Introduction to econometrics
Economics 208D: Fall 2012
Duncan Thomas
Problem Set 2
Due: Beginning of class, September 25, 2012
The data for this problem set are available on the class webpage. Click on the link to the data file
and save the data on your l
ECON 208D Introduction to Econometrics
Solutions to Problem Set 6
Question 1 (Graded by C.Y. Lee)
(a) The regression results are presented in the following table. The estimate of coecient on
lnPCE is 1 and the p-value (0.001) is less than 5%. Thus housing
ECON 208D Introduction to Econometrics
Solutions to Problem Set 3
Question 1 (Graded by James Thomas)
Part a
One way to solve this problem is to rst calculate the expected prot from one randomly
drawn worker and then multiply this by 10 to obtain the expe
Introduction to econometrics
Economics 208D: Fall 2012
Duncan Thomas
Problem Set 6
Due: Beginning of class, November 27, 2012
Question 1
The data for this problem set, ps6q1.dta, are available on the class webpage. Click on the link to
the data file and s
ECON 208D Introduction to Econometrics
Solutions to Problem Set 2
Question 1 (Graded by Veronica Montalva)
(a) Using STATA we calculate the correlation coecient: 0.561.
(b) Using STATA we run the regression of T on D and obtain:
Table 1: Regression of T o
Introduction to econometrics
Economics 208D: Fall 2012
Duncan Thomas
Problem Set 3
Due: Beginning of class, October 9, 2012
The data for this problem set are available on the class webpage. Click on the link to the data file
and save the data on your loca
ECON 208D Introduction to Econometrics
Solutions to Problem Set 5
Question 1 (a-d graded by James Thomas, e-h
graded by CY Lee)
Part a
Item 1
wage
lnwage
Mean
47,852
10.09
Median
28,923
10.27
Variance 3,440,000,000 1.7956
Skewness
2.443
-.8232
Kurtosis
9.
ECON 139
Duke University
Spring 2008
Midterm 1
Instructions (PLEASE READ CAREFULLY before starting):
This test has a total of 3 problems and 72 points.
You have 75 minutes to nish it.
Remember to Show Your Work and please be sure that your handwriting is
ECON 208D Introduction to Econometrics
Solutions to Problem Set 4
Question 1 (Graded by Veronica Montalva)
(a) See the picture below.
Figure 1: joint pdf: f (x, y )
(b) Keep in mind that we need to take into account that x y :
1/2
1/2
y
Pr[0 x 1/2, 0 y 1/
ECON 139
Duke University
Spring 2007
Midterm 1
Instructions (PLEASE READ CAREFULLY before starting):
This test has a total of 4 problems and 70 points.
You have 75 minutes to nish it.
Remember to Show Your Work and please be sure that your handwriting is
Econ 208D
INTRODUCTION TO
ECONOMETRICS
Duncan Thomas
Duncan Thomas: Introduction to Econometrics - Lecture 01
-1-
INTRODUCTION TO
ECONOMETRICS
Measurement of economic phenomena
Duncan Thomas: Introduction to Econometrics - Lecture 01
-2-
ECONOMETRICKS
THE
Duke University, practice problems for Introduction to Econometrics
February 16, 2011
1
Properties of Expectations, Variances and Covariances, Distributions
1. (5 points) You know that income per head in Italy (denoted by y ), expressed in Euros, is norma
An Example
Duncan Thomas: Introduction to Econometrics - Lecture 18
-1-
Testing joint hypotheses: Example
wif 0 1nPCE i 2nPCE i 2 + 3nN i + u i
H 0 : 1 0 and 2 =0
H1 : 1 0 and/or 2 0
Model under H0 : Restricted model (food shares unrelated to lnPCE)
wif 0
An Example
Duncan Thomas: Introduction to Econometrics - Lecture 17
-1-
Linear Regression Model: An Example
Food Engel Curve
n X if 0 1nX i u i
n X if =log(food expenditure)
nX i = log(total expenditure)
Elasticity of demand = 1
Engels Law
Share of budg
Variance of
Duncan Thomas: Introduction to Econometrics - Lecture 16
-1-
Variance of
var(1 ) E [1 E 1 ]2
E [1 1 ]2
i x i u i
1 1
i x i2
1 1
i x i u i
i x i2
Why?
Why?
[7]
2
) E [ ]2 E i x i u i = ?
var(1
1
1
2
i xi
Let w i x i / x i2
E[ i w i u
When do you really need dummy variables?
Earnings and occupation
Say you are interested in relationship between occupation and earnings.
Consider three occupation groups:
1. unskilled workers
2. skilled workers
3. management
600
0
200
wage
400
Let occup =
Estimation of 0
In the model
Yit 0 1 X it i uit
[3']
0 and i are not separately identified. Need a convention.
Estimate model with dummy variable for every twin pair:
Yit 1D1t 2 D2t 3 D3t X it uit
Drop intercept. Why?
Duncan Thomas - Introduction to Eco
What properties do we want a hypothesis test to have?
Low ?
pr(reject H0 when correct)
No, pick a priori
i.e. by design only hope to hit the nail on the head (1-)% of time
(1-) chosen
What about ?
1-
pr(fail reject H0 when correct)
pr(reject H1 when it is
Example: Construction of a test
Is this class worth it?
wP w
[1]
w P = average starting wage of students who pass class
w = average starting wage of students who dont pass
The class is worth the effort w P w = = $20K
w P w = $0
The class is worthless
1.
Why is Maximum Likelihood Estimation (MLE) useful?
General principle that can be applied to any random variable
Under many (plausible) conditions
any MLE is best estimator given assumptions about underlying pdf
In large samples,
MLE is the MVUE (minimum v
3. Ratio of standard normal and
2
is distributed as a t statistic
r
Standard normal
Z N(0,1)
Chi squared
U X2
r
-<Z<
0<U<
If Z and U are independent
T
Z
tr
Ur
-<T<
T is distributed as a t statistic with r degrees of freedom
(dof=# squared standard normal
Sampling distributions
If we draw a sample from a population and for each draw, every individual
in the population has an equal chance of being selected, we have a simple
random sample.
Note the probability distribution associated with each observation is
Expectations of continuous random variable
E[X] x f (x) dx
Rx
Discrete case
E[X]= x x f (x)
Mean is average of all values weighted by probability that value is observed
var[X] 2 (x ) 2 f (x)dx
Rx
var[X] =
(x ) 2 f (x)
x
Variance is sum of squared deviatio
Multicollinearity
Called multicollinearity (because can involve many covariates)
Often "perfect multicollinearity" arises because included covariate
that should not have been included
(e.g. dummy variable trap)
STATA drops one of the covariates and procee
Estimation in the multivariate model
Yi = 0 1X1i + u i
Bivariate model
Multivariate model
Yi = 0 1X1i + 2 X 2i + u i
i xi yi
1 xi2
i
[1]
i r i yi
1
i r i 2
[2]
1
1
where X1i = 0 1X 2i + r1i
r1i =X1i 0 1X 2i
so
Derivation of [2]
Rewrite
Yi = 0 1X1i + 2
Heteroskedasticity: Summary
Arises if there is failure of A3:
E[ui]2 = 2
"i
Coefficient estimates unbiased
E 1 1
Variance
Inference with
2
messed up
i x
Test: (Breusch-Pagan or White) u i2 = f(Xi)+ i
H0: Xs are not significant predictors of u i2
2
i
If re
Introduction to econometrics
Economics 208D: Fall 2012
Duncan Thomas
Problem Set 5
Due: Beginning of class, November 6, 2012
Question 1
The data for this problem set, ps5q1.dta, are available on the class webpage. Click on the link to
the data file and sa
Introduction to econometrics
Economics 208D: Fall 2012
Duncan Thomas
Problem Set 4
Due: Beginning of class, October 23, 2012
Question 1
Let X and Y be continuous random variables with joint pdf
f(x y) = 2
0<x<y<1
(a) Draw the pdf of f(x y)
(b) What is Pr[
Introductory Econometrics
Handout 6
Duncan Thomas
LINEAR REGRESSION: ESTIMATION AND TESTING
1. SIMPLE REGRESSION MODEL
We wish to fit:
Yi = 0 + 1Xi + ui , i = 1, ., n
for n observations.
Xi is the independent variable; it is a fixed regressor or covariate
Introductory Econometrics
Handout 5
Duncan Thomas
ESTIMATION AND INFERENCE
1. POINT ESTIMATION
Until now, we have assumed the distribution generating a batch of data is known and we asked
what is the probability of observing a particular outcome. What hap