an
important variable, CA, has been omitted from the specification. The
researcher replies that the standard error will remain valid if CA can be
assumed to have a normal distribution. Explain whether the commentator or
the researcher is correct.
(iii) [3 marks] Another commentator says that whether the distribution of CA is normal or not
makes no difference to the validity of the standard error. Evaluate this
assertion.
(b) The extra course is remedial in nature and the researcher thinks that its impact on the literacy
test scores is likely to be inversely related to the ability of the children.
(i) [3 marks] Show how the model may be extended to allow for this effect.
(ii) [5 marks] Provide an interpretation of each of the parameters in the extended model,
stating its likely sign, if this is possible.
(iii) [4 marks] Explain why b2 in the simple regression in part (a) would now be a biased
estimator of 2, and evaluate the direction of the bias.
(c) [2 marks] The next year, the course becomes optional. The researcher takes a sample of data
and again performs a simple regression of Y on D. Explain how the estimate of the
slope coefficient is likely to be affected by the fact that the course is optional.
 LSE 2011/EC220
Page 7 of 14
5. (a) A variable Y is determined by a variable Z, the relationship being
Y  2Z  v
where v is an iid (independently and identically distributed) disturbance term with mean 0 and
variance  v2 and the observations for Z are randomly drawn from a population with mean  Z
and variance  Z2 . The observations on Z are subject to measurement error, the observed
variable in observation i being Xi where
X i  Z i  wi
and wi is the measurement error. It may be assumed that w is iid with mean 0 and variance
 w2 and that Z, v, and w are distributed independently. When Y is regressed on X, given a
sample of n observations, the OLS (ordinary least squares) estimator of 2 is
n
X Y
i i

b2OLS
i 1
n
X
2
i
i 1
(i) [1 mark]
Show how Y is related to X, stating mathematically the relationship.
(ii) [1 mark]
Explain why it is not possible to obtain a closed-form expression for the
expectation of b2OLS .
(iii) [5 marks] Demonstrate that b2OLS is an inconsistent estimator of 2 and that the limiting
value is
2  2
 w2
 Z2   w2   Z2
For this purpose, you may use, without proof, the identity
 X i  X Yi  Y    X i Yi  nXY
n
n
i 1
i 1
(iv) [2 marks] The large-sample bias is an inverse function of  Z . Give an intuitive
explanation for this.
Y
is a consistent estimator of 2.
X
(vi) [2 marks] Give an intuitive explanation for this.
(v) [4 marks] Demonstrate mathematically that
This question continues on the next page.
 LSE 2011/EC220
Page 8 of 14
(b) In a similar model, the dependent variable is also subject to measurement error. The true
relationship is
Q  2Z  v
where Z and v are as in part (a). Z is subject to measurement error w, the observed variable
being X, with X and w being determined as in part (a). The dependent variable Q is subject to
measurement error, the observed dependent variable Y being affected by the same
measurement error, w, with factor 2:
Yi  Qi   2 wi
(i) [5 marks] Explain whether, in this case, it is possible to determine whether b2OLS is (1)
an unbiased estimator of 2, (2) a consistent estimator of 2. [Note: Detailed
mathematical proofs are not required and no credit will be given for them.]
(ii)` [2 marks] Give an intuitive explanation of why it is consistent, despite both variables
being subject to measurement error.
(iii) [3 marks] Explain whether, in this special case, having to regress Y on X instead of Q on
Z has any advantages or disadvantages. Note: If it had been possible to
regress Q on Z, the variance of the estimator would have been
 v2
n MSDZ 
where MSD(Z), the mean squared deviation of Z, is given by
MSDZ  
 LSE 2011/EC220
1
n
 Z
n
Z
2
i
i 1
Page 9 of 14
6. A researcher has annual data for 30 years on aggregate output, Yt, aggregate consumption Ct,
aggregate investment, It, and aggregate government expenditure, Gt, for a closed economy and
wishes to fit the model
C t   1   2 Yt  u t
(1)
Yt  C t  I t  Gt
(2)
Equation (2) is an income identity. The disturbance term ut in equation (1) may be assumed to be
iid (independently and identically distributed) and, for the purposes of this question, any potential
time series problems such as autocorrelation and nonstationarity may be ignored. The researcher
believes that both I and G are exogenous variables and initially combines them into a single
variable Z defined as their sum: Z = I + G. The revised identity is then
Yt  C t  Z t
(2*)
(a) (i) [3 marks] Show how IV (instrumental variables) estimation, using Z as an instrument,
can be used to fit equation (1) and demonstrate that the estimator is consistent.
(ii) [3 marks] The variance of the IV estimator of 2 can be approximated as
 b2IV 
2
 u2
2
 Yt  Y 
T

