1. Ch. 13. Pooled Cross Sections Across Time: Simple Panel Data.
Difference-in-Difference for treatment effects
How DiD can eliminate bias in cross-sectional OLS.
Potential sources of bias after DiD
Panel Data
First Difference for two period panel data.
Fixed effects for multi-period panel data.
How first differencing or fixed effects can eliminate bias in OLS
Potential issues with FD and FE models

2. Pooling Cross Sections across Time: Simple Panel Data Methods
Policy analysis with pooled cross sections
Two or more independently sampled cross sections can be used to evaluate the impact of a certain event or policy change
Effect of new garbage incinerator’s location on housing prices
Examine the effect of the location of a house on its price before and after the garbage incinerator was built:
After incinerator was built
Before incinerator was built

3. Pooling Cross Sections across Time: Simple Panel Data Methods
Garbage incinerator and housing prices
Note: near incinerator had negative effect on housing prices before incinerator was built? Why?
Would be inappropriate to interpret negative effect of incinerator after it‘s built as a causal effect. Some of effect is due to fact that incinerator was built near lower price homes.
More appropriate to look at difference-in-difference (DiD) after incinerator was built: p near – p far = , before incinerator was built: = p near – p far = -18, difference in differences (DiD) = ,863.9

4. Pooling Cross Sections across Time: Simple Panel Data Methods
Difference-in-differences in a regression framework
Show how 𝛿 1 is the DiD estimator derived above
DiD regression allows for standard errors and t-stat of DiD effect.
If houses sold before and after the incinerator was built were systematically different, further explanatory variables should be included
Adding housing characteristics will also reduce the error variance and thus standard errors
Before/After comparisons in “natural experiments”
DiD can be used to evaluate policy changes or other exogenous events
Difference in change of house price for those near vs not near incinerator (diff-in-diff)

5. Pooling Cross Sections across Time: Simple Panel Data Methods
Policy evaluation using difference-in-differences
Compare outcomes of the two groups before and after the policy change
Suppose that something happens in the treated group causing its growth to differ by 𝜃 relative to the control group. DiD estimator will then include true effect of treatment ( 𝛿 1 ) and the effect of the other factors causing growth to differ by 𝜃 in the treated group.
Examples Minimum wage increase is the treatment. How is DiD estimate of employment effect biased if the state that passes the minimum wage has unusually high economic growth? Unusually low economic growth?
Might use placebo test to be sure that DiD estimator isn‘t picking up effect of some other factor.
Minimum wage hike shouldn‘t affect employment growth of college graduates.

6. Pooling Cross Sections across Time: Simple Panel Data Methods
Two-period panel data (Fixed Effect) analysis
Example: Effect of unemployment on city crime rate
Assume that no other explanatory variables are available. Will it be possible to estimate the causal effect of unemployment on crime?
Yes, if cities are observed for at least two periods and other factors affecting crime stay approximately constant over those periods:
Time dummy for the second period
Unobserved city specific time-invariant factors (= fixed effect)
Examples of time-constant variables that might affect city crime?
Other unobserved factors (= idiosyncratic error)

7. Pooling Cross Sections across Time: Simple Panel Data Methods
Effect of unemployment on city crime rate
Estimate differenced equation by OLS:
Fixed effect drops out
+ 1 percentage point unemploy-ment rate leads to 2.22 more crimes per 1,000 people
Secular increase in crime across all cities.

8. Pooling Cross Sections across Time: Simple Panel Data Methods
Discussion of first-differenced panel estimator
Further explanatory variables may be included in original equation
There may be arbitrary correlation between the unobserved time-invariant characteristics and the included explanatory variables
For example, suppose cities with less educated workers (virtually a time-invariant characteristic) have higher crime and also higher unemployment – how would this bias OLS estimate of effect of unemployment?
First differences cause effect of any time-invariant variables to be differenced out of the regression. Eliminates bias from exclusion of important time- invariant variables that would emerge in OLS.
First-differenced estimates will be imprecise if explanatory variables vary little over time (no estimate possible if time-invariant)

9. Panel Data Methods with More than 2 Periods.
Fixed effects estimation
Estimate deviations from i-specific means using OLS
Estimates rely on time variation within cross-sectional units
(= within estimator)
xtset & xtreg in Stata.
Fixed effect, potentially correlated with explanatory variables
Form time-averages for each individual
Because (the fixed effect is removed)

10. Advanced Panel Data Methods
Example: Effect of training grants on firm scrap rate (number of defective items per 100 produced)
Time-invariant reasons why one firm is more productive than another are controlled for. The important point is that these may be correlated with the other explanatory variables.
Fixed-effects estimation using the years 1987, 1988, and 1989:
Stars denote deviations from i-specific means
Training grants significantly improve productivity (with a time lag)

11. Advanced Panel Data Methods
Discussion of fixed effects estimator
Strict exogeneity in the original model has to be assumed
The R2 of the demeaned equation is inappropriate measure of R2 – will be within group R2
The effect of time-invariant variables cannot be estimated
The effect of interactions of controls with time-invariant variables can be estimated (e.g. Interact a person‘s sex with year)
If a full set of time dummies are included, the effect of variables whose change over time is constant cannot be estimated (e.g. age)
Degrees of freedom have to be adjusted because the individual specific averages are estimated in addition to other coefficients (resulting degrees of freedom = NT-N-k)

12. Advanced Panel Data Methods (Ch.14)
Applying panel data methods to other data structures
Panel data methods can be used in other contexts where constant unobserved effects have to be removed
Example: Wage equations for twins
Unobserved genetic and family characteristics that do not vary across twins
Equation for twin 1 in family i
Equation for twin 2 in family i
Estimate differenced equation by OLS