May Examination Period 2022
ECN225 Econometrics 2
Duration: 3 hours
Answer ALL questions. The questions carry equal marks. JUSTIFY
ALL YOUR ANSWERS. There are tables of critical values from the standard normal, Fm,1 , ADF, !2 and QLR distribution provided on pages
9—10. Delete any answers that you do not wish to be marked. The final submission on QMplus is the version that will be marked. All nonmathematical questions should be typed and not handwritten
unless the student has obtained an o¢cial exemption.
This examination paper MUST NOT be shared with anyone else. Doing so
will be considered a very serious assessment o§ence under the Queen Mary
Academic Misconduct Policy.
This examination is an individual assessment and must be entirely your own
work. All work will be run through the plagiarism software, Turnitin. The
software will also compare your script against all other student submissions.
Any evidence of plagiarism or collusion will be taken forward as academic
misconduct.
Calculators are permitted in this examination.
Please ensure that your working is clearly shown with all steps of your calculation included in your answer document, including any formula used.
When writing formulas, please note the following:
- It is acceptable to use the standard alphabet rather than greek letters. The
following are recommended: m for µ, s for ", w for !, r for $, d for !, b for %
- For mathematical operators: add +, subtract -, multiply *, and divide /
- Where appropriate, use an underscore to indicate a subscript, e.g. r_f for
rf
p
- Use the ^character for power, e.g. x^2 for x2 , x^0.5 for x
- As an alternative to x^.5 you may type sqrt(x)
- Use brackets as necessary. To make your answer clearer use di§erent brackets where appropriate, e.g. [] {} ()
Examiner: Dr. ätěpána Lazarová
c Queen Mary University of London 2022
"
Page 2
ECN225 (2022)
Question 1
A dataset contains data on birth weights of 65535 babies born in June 1997
along with variables that are potentially related to birth weights. The sample
is restricted to singleton births, with mothers recorded as either black or
white, between the ages of 18 and 45, resident in the United States.
With the aim to investigate factors influencing the weight of babies at
birth, a regression of W eight on a range of variables has been carried out.
A description of variables is given below and the estimation results can be
found on the following page.
Referring to these results, answer the following questions. Keep in mind
that two of the regressors, M om_Age and M _W tGain, are entered in the
regression in terms of the deviations from their mean.
(a) Interpret the estimated intercept.
(5 marks)
(b) Are boys, on average, heavier than girls at birth, all other factors being
equal? If so, by how much? Is the di§erence between boys and girls
estimated from this regression statistically significant at the 5% level?
(5 marks)
(c) Would you say the fact that the mother is a smoker is an important
determinant of the birth weight of the baby?
(5 marks)
(d) What is the expected di§erence in birth weight of babies whose mothers
have a high school and babies whose mothers are college graduates,
ceteris paribus? Explain your reasoning.
(5 marks)
(e) Is there evidence of a nonlinear relationship between birth weight and
mother’s weight gain?
(5 marks)
Description of variables: W eight is birth weight in grams, Boy is equal
to 1 if the baby is a boy, 0 if it is a girl, M arried is 1 if the mother is married,
0 if unmarried.
Education of the mother is divided into four categories: less than high
school, high school (dummy variable Ed_Hs), some college (dummy variable
Ed_SmCol), and college graduate (dummy variable Ed_Col). The omitted
category is “less than high school,” so coe¢cients must be interpreted relative to this category.
Smoke is equal to 1 if the mother smokes, 0 if she does not, CigsP er is
cigarettes smoked per day, Black is 1 if the mother is black, 0 otherwise.
M om_Age is age in years, M _W tGain is mother’s weight gain in pounds.
ECN225 (2022)
Page 3
The last two variables are entered into the regression as centered around their
mean, M om_Age_Cntrd and M _W tGain_Cntrd, i.e. M om_Age_Cntrd
is the variable M om_Age minus the average age of M om_Age, and similarly for M _W tGain_Cntrd. The average age of mothers is 27.2 years and
the average weight gain is 30.8 pounds.