1
rY2, Z

 u2
T MSDY 

1
rY2, Z
t 1
where rY,Z is the sample correlation between Y and Z, MSD(Y) is the sample
mean square deviation of Y, and T is the number of observations in the sample.
Explain the sense in which the expression for the variance is an
approximation.
(iii) [3 marks] Explain the implications of the fact that it is an approximation.
(iv) [1 mark]
The researcher now returns to equation (2), separating Zt into its two
components It and Gt. Derive the reduced form equation for Yt using
equations (1) and (2).
(v) [2 marks] At a workshop, the researcher says that equation (1) could be fitted with either
I or G as instruments, but it would be better to use TSLS (Two-Stage Least
Squares). Explain in general terms why he said this.
(vi) [3 marks] A commentator notes that the theoretical coefficients of It and Gt in the
reduced form equation are the same. He says that, in this case, TSLS has no
theoretical advantage. Evaluate this assertion.
(vii) [2 marks] Another commentator says that, since the coefficients are the same, it will be
impossible to tell whether variations in Y are due to variations in I or to
variations in G, and hence one will have a problem of exact multicollinearity.
Evaluate this assertion.
(viii) [3 marks] Another commentator agrees, and says that the problem could be avoided by
combining I and G into a single variable. Since their theoretical coefficients
are the same, they should be given equal weight. Thus, in fact, one should use
the variable Z defined earlier in this question. Evaluate this assertion.
This question continues on the next page.
 LSE 2011/EC220
Page 10 of 14
(b) Suppose that, in fact, Y is determined exogenously and that I is determined endogenously,
being equal to the rest of output after consumption and government expenditure have taken
their share. Equation (2) should then be written
I t  Yt  C t  Gt
(2**)
with Yt and Gt exogenous. Equation (1) is as before.
(i) [3 marks] Analyze the implications for the IV estimation of 2, using Z as an instrument,
in part (a) (i).
(ii) [2 marks] Explain how the model should be fitted in this case.
 LSE 2011/EC220
Page 11 of 14
7. Consider the model
Yt   1   2 X t   t
(1)
X t   2 X t 1   t
(2)
where  2  1 , t is an iid (independently and identically distributed) disturbance term, with zero
mean and variance  2 , common to both equations, and X0 is generated randomly from the
ensemble distribution for X.
(a) (i) [2 marks] Explain what is meant by the ensemble distribution for X.
(ii) [2 marks] Show that the ensemble distribution for X has zero mean.
(iii) [3 marks] Show that the ensemble distribution for X has variance
 2
1   22
(iv) [3 marks] Describe in general terms the properties of the OLS (ordinary least squares)
estimator of 2, given a sample of data. [Note: Mathematical analysis is not
required]
(b) Equation (1) is fitted using a sample of T observations using OLS, the fitted model being
Yˆ  b  b X
t
(i) [1 mark]
1
2
t
Explain why one cannot evaluate E b2  for a finite sample, if 0 < 2 < 1.
(ii) [3 marks] Derive the large sample bias for 2, if 0 < 2 < 1.
(c) In a simulation, 1 = 5, 2 = 2, and t is generated as an iid variable with normal distribution
with mean 0 and variance 1. The simulation is conducted for three sample sizes, T = 25, T =
100, and T = 400, and three different values of 2: 2 = 0, 2 = 0.5, and 2 = 1.0. The figure
shows the distributions of b2 for 2 = 0.5 and 2 = 1.0 for each sample size, for 10 million
samples.
50
T = 400,  2 = 1
40
30
20
T = 400,  2 = 0.5
10
T = 100,  2 = 1
T = 25,  2 = 1
T = 100,  2 = 0.5
T = 25,  2 = 0.5
0
1
1.5
2
2.5
3
3.5
Distribution of b 2
(i) [3 marks] When 2 = 0, b2 is exactly equal to 3 for all samples, irrespectively of whether
the sample size is 25, 100, or 400. Explain why this should be the case.
This question continues on the next page.
 LSE 2011/EC220
Page 12 of 14
(ii) [2 marks] Explain whether the distributions in the figure support your asymptotic
analysis in part (b) (ii).
(iii) [3 marks] Explain why Y is cointegrated with X if 2 = 1.0.
(iv) [3 marks] In the figure it is clear that the speed of the convergence of the distribution of
b2 to its limiting spike is different for 2 = 0.5 and 2 = 1.0. Give an
explanation
There is a further question on the next page.
 LSE 2011/EC220
Page 13 of 14
8.
A variable Yt is generated by the process
Yt   1  u t
where 1 is a fixed, unknown parameter and ut is a disturbance term. A researcher wishes to
estimate 1 using a sample of T observations on Yt.
(a) Assuming that ut is generated as an iid process.
(i) [3 marks] Show from first principles that b1OLS  Y is the OLS (ordinary least squares)
estimator of 1.
(ii) [1 mark]
Show that it is an unbiased estimator of 1.
(iii) [1 mark]
Show that it is a consistent estimator of 1.
(b) Assuming that the disturbance term ut is subject to AR(1) autocorrelation and is generated by
the process
u t  u t 1   t
where t is iid and 0 <  < 1,
(i) [2 marks] Show how the model may be rewritten
Yt   1 1     Yt 1   t
and explain the point of rewriting the relationship in this way.
(ii) [2 marks] Explain in general terms why OLS would not yield an unbiased estimate of .
(iii) [3 marks] Demonstrate mathematically that OLS would yield a consistent estimate of .
(iv) [2 marks] Assuming that OLS will also provide a consistent estimate of the
intercept  1 1    , show how one may obtain a consistent estimate of 1.
(v) [2 marks] Demonstrate that Y remains an unbiased and consistent estimator of 1.
(vi) [2 marks] Discuss whether it is possible to determine which of the two estimators of 1
has more desirable properties.
(c) Assuming that the disturbance term ut is generated by the process
u t  u t 1   t
where t is iid, and that the unobserved initial value of Y at time zero, Y0, happens to be equal
to 1.
(i) [1 mark] Explain why the technique in part (b) (iv) cannot be used to obtain an estimate
of 1.
(ii) [3 marks] Demonstrate that Y is an unbiased but inconsistent estimator of 1. Note: The
variance of Y is
T  12T  1  2

(iii) [1 mark]
6T
Demonstrate that Y1, the value of Y in the first observation, is an unbiased but
inconsistent estimator of 1.
(iv) [2 marks] Explain which would be preferable as an estimator of 1: Y or Y1.
 LSE 2011/EC220
Page 14 of 14