### Econometrics Course Review

```Old Stuff
New Stuff on Exam 3
Econometrics Course Review
Jeff Borowitz
Georgia State University
Jeff Borowitz
Econometrics Review
1 / 22
Old Stuff
New Stuff on Exam 3
Ordinary Least Squares
Hypothesis Testing
Logistics
We will have no final exam in this class - just a third exam on
Thursday, December 5
The third exam will not be cumulative
Material since (and including) dummy variables
There will be no 7th Problem Set
Jeff Borowitz
Econometrics Review
2 / 22
Old Stuff
New Stuff on Exam 3
Ordinary Least Squares
Hypothesis Testing
Plan for Day
Review of Course Topics
Next class, we will do an applied problem like for the other exams
Jeff Borowitz
Econometrics Review
3 / 22
Old Stuff
New Stuff on Exam 3
Ordinary Least Squares
Hypothesis Testing
Ordinary Least Squares (Single/Multi-Variate)
Gauss Markov Assumptions
1
2
3
4
5
6
Linear in parameters
Data is a random sample
No multicollinearity
Zero conditional mean of variance
Homoskedasticity
Normality
Formulas to calculate βˆ in univariate case
Ceteris paribus coefficient interpretation
Jeff Borowitz
Econometrics Review
4 / 22
Old Stuff
New Stuff on Exam 3
Ordinary Least Squares
Hypothesis Testing
Analysis of Variance
SSE , SST , SSR
R2
Fraction of explained variation
¯2
Larger SST means parameters are estimated with less noise
Jeff Borowitz
Econometrics Review
5 / 22
Old Stuff
New Stuff on Exam 3
Ordinary Least Squares
Hypothesis Testing
Where does OLS coefficient variance come from?
Increase in error variance
Decrease in sample size (SST )
Amount of independent variation (1-Rj2 )
Omitted variable bias
What effects on parameters of interest?
Jeff Borowitz
Econometrics Review
6 / 22
Old Stuff
New Stuff on Exam 3
Ordinary Least Squares
Hypothesis Testing
Hypothesis Testing: Steps
Test whether βˆ = 3 against the alternative that βˆ = 3 at the two-sided
α = .05 confidence level
1 State null/alternative hypothesis:
H0 :βˆ = 3
H1 :βˆ = 3
2
Form t-statistic:
t=
3
4
βˆ − 3
ˆ
se(β)
Look up tcrit from a table based on degrees of freedom, α, and
number of sides
Reject if magnitude of t is bigger than tcrit (if two-sided)
Jeff Borowitz
Econometrics Review
7 / 22
Old Stuff
New Stuff on Exam 3
Ordinary Least Squares
Hypothesis Testing
Confidence Intervals
Since βˆ is unbiased, the true β is normally distributed around βˆ
To calculate the α confidence interval
1
2
3
Find c as the critical t value in a two sided test at the α level
ˆ
Upper bound is βˆ + cse(β)
ˆ
ˆ
Lower bound is β − cse(β)
There is a 1 − α chance that the true β is in this range
If a value a lies in the confidence interval, you would not reject the
test of βˆ = a
Jeff Borowitz
Econometrics Review
8 / 22
Old Stuff
New Stuff on Exam 3
Ordinary Least Squares
Hypothesis Testing
Units
Analyzed how our estimation depends on units
Scaling RHS variable by α:
Scales β, se(β) by 1/α
t, p, R 2 stay the same
Other coefficients also unchanged
Scaling LHS variable by α
Scales all β’s, se(β)’s by α
t, p, R 2 stay the same
Jeff Borowitz
Econometrics Review
9 / 22
Old Stuff
New Stuff on Exam 3
Ordinary Least Squares
Hypothesis Testing
F-Testing
Compare nested models
y =β0 + β1 x1 + β2 x2 + u
vs.
y =β0 + β1 x1 + β2 x2 + β3 x3 + β4 x4 + u
Test hypotheses like: β3 = β4 = 0
Calculate the special F statistic for the test of all non-intercept
coefficients in the model
Jeff Borowitz
Econometrics Review
10 / 22
Old Stuff
New Stuff on Exam 3
Ordinary Least Squares
Hypothesis Testing
Interpreting Models with Different Functional Forms
Logged left and right hand side variables, and their interpretations
Compute marginal effects
Determine and interpret turn around point
Interaction terms
Compute Marginal Effects
Interpret coefficients
Jeff Borowitz
Econometrics Review
11 / 22
Old Stuff
New Stuff on Exam 3
Heteroskedasticity
Miscellaneous Issues
Time Series/Panel/Instruments
New Stuff
The stuff from this point on could be on the test
Jeff Borowitz
Econometrics Review
12 / 22
Old Stuff
New Stuff on Exam 3
Heteroskedasticity
Miscellaneous Issues
Time Series/Panel/Instruments
Dummy Variables
This will include some slightly more complicated dummy variable stuff
Dummy/continuous interactions
Where is there equivalence between the groups?