Page 4
ECN225 (2022)
Question 2
(a) Consider the linear model Yi = % 0 + % 1 X1i + % 2 X2i + ui where ui is the
error term and where E (ui |X1i , X2i ) = 0, observations (Yi , X1i , X2i )
are independent and identically distributed, 0 < E (Y1i4 ) < 1, 0 <
4
4
2
E (X1i
) < 1, 0 < E (X2i
) < 1 and where X2i = X1i
. Are the
OLS estimators %̂ 1 and %̂ 2 of the coe¢cients % 1 and % 2 unbiased and
consistent? Explain.
(4 marks)
(b) Suppose that in the linear model Yi = % 0 + % 1 Xi + ui the error has zero
mean, E (ui ) = 0, that it has a non-zero conditional mean, E (ui |Xi ) 6=
0, that observations (Yi , Xi ) are independent and identically distributed
and that the fourth moments of Yi and Xi are finite. Is the OLS
estimator %̂ 1 biased? Is it consistent? In case it is biased and there is
positive correlation between the regressor and the error, does %̂ 1 tend
to overestimate or underestimate the true value of % 1 ? Discuss briefly
but carefully.
(6 marks)
(c) Consider a polynomial regression model
Yi = % 0 + % 1 Xi + % 2 Xi2 + . . . + % r Xir + ui ,
where E (ui |Xi , Xi2 , . . . , Xir ) = 0, observations (Yi , Xi ) are independent
and identically distributed and 0 < E (Yi4 ) < 1, 0 < E (Xi4r ) < 1.
How is the OLS estimation a§ected if we estimate a linear regression
(r = 1) when the true form of the regression function is quadratic
(r = 2) or cubic (r = 3)?
(3 marks)
(d) A researcher needs to select the degree of polynomial in her regression
model. A colleague advises her that a higher-degree polynomial is more
flexible and has a better chance to fit the data, and that therefore she
should consider a polynomial of a very high degree. Is this a sound
advice from the econometrician’s point of view? Explain. (4 marks)
Question continues on next page
ECN225 (2022)
Page 5
(e) Consider the model
Yi = % 0 + % 1 Xi + ui
where E (ui |Xi ) = 0, observations (Yi , Xi ) are independent and identically distributed and 0 < E (Yi4 ) < 1, 0 < E (Xi4 ) < 1. Suppose Xi is
measured with error and so X̃i is observed instead of Xi , X̃i = Xi + wi ,
where Ewi = 0, var (wi ) = " 2 , cov (wi , Xi ) = 0 and cov (wi , ui ) = 0. In
this case, it can be shown that
p
%̂ 1 !
" 2X
% .
" 2X + " 2w 1
Does %̂ 1 overestimate or underestimate the true value of % 1 ? What
can be said about %̂ 1 if there is a very small or very large amount of
measurement error?
(5 marks)
(f) In part (e), if we could estimate " 2w /" 2X , how could we correct for the
! 2X
bias in !2 +!
(3 marks)
2 % 1?
X
w
Page 6
ECN225 (2022)
Question 3
In this question, use the macroeconomic time series data for the US in the file
macro_2eSW.xls available on QM+ and carry out computations in EViews.
The description of the data set can be found in file macro_2eSW.pdf. Answers
to this question must be supported by printouts of EViews graphs and tables
of estimation and testing results.
(a) Create series Inft of quarterly data on the annualized US inflation rate.
(3 marks)
(b) Display a plot of Inft series. Would you say this series is stationary?
(4 marks)
(c) Using the sample period 1958:1-2005:1, carry out an augmented DickeyFuller statistic to test for a unit autoregressive root in the AR model for
Inft . Use the Schwarz selection criterion for determining the lag order
and test against the alternative that the series is stationary without a
trend.
(6 marks)
(d) Using the sample period 1957:1-2005:1, estimate an OLS regression of
3-month treasury bill interest rate, F Y GM 3 t , on 1-year treasury bond
interest rate, F Y GT 1 t . Use heteroskedasticity robust standard errors.
Comment on the statistical significance of the coe¢cients and the fit of
the regression. What is the conclusion about the relationship between
the two interest rates?
(6 marks)
(e) What is a spurious regression? Is the regression in part (d) a spurious regression? Carefully explain and accompany your reasoning with
additional tests.
