10
Regression Chapter with Panel Data (Part 1)
Trafc Deaths and Alcohol Taxes
There are approximately 40,000 highway trafc fatalities each
year in US.
Approximately one-third of fatal crashes involve a driver who
was drinking.
One study estimates that as many as 25% of drivers on the road
between 1AM and 3AM have been drinking.
A driver who is legally drunk is at least 13 times as likely to
cause a fatal crash as a driver who has not been drinking.
We study how effective government policies designed to
discourage drunk driving actually are in reducing trafc
deaths.
Panel Data
The state trafc fatality data we will study are panel data.
Data are for n = 48 states, where each state is observed in T = 7
time periods (1982, ..., 1988).
Panel data consist of observation on the same n entities at two
or more time periods T.
If the data set contains observations on the variables X and Y,
then the data are denoted
(Xit , Yit ) , i = 1, ..., n and t = 1, ..., T,
where the rst subscript, i, referers to the entity being observed,
and the second subscript, t, refers to the date at which it is
observed.
Trafc Deaths and Alcohol Taxes
The OLS regression lines obtained by regressing the fatality
rate on the beer tax are:
\
FatalityRate = 2.01 + 0.15BeerTax (1982 data).
(0.15) (0.13)
\
FatalityRate = 1.86 + 0.44BeerTax (1988 data).
(0.11) (0.13)
Should we conclude that an increase in the tax on beer leads
more trafc deaths?
Consider the omitted variable bias. Some variables, such as the
cultural acceptance of drinking and driving might be hard to
measure.
Trafc Deaths and Alcohol Taxes
Panel Data with Two Time Periods: "Before and After"
Comparisons
Let Zi be a variable that determines the fatality rate in the ith
state, but does not change over time.
FatalityRateit = 0 + 1 BeerTaxit + 2 Zi + uit ,
where uit is the error term and i = 1, ..., n and t = 1, ...T.
The inuence of Zi can be eliminated:
FatalityRatei1982 = 0 + 1 BeerTaxi1982 + 2 Zi + ui1982
FatalityRatei1988 = 0 + 1 BeerTaxi1988 + 2 Zi + ui1988
Subtracting the rst from second eliminates the effect of Zi :
FatalityRatei1988
= 1 (BeerTaxi1988
FatalityRatei1982
BeerTaxi1982 ) + ui1988
ui1982 .
Panel Data with Two Time Periods: "Before and After"
Comparisons
Figure presents 10.2 a scatterplot of the change in the fatality
rate between 1982 and 1988 against the change in the beer tax.
The estimated OLS regression line is
\
FatalityRatei1988 FatalityRatei1982
=
0.072
1.04 (BeerTaxi1988
BeerTaxi1982 ) .
(0.065) (0.36)
An increase in the beer tax by $1 per case reduces the trafc
fatality rate by 1.04 deaths per 10,000 people.
Consider that the avarage fatality rate is 2 in the data, this effect
is very large.
Panel Data with Two Time Periods: "Before and After"
Comparisons
Fixed Effects Regression
Unlike the "before and after" comparisons, xed effects
regression can be used with more than two time observations
for each entity.
Yit = 0 + 1 Xit + 2 Zi + uit ,
where Zt is an unobserved variable that varies from one state to
the next but does not change over time.
We want to estimate the 1 , the effect on Y on X controlling the
unobserved Zi .
Because Zi varies across states but is constant over time, can be
interpreted as having n intercepts, one for each state.
Fixed Effects Regression
Let i = 0 + 2 Zi ,
Yit = 1 Xit + i + uit ,
is the xed effects regression model, in which 1 , ..., n are
treated as unknown intercepts to be estimated, one for each
state.
1 , ..., n are known as entity xed effects.
Equivalently, the xed regression model can be written using
binary variables:
Yit = 0 + 1 Xit + 2 D2i + 3 D3i +
+ n Dni + uit ,
where D2i = 1 if i = 2 and D2i = 0 otherwise, and so forth.
Fixed Effects Regression
In general, the xed effects regression model is
Yit = 1 Xit +
+ k Xk,it + i + uit ,
where i = 1, ..., n and t = 1, ..., T, where X1,it is the value of the
rst regressor for entity i in time period t, X2,it is the value of
the second regressor, and so forth, and 1 , ..., n are entity
specic intercepts.
Equivalently, the xed regression model can be written using
binary variables:
Yit = 0 + 1 Xit +
+3 D3i +
+ k Xk,it + 2 D2i
+ n Dni + uit ,
where D2i = 1 if i = 2 and D2i = 0 otherwise, and so forth.
Fixed Effects Regression
The OLS estimate of the xed