Dummy-dummy interactions: e.g. the interpretation of gender/race
and interacted coefficients
Be able to determine the difference in outcomes between black males
and white females, or any other combination:
y = β0 + δ0 female + δ1 black + δ2 female ∗ black+ u
Jeff Borowitz
Econometrics Review
13 / 22
Old Stuff
New Stuff on Exam 3
Heteroskedasticity
Miscellaneous Issues
Time Series/Panel/Instruments
Interaction Terms
We didn’t really interpret interaction terms very heavily on the last
test
Returns to education vary with experience
Or returns to experience vary with education. . .
log(wage) =β0 + β1 educ + β2 exper + β3 educ · exper + u
Returns to education vary with a dummy variable:
log(wage) =β0 + β1 educ + β2 female + β3 educ · female + u
Jeff Borowitz
Econometrics Review
14 / 22
Old Stuff
New Stuff on Exam 3
Heteroskedasticity
Miscellaneous Issues
Time Series/Panel/Instruments
Linear Probability Models
Just a regression where LHS variable is binary
Would be a big problem if you have lots of units predicted to have
probabilities not from 0 to 1
yˆ has the interpretation as a probability
Changes in x have the interpretation of increasing the probability that
y = 1.
Jeff Borowitz
Econometrics Review
15 / 22
Old Stuff
New Stuff on Exam 3
Heteroskedasticity
Miscellaneous Issues
Time Series/Panel/Instruments
Heteroskedasticity
What is it?
Different points have different error variances
How to tell if it is a problem in your data?
Breusch-Pagan test: regress uˆ2 on X ’s
White test: Regress uˆ2 on yˆ and higher powers
What are the problems?
Standard errors are invalid
Can improve on power using WLS/GLS/FGLS
Walk through applying the test or FGLS
Find the modified WLS model as in problem set 5
Show some R results where models necessary to calculate BP/White
tests are shown, and you decide whether to reject and why.
Jeff Borowitz
Econometrics Review
16 / 22
Old Stuff
New Stuff on Exam 3
Heteroskedasticity
Miscellaneous Issues
Time Series/Panel/Instruments
Weighted Least Squares
Weighted Least Squares
We can transform a heteroskedastic model so that it is homoskedastic
Like on the last problem set. . .
Feasible Generalized Least Squares
If you don’t know the form of heteroskedasticity, you estimate it
uˆ2 = α0 + α1 x1 + . . . + ε
ˆ2
Then divide each observation by: uˆ
You should be able to do a WLS transformation, and describe how
you would do FGLS specifically
Jeff Borowitz
Econometrics Review
17 / 22
Old Stuff
New Stuff on Exam 3
Heteroskedasticity
Miscellaneous Issues
Time Series/Panel/Instruments
Miscellaneous Stuff
Proxy variables
What makes a good proxy (correlated with important part of
unobservable)?
Ramsey RESET test
Use an F-test for whether yˆ and higher powers help explain y in initial
regression
Measurement error
Measurement error on LHS is no big deal, just increases error variance
Measurement error on RHS biases coefficients towards 0
Missing data
If data is missing at random, or in a way that just depends on x’s
that’s OK
If data is missing in a way that depends on y , that’s bad
Jeff Borowitz
Econometrics Review
18 / 22
Old Stuff
New Stuff on Exam 3
Heteroskedasticity
Miscellaneous Issues
Time Series/Panel/Instruments
Time Series
What are the new Gauss-Markov assumptions
Strict exogeneity: xt is uncorrelated with future and past ut
This rules out e.g. police levels responding to murder rates
No serial correlation: if murder is high for some reason today, that
can’t make it more likely to be high tomorrow too
Finite distributed lag models
What does the lag mean?
Impact propensity: contemporaneous coefficient
Long run propensity: sum of coefficients
Jeff Borowitz
Econometrics Review
19 / 22
Old Stuff
New Stuff on Exam 3
Heteroskedasticity
Miscellaneous Issues
Time Series/Panel/Instruments
Time Series: Seasonality and Trends
If variables are trending together this will bias coefficients in a time
series regression.
You can detrend a variable by regressing it on t and taking residuals
You can get the same regression results by first detrending the variables
or by controlling for t in a regression
You can seasonally adjust a variable by regressing it on a set of
dummy variables for each season and taking residuals.
Jeff Borowitz
Econometrics Review
20 / 22
Old Stuff
New Stuff on Exam 3
Heteroskedasticity
Miscellaneous Issues
Time Series/Panel/Instruments
Difference in differences
Explain how to use treatment group and post event dummies to get
difference in differences estimate
y =β0 + β1 post + β2 T + β3 post · T + u
Interpret the interaction coefficient as the effect of treatment
Fixed effects
Use variation only within an individual: how does joining a union affect
wages compared to an individual’s history
Instrumental Variables
A good instrument affects x but not y (proximity to college affects
college attendance but not wages)
Example question: would IQ make a good instrument for education?
Why or why not?
Jeff Borowitz
Econometrics Review
21 / 22
Old Stuff
New Stuff on Exam 3
Heteroskedasticity
Miscellaneous Issues
Time Series/Panel/Instruments
Perspective
Econometrics are only going to get more important in economics and
in the world, as more stuff get measured
OLS is a useful framework to understand a range of econometric
issues