Problem Set 4 Solutions
ECON 837
Prof. Simon Woodcock, Spring 2014
1. We have Z = [y X] ; so that
y0
X0
Z0 Z =
y
X
y0 y
X0 y
=
y0 X
X0 X
and hence
10
25
y0 y = 100 X0 y =
20
0
X0 X =
0
75
:
P
Since the model includes an intercept, we know the (1; 1) eleme
Problem Set 6
ECON 837
Prof. Simon Woodcock, Spring 2012
Due: March 28 in class
1. [Final Exam, 2004] Suppose fxi ; yi gn=1 are an iid random sample from some distribution
i
with at least 4 moments. Denote E [x] by x and E [y ] by y : Let = x
y and let
=
Problem Set 5 Solutions
ECON 837
Prof. Simon Woodcock, Spring 2014
1. We know plimXn Yn = plimXn plimYn = 6: Likewise, plim(Xn + Yn ) = plimXn + plimYn = 5: Finally, by the Slutsky
2
2
2
2
Theorem, plim Xn + Yn = (plimXn ) + (plimYn ) = 13:
2. Since the x
Problem Set 6 Solutions
ECON 837
Prof. Simon Woodcock, Spring 2012
1. Dene x =
1
n
P
i
xi and y =
P
1
n
i
yi : From Khinchine WLLN we know that:
s
p
x!
and
x
p
y!
y:
From the CLT we know that:
x
a
N
2
x
x;
and
n
where
xi
yi
V ar
p
(a) Let d = x
y : Then d
Marketing Mix
1
Oh iPhone, oh iPhone Ashley Dyal BUS 235 Mr. Vasquez December 13, 2010
Marketing Mix
2
Oh iPhone, Oh iPhone The cell phone market is changing almost daily. There are always more desired products out that everyone is just dying to get their
8: The k-Variable Linear Model 3
ECON 837
Prof. Simon Woodcock, Spring 2014
Constrained Estimators
Last day we derived a test statistic for nested linear hypotheses under normality of y: We
stated the conditions under which the test statistic had a known
7: The k-Variable Linear Model 2
ECON 837
Prof. Simon Woodcock, Spring 2014
Normality
To this point we have made two assumptions: linearity and spherical errors. We now add
a third assumption so that we can derive sampling distributions for ^ and s2 and d
6: The k-Variable Linear Model 1
ECON 837
Prof. Simon Woodcock, Spring 2014
Matrix Formulation
Now we turn our attention to the matrix formulation of the linear model. We will see that
the k -variable case is just a generalisation of two-variable simple r
5: More About Simple Regression
ECON 837
Prof. Simon Woodcock, Spring 2014
Recall from last time that ^ and ^ are the least squares estimators of
and : We call
yi = ^ + ^ xi the predicted values.
^
We turn now to the fraction of variation in y explained b
4: Simple Linear Regression
ECON 837
Prof. Simon Woodcock, Spring 2014
We will begin our discussion of linear regression by considering the simplest possible
model specication, namely, a single equation in two variables. The general multiple regression is
3: Introduction to Estimation and Inference
ECON 837
Prof. Simon Woodcock, Spring 2014
Typically, the data we observe consist of repeated measurements on one or more variables
of interest. We usually think of these as being the outcome of a DGP. Underlyin
2: Joint Distributions
ECON 837
Prof. Simon Woodcock, Spring 2014
The last lecture focused on probability and distribution in the case of a single random variable. However, in econometrics
we are almost always interested in the relationship between two or
1: Probability and Distribution Basics
ECON 837
Simon Woodcock, Spring 2014
Random Variables
Econometrics is the application of economic models to economic data. Economic data are measurements of some aspect of the
economy. We usually think of them as bei
Problem Set 8 Solutions
ECON 837
Prof. Simon Woodcock, Spring 2012
1.