(6 marks)
ECN225 (2022)
Page 7
Question 4
In a study of the relationship between real aggregate consumption and prices
in a specific country based on a time series of 104 observations, the following
OLS regression function was obtained (standard errors in parentheses),
Ct = 13.74 + 0.24 Pt ,
(0.35)
(0.05)
R2 = 0.12,
SER = 2.85,
(1)
where Ct is logarithm of consumption and Pt is logarithm of price.
(a) Is the coe¢cient on Pt statistically significant?
(3 marks)
(b) Compute the p-value for the test of significance of the coe¢cient on Pt .
(3 marks)
(c) Comment on the sign of the coe¢cient on Pt .
(2 marks)
Suppose that, in addition to a consumption equation
Ct = % 0 + % 1 Pt + ut ,
where the error ut is independently and identically distributed with mean 0
and variance " 2u , a supply equation
Qt = & 0 + & 1 Pt + & 2 Zt + vt
is considered, where Qt is logarithm of production, Zt is a productivity variable independent of ut and vt , and the error term vt is independent of ut and
is independently and identically distributed with mean 0 and variance " 2v .
(d) Suppose that the equilibrium condition Qt = Ct holds. Show that, if
% 1 6= & 1 , then
Pt =
1
(& & % 0 + & 2 Zt + vt & ut ) .
%1 & &1 0
What is the implication of this relationship for the OLS estimation of
the slope parameter % 1 reported in (1)?
(6 marks)
(e) Propose an alternative method for estimating % 1 . Explain when this
estimator improves on the OLS estimator.
(6 marks)
Question continues on next page
Page 8
ECN225 (2022)
(f) The OLS regression of Pt on Zt for t = 1, . . . , 104 gives
Pbt = &3.01 & 0.74 Zt ,
(0.45)
(0.07)
R2 = 0.31,
SER = 3.66.
The researcher also computes
IV
%̂ 1 = &0.66,
IV
%̂ 0 = 10.91.
IV
Is Zt a relevant instrument? Comment on the sign of %̂ 1 .
(5 marks)
End of Paper - An Appendix of 2 pages follows
ECN225 (2022)
Page 9
Large Sample Critical Values for the t-statistic
from the Standard Normal Distribution
Significance level
10%
5%
1%
2-Sided Test (6=)
Reject if |t| is greater than 1.64
1.96
2.58
1-Sided Test (>)
Reject if t is greater than
1.28
1.64
2.33
1-Sided Test (<)
Reject if t is less than
&1.28 &1.64
&2.33
Critical Values for the Fm,1
Distribution
Significance level
Degrees of freedom 10%
5%
1%
1
2.71 3.84 6.63
2
2.30 3.00 4.61
3
2.08 2.60 3.78
4
1.94 2.37 3.32
5
1.85 2.21 3.02
6
1.77 2.10 2.80
7
1.72 2.01 2.64
8
1.67 1.94 2.51
9
1.63 1.88 2.41
10
1.60 1.83 2.32
Large Sample Critical Values of the
Augmented Dickey-Fuller Statistic
Significance level
Deterministic regressors 10%
5%
1%
Intercept only
&2.57 &2.86 &3.43
Intercept and time trend
&3.12 &3.41 &3.96
Page 10
ECN225 (2022)
Critical Values for the !2
Distribution
Significance
Degrees of freedom 10%
5%
1
2.71 3.84
2
4.61 5.99
3
6.25 7.81
4
7.78 9.49
5
9.24 11.07
6
10.64 12.59
7
12.02 14.07
8
13.36 15.51
9
14.68 16.92
10
15.99 18.31
level
1%
6.63
9.21
11.34
13.28
15.09
16.81
18.48
20.09
21.67
23.21
Critical Values of the QLR Statistic
with 15% Trimming
Number of Restrictions (q) 10% 5%
1
7.12 8.68
2
5.00 5.86
3
4.09 4.71
4
3.59 4.09
5
3.02 3.66
6
2.84 3.37
7
2.69 3.15
8
2.58 2.98
9
2.48 2.84
10
2.40 2.71
End of Appendix
1%
12.16
7.78
6.02
5.12
4.53
4.12
3.82
3.57
3.38
3.23