UBC, ECONOMICS 527
2013 MIDTERM EXAMINATION
Suggested solutions
1. (a) Let z denote the -the quantile of the standard normal distribution. Since
Y X
X N (0, 1),
we have that
Y X
z X
= P Y z + X X .
=P
Thus, the -th conditional quantile of Y given X is:
Econ 527
Assignment 1
The due date for this assignment is Thursday September 17.
1. Let
be an arbitrary sample space, and P be a probability function dened on a
collection of the events. Is there an event independent of itself? If not, prove. If yes,
give
U P D AT E D : M A R C H 9 , 2 0 0 6
LECTURE 16
TIME SERIES TOPICS
Denitions
Often econometricians have to deal with data sets that come in the form of a time series or stochastic process,
a collection of observations on the same variable (or vector of va
NOVEMBER 24, 2008
GMM I
Denition
Suppose that an econometrician observes the data fWi : i = 1; : : : ; ng where Wi is a random p-vector. Let g
be a l dimensional function depending on Wi and the k-vector of parameters b:
0
1
g1 (Wi ; b)
B
C
.
.
g (Wi ; b)
DECEMBER 8, 2008
LECTURE 8
LARGE PROPERTIES OF OLS, ASYMPTOTIC CONFIDENCE INTERVALS AND
HYPOTHESIS TESTING
In this lecture, we discuss large sample properties of the OLS estimator for the linear regression model
dened by the following assumptions:
0
(A1)
DECEMBER 8, 2008
LECTURE 10
ENDOGENEITY AND INSTRUMENTAL VARIABLES ESTIMATION
Endogeneity
Consider a partitioned regression model:
Yi
0
= Xi + Ui
0
0
= X1i 1 + X2i
2
+ Ui ;
(1)
where X1i is a k1 -vector and X2i is a k2 -vector of random regressors, 1 is k
NOVEMBER 19, 2009
LECTURE 6
2
PROPERTIES OF R , MODEL MISSPECIFICATION, TEST OF STRUCTURAL
CHANGE, DUMMY VARIABLES, FORECASTS
Properties of R
2
2
In this section, we will show that, when adding new regressors, R will rise/fall if the F -statistic associat
NOVEMBER 19, 2009
LECTURE 7
LARGE SAMPLE THEORY
Limits and convergence concepts: almost sure, in probability and
in mean
Let fan : n = 1; 2; : : :g be a sequence of non-random real numbers. We say that a is the limit of fan g if for
all real > 0 we can nd
SEPTEMBER 24, 2009
LECTURE 2
LINEAR REGRESSION MODEL AND OLS
Denitions
A common question in econometrics is to study the eect of one group of variables Xi , usually called the
regressors, on another Yi ; the dependent variables. An econometrician observes
Econ 527
Assignment 2
The due date for this assignment is Thursday September 24.
1. Davidson and MacKinnon, Chapter 1, Exercises 1.1, 1.2 (the simulation part of the
exercise is optional), 1.5, 1.10, 1.19, 1.20 on pages 38-40.
2
2. Let X
n : Show that EX
Econ 527
Assignment 3
The due date for this assignment is Wednesday September 30.
1. Davidson and MacKinnon, Chapter 1, Exercise 1.23.
2. Davidson and MacKinnon, Chapter 2, Exercises 2.1, 2.2, 2.3.
3. Davidson and MacKinnon, Chapter 3, Exercise 3.6.
4. Co
Econ 527
Assignment 10
The due date for this assignment is Thursday December 3.
1. Davidson and MacKinnon, Exercises 8.1, 8.2, 8.3, 8.4, and 8.9 on pages 347-348.
2. The data for this exercise (available from the course web page) contains the following
in
Econ 527
Assignment 9
The due date for this assignment is Wednesday November 25.
1. Let bn = bn;1 ; : : : ; bn;k
0
be an estimator of the k-vector of parameters = ( 1 ; : : : ;
0
k) .
!d W N (0; ), where is a positive defSuppose that bn !p , and n1=2 bn
i
Assignment 9 : Suggested Solutions
November 25, 2009
Q1.
1
a) Dene Xn = n 2 (n ) and h(Xn ) = Xn c. Apply the Continuous Mapping Theorem we have
1
n 2 (n ) c = h(Xn ) d h(X) = W c
By the property of normal density we have
1
n 2 (n ) c d N (0, c c).
b) Set
Econ 527
Assignment 8
The due date for this assignment is Thursday November 12.