(a) The endogenous variables are Ct and Yt ; the only exogenous variable is It : The reduced form is given by
Yt
Ct
(b) The least squares estimator is
P
Ct
^=
P
=
=
From (1) we see
P
P
Problem Set 8
ECON 837
Prof. Simon Woodcock, Spring 2012
Do not hand in
1. Consider a simple Keynesian model of national income determination:
(1)
(2)
Ct =
+ Y t + "t
Yt = Ct + It
where Ct is consumption, Yt is income, and It is investment. Assume "t
iid
10: Introduction to Asymptotic Theory
ECON 837
Prof. Simon Woodcock, Spring 2014
To this point, our discussion has focused on the nite sample properties of estimators
(primarily least squares, but also sample means, variances, and the like), and nite samp
9: k-Variable Linear Model Miscellany
ECON 837
Prof. Simon Woodcock, Spring 2014
Dummy Variables
Dummy variables are regressors that take value 0 or 1. Usually these indicate the absence or
presence of a characteristic. Dummy variables allow the intercept
11: Asymptotic Properties of Maximum Likelihood
Estimators
ECON 837
Prof. Simon Woodcock, Spring 2014
We discussed maximum likelihood estimation at several points already. It is a very
ve
popular estimation method. It assumes a parametric distribution for
Problem Set 1 Solutions
ECON 837
Prof. Simon Woodcock, Spring 2012
1.
(a) The mgf is
MZ (t)
tz
Z
1
Z
tz
1
1
tz
e fZ (z ) dz =
= Ee =
ee
0
0
Z1
Z1
1 z =( 1 t )
1 z( 1 t )
dz =
dz
e
e
=
0
0
Z1
1
1
t z =( 1 t )
=
dz
e
1
t0
1
=
1
t
since the nal integrand is
Problem Set 1
ECON 837
Prof. Simon Woodcock, Spring 2014
Due: Wednesday, Jan. 29
(Some problems are from Casella and Berger Statistical Inference, 1990).
1. [Partial question from 2005 Midterm] A random variable z has an exponential( ) distribution if it
19: The Generalized Method of Moments
ECON 837
Prof. Simon Woodcock, Spring 2014
Generalized Method of Moments (GMM) estimators are extremely popular among applied researchers. GMM is a very
exible estimation framework that yields consistent and
asymptot
18: Introduction to Nonlinear Models
ECON 837
Prof. Simon Woodcock, Spring 2014
The regression models we have considered to this point have all been linear in parameters.
Today we brie consider nonlinear models. There are three primary estimation framell
17: SEM Identication and Estimation
ECON 837
Prof. Simon Woodcock, Spring 2014
Last day we introduced Simultaneous Equations Models (SEMs). Today, we discuss how
to determine whether structural parameters are identied, and how to estimate SEMs.
Classical
16: Introduction to Simultaneous Equations Models
ECON 837
Prof. Simon Woodcock, Spring 2014
One of the things that distinguishes econometrics from statistics is our reliance on economic theory to guide empirical work. In many cases, economic theory predi
15: GLS Applications
ECON 837
Prof. Simon Woodcock, Spring 2014
Recall that when V ar ["jX] = V and V
(X0 V 1 X)
1
1
= P0 P; we obtain the GLS estimator ^ G =
X0 V 1 y by applying least squares to the transformed model
Py = PX + P":
Today, we discuss some
14: Generalized Least Squares
ECON 837
Prof. Simon Woodcock, Spring 2014
We now return to our linear regression model, but depart from the assumption of spherical
errors. We call this the generalized regression model. We retain the linearity assumption,
E
13: Asymptotic Testing
ECON 837
Prof. Simon Woodcock, Spring 2014
The main point of asymptotic theory is to develop approximate sampling distributions
that we can use for inference. Today we talk about three asymptotically equivalent testing
ll
procedures
12: Asymptotics of Least Squares
ECON 837
Prof. Simon Woodcock, Spring 2014
We discussed the nite sample properties of the least squares estimator in detail. We
ve
ve
also discussed the asymptotic properties of maximum likelihood estimators. We know that
Problem Set 7 Solutions
ECON 837
Prof. Simon Woodcock, Spring 2012
1. (Note: the expectations given in the question should all have been conditional, not unconditional. My apologies for the
oversight. Working with unconditional expectations is equivalent
Problem Set 7
ECON 837
Prof. Simon Woodcock, Spring 2012
Due: April 4
1. Suppose we have the regression model
y = X +"
E ["] = 0; E ["0 ] =
2
; tr ( ) = n:
(1)
(2)
Consider the OLS and GLS estimators,
^ OLS = (X0 X) 1 X0 y
^ GLS = X0 1 X 1 X0
h
(a) What i