M Phil Econometrics Problem Set for Week 5

M Phil Econometrics
Problem Set for Week 5
1) Properties of matrices used in OLS
Consider the 2
2 matrix
A=
a11 a12
a21 a22
i) What is the determinant of A, denoted jAj ?
ii) What is the inverse of A, denoted A
1
?
iii) What is nA 1 , for some scalar n ?
Now consider the matrix
B=
a11
n
a21
n
A
=
n
a12
n
a22
n
iv) What is jBj ?
v) What is B
1
?
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vi) Comment brie‡y on the relationship between nA 1 and An
illustrated by your answers to parts (iii) and (v). Where is this relationship useful
in studying properties of the OLS estimator in the linear model?
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2) Reverse regression
Consider the simple linear regression model
yi =
+ xi + ui
where yi and xi are two scalar variables, observed for a sample i = 1; 2; :::; N:
Recall that the OLS estimator of the slope parameter
where Sxy =
1
N
XN
i=1
(xi
x)(yi
b = Sxy
Sxx
y) and Sxx =
Now consider the reverse regression
1
N
XN
i=1
(xi
is given by
x)2 :
xi = a + byi + ei
i) Relate the parameters a and b and the error term ei to the parameters
and and the error term ui :
ii) De…ne the OLS estimator bb of the slope parameter b in the reverse
regression.
iii) What is the product bb b? What are the upper and lower bounds for
this product? How is this product related to the R2 measure of goodness of
…t for either of these regressions?
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3) Use of dummy variables
A researcher is interested in the e¤ect of school class sizes on the educational attainment of students. She estimates the linear model
+ CSi + xS0
i
Ai =
S
+ xFi 0
F
for i = 1; 2; :::; N
+ ui
where Ai is a scalar measure of attainment for student i (e.g. a standardized
test score), CSi is a measure of the average size of classes attended by student
i (e.g. average number of students per class), xSi is a vector of observations
on other characteristics of the school attended by student i (e.g. school size,
school location, school type, etc.) and xFi is a vector of observations on family background variables for student i (e.g. number of siblings, educational
attainment of parents, etc.). The model is estimated by OLS, using a random
sample of 5000 students.
i) The researcher reports that the OLS estimate of the parameter
0:017 with a standard error of 0:011.
is
Use this information to test the null hypothesis that class size has no
e¤ect on educational attainment, against a two-sided alternative, at the 5%
signi…cance level. What do you conclude?
ii) It is suggested that the e¤ect of class size on educational attainment
may be di¤erent for boys and girls. De…ne the zero/one dummy variable Di
to be equal to one if student i is a boy, and to be equal to zero if student i
is a girl.
B
Interpret the coe¢ cients
Ai =
B
Di +
G
Ai =
B
Di +
G
(1
Di ) +
B
G
in the model
Di ) + CSi + xS0
i
(1
Interpret the coe¢ cients
and
B
and
G
(Di CSi ) +
3
S
+ xFi 0
F
+ ui
in the model
G
[(1
Di )CSi ] + xS0
i
S
+ xFi 0
F
+ ui
iii) The researcher reports the OLS estimates of the parameters B and
G
to be 0:009 and 0:042, with standard errors of 0:010 and 0:012 respectively. What does this suggest about the e¤ect of class size on the educational
attainment of boys? What does this suggest about the e¤ect of class size on
the educational attainment of girls?
iv) Re-parameterize this model to obtain a test of the null hypothesis
H0 : B = G as a test of a single exclusion restriction in the re-parameterized
model.
G
v) The researcher reports the OLS estimate of ( B
) to be 0:033,
with a standard error of 0:016: What does this information suggest about the
claim that the e¤ect of class size on educational attainment is the same for
boys and girls?
vi) The researcher concludes on the basis of this evidence that the effect of class size on educational attainment is signi…cantly di¤erent for boys
and girls, and that bigger class sizes have a signi…cant negative e¤ect on the
educational attainment of girls. Suggest how the model that has been estimated could be extended to provide further evidence on the validity of these
conclusions.
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