1. Davidson and MacKinnon, Exercises 4.18 and 5.10.
2. Consider the following two regression models.
Model 1: Yi = Xi0 1 + Ui if i = 1; : : : ; n1 , and Yi = Xi0 2 + Ui if i =
Econ 527
Assignment 7
The due date for this assignment is Wednesday November 4.
1. Davidson and MacKinnon, Exercises 4.7, 4.10, 4.11.
2. Davidson and MacKinnon, Exercises 3.23 and 4.25. In 4.25, perform only the nite
sample test discussed in Lecture 5.
3.
Econ 527
Assignment 6
The due date for this assignment is Wednesday October 28.
1. Consider the model
Yi =
+ Ui ;
where Ui are iid N (0; 1) random variables, i = 1; : : : ; n:
(a) Find the OLS estimator of and its mean, variance, and distribution.
(b) Sup
Econ 527
Assignment 4
The due date for this assignment is Thursday October 8.
1. Davidson and MacKinnon, Chapter 2, Exercises 2.5, 2.6, 2.11, 2.13.
2. Chapter 3, Exercises 3.7-3.10. Hint: In Exercise 3.8, A is symmetric and positive
denite, and therefore
Econ 527
Assignment 5
The due date for this assignment is Wednesday October 14.
1. Davidson and MacKinnon, Chapter 2, Questions 2.15, 2.16, 2.17, 2.18, 2.20, 2.23, 2.24.
2. Davidson and MacKinnon, Chapter 3, Exercises 3.11, 3.15, 3.16.
1
OCTOBER 13, 2009
LECTURE 5
HYPOTHESIS TESTING
Basic concepts
In this lecture we continue to discuss the normal classical linear regression dened by Assumptions (A1)-(A5).
A statistical hypothesis is usually an assertion about the population parameters, fo
OCTOBER 18, 2009
LECTURE 4
CONFIDENCE INTERVALS
In this lecture we consider the normal regression model dened by Assumptions (A1)-(A5).
The point estimator b of the vector of parameters is not very informative, since P b =
= 0. In
this lecture, we conside
NOVEMBER 7, 2013
LECTURE 7
LARGE SAMPLE THEORY
Limits and convergence concepts: almost sure, in probability and in
mean
Let cfw_an : n = 1, 2, . . . be a sequence of non-random real numbers. We say that a is the limit of cfw_an if for
all real > 0 we can
October 17, 2012
LECTURE 5
HYPOTHESIS TESTING
Basic concepts
In this lecture we continue to discuss the normal classical linear regression dened by Assumptions (A1)-(A5).
Let Rd be a parameter of interest. Some examples of include:
The coecient of one of
SEPTEMBER 20, 2012
LECTURE 2
LINEAR REGRESSION MODEL AND OLS
Denitions
A common question in econometrics is to study the eect of one group of variables Xi , usually called the
regressors, on another Yi , the dependent variables. An econometrician observes
OCTOBER 14, 2010
LECTURE 4
CONFIDENCE INTERVALS
In this lecture we consider the normal regression model dened by Assumptions (A1)-(A5).
The point estimator of the vector of parameters is not very informative, since P = = 0. In
this lecture, we consider co
DECEMBER 8, 2008
LECTURE 9
HETEROSKEDASTICITY AND GENERALIZED LS
Generalized LS
In this lecture, we consider the same model as in Lecture 8, dened by Assumptions (A1), (A6) - (A9).
However, we will assume that
(A2*) E (Ui jXi ) = 0:
The assumption is stro
NOVEMBER 2, 2010
LECTURE 6
2
PROPERTIES OF R , MODEL MISSPECIFICATION, TEST OF STRUCTURAL
CHANGE, DUMMY VARIABLES, FORECASTS
Properties of R
2
2
In this section, we will show that, when adding new regressors, R will rise/fall if the F -statistic associate
DECEMBER 17, 2010
LECTURE 10
ENDOGENEITY AND INSTRUMENTAL VARIABLES ESTIMATION
Endogeneity
Consider a partitioned regression model:
Yi
= Xi + Ui
= X1i 1 + X2i 2 + Ui ,
(1)
where X1i is a k1 -vector and X2i is a k2 -vector of random regressors, 1 is k1 1 a