Homework 3 Solutions
Ryan Abman, Chris Severen, Dan Argyle
February 13, 2014
1
MLE
A) Seeing as how cfw_Xt n is a sequence of independent and identically distributed (iid) exponential random
t=1
variables with parameter 0 , our likelihood function is:
n
n
Homework 4 Solutions
Ryan Abman, Chris Severen, Dan Argyle
February 24, 2014
1
Asymptotic Normality to Consistency
A) Consistency is a basic requirement of an estimator. As long as our estimator is consistent, an arbitrarily
large sample will allow us to
Homework 1 Solutions
Ryan Abman, Chris Severen, Dan Argyle
January 21, 2014
1
Scaling and R2
A) In this context, you would explain to the board member that R2 measures the proportion of test score
variation explained by the model and there are likely many
University of California
Department of Economics
D. Steigerwald
Economics 241B
Exercise 3
1.
Let (X1,Xn) be an independent and identically distributed sample of exponential
random variables with parameter 0 , so that
f X t ( x; 0 ) 0e 0 x where 0 x< and 0
University of California
Department of Economics
D. Steigerwald
Economics 241B
Exercise 7
1.
Consider the population model (in deviation-from-means form)
Yt X t* U t ,
in which X t* is latent. The observed regressor is
X t X t* Vt ,
in which Vt is indepen
University of California
Department of Economics
D. Steigerwald
Economics 241B
Exercise 5
1.
We will investigate the following claim (which appears in nearly this form in
Studenmund)
If an equation that contains a lagged dependent variable as a regressor
University of California
Department of Economics
D. Steigerwald
Economics 241B
Exercise 2
1.
Consider the bivariate regression model
Yt X t U t ,
in which the regressor is exogenous and all observations are measured as deviations from
means. You wish to t
241B Lecture
Ergodic Stationarity
The key concept in extending analysis to time series, is a stochastic process. A
stochastic process is simply a formal name given to a sequence of random variables.
If the index denotes time, then the stochastic process i
University of California
Department of Economics
Doug Steigerwald
Econometrics
Economics 241B
Course Goals:
To provide training in linear regression models, with special focus on the issues of endogeneity
and consistent standard error estimation, and basi
Key Topics
Probability Concepts
Understand and verify the following implications, with X = fx1 ; : : : ; xn g,
E (ut jX ) = 0 ) E (ut jxt ) = 0 ) E (ut ) = 0
E (ut jxt ) = 0 ) E (xt ut ) = 0
You should know this from prior coursework, but it is covered in
ECON 241B
Additional Problems
Question 1 Winter 2011 Q1
You wish to estimate the eect of unemployment on criminal behavior in Gotham City. In
your model, Crimet , is a function of the unemployment rate, U nempt , and the presence
of Batman, Batmant . You
Homework 5 Solutions
Ryan Abman, Chris Severen, Dan Argyle
February 27, 2014
1
The Studenmund Claim
A) Because Yt is strictly stationary, E(Yt ) does not depend on t. Thus E(Yt ) = E(Yt1 ) = 0. Therefore:
E(Ut ) = E(Yt ) E(Yt1 ) = 0
Because strict station
Homework 6 Solutions
Ryan Abman, Chris Severen, Dan Argyle
March 5, 2014
1
Asymptotic Behavior of Hypothesis Tests
A) Note that our test statistic is:
Bk k
n(Bk k )
=
SE (Bk )
Avar(Bk )
1 1
where Avar(Bk ) = Sxx SSxx and the subscripted k implies that wer
Homework 7 Solutions
Ryan Abman, Chris Severen, Dan Argyle
March 14, 2014
1
Instrumental Variables
A) To show that 1 can be written as a function of the average wage for law degree holders and non law
degree holders, lets reintroduce the idea of the condi
Economics 241B
Population Regression Models
A starting point for work in econometrics is to select a variable of interest,
often termed the dependent variable, most frequently denoted Yt . The phrase
dependent variable arises, because we attempt to model
Economics 241B
Endogeneity Bias - The Example of Working
The classic illustration of the biases created by endogeneity dates to Working in
1927. Consider the simple model of demand and supply (for coee, say)
Qdt = 0 +
Qst = 0 +
Qdt = Qst :
1 Pt
+ Ut
1 Pt
Heteroskedasticity-Consistent Standard Errors
Hayashi 2.5
Goal:
accurate standard error estimates with heteroskedasticity
Scalar model
V ar ^ =
(
P
xt u t ) 2
P 2 2
xt
martingale dierence sequence assumption implies
P 2 2
xt u t
^
V ar = P
2
x2t
(no seria
Conditional Expectation Functions
Econometrics II
Douglas G. Steigerwald
UC Santa Barbara
D. Steigerwald (UCSB)
Expectation Functions
1 / 23
Overview
Reference: B. Hansen Econometrics Chapters 1 and 2.0 - 2.8
most commonly applied econometrics tool
I
leas
Population Regression Models
Econometrics II
Douglas G. Steigerwald
UC Santa Barbara
Winter 2011
D. Steigerwald (UCSB)
Population Models
Winter 2011
1 / 12
Overview
Reference: F. Hayashi Econometrics Chapter 1.1
(yt , xt1 , . . . , xtK ) := (yt , xt0 )
da
Economics 241B 2014 Final
Question 1
An ongoing debate concerns the economic returns to schooling. The goal is to ensure that
the observed correlation between schooling and wage rates is not due to correlation between
schooling and a workers ability or ot
Economics 241B 2014 Final
Question 1
An ongoing debate concerns the economic returns to schooling. The goal is to ensure that the
observed correlation between schooling and wage rates is not due to correlation between schooling
and a workers ability or ot
Homework 2 Solutions
Ryan Abman, Chris Severen, Dan Argyle
January 30, 2014
1
Distribution of estimator
A) Rejecting the null hypothesis tells us (with a fairly high degree of certainty) that the population value
is not the null hypothesized value. If we
ECON 241B
Additional Problems
Question 1 Goes with HW6 (from Dougs old HW)
Consider the model:
Yt = 01 Xt1 + 02 Xt2 + Ut
t = 1, . . . , n
(1)
where Ut i.i.d.N (0, 2 ) while Xt1 and Xt2 are scalars such that E[Xt1 Xt2 ] = 0 but
where E[Xt1 Ut ] = E[Xt2 Ut
Economics 241B
Exercise 6
1.
Asymptotic Behavior of Hypothesis Tests
Your eldwork requires that you test H0 :
a.
You know that under H0 ,
p
n Bk
k
=
k
vs. H1 :
k
6=
d
k
p
! N (0; Avar (Bk )
k.
(1)
\
Avar (Bk ) ! Avar (Bk ) ;
\
^
where Avar (Bk ) = Sxx1 SS
Economics 241B
Exercise 4
1.
Asymptotic Normality to Consistency
You undertake a research project on carbon sequestration with a colleague from
the geology department. He defers to you as the expert on statistics.
a.
In one discussion, you note that the e
Economics 241B
Problem Set 1
1.
Scaling and R2
You are studying student test scores. To determine the impact of teachers on
test scores, you regress student specic test scores against: parents income and
education, school characteristics and teacher chara
Economics 241B
Review of Limit Theorems for Sequences of Random Variables
Convergence in Distribution
The previous denitions of convergence focus on the outcome sequences of a random variable. Convergence in distribution refers to the probability distribu
Economics 241B
Modes of Convergence
We have concentrated on the (exact) nite-sample distribution for the OLS estimator. The nite-sample theory breaks down if one of the following three
assumptions is violated: 1) the exogeneity of the regressors, 2) the n
241B Lecture
Application: Returns to Scale in Electricity Markets
We work from Nerlove (1963): a classic study of returns to scale in a regulated
industry.
The Electricity Supply Industry
In 1963, the US electricity industry was characterized by:
Privatel
241B Lecture
Generalized Least Squares
One classic assumption (1.4) states that the errors have a spherical conditional
covariance matrix,
E (U U 0 jX ) = 2 I:
Today we relax the assumption and allow
E (U U 0 jX ) =
2
(1)
V (X ) ;
where V (X ) is an n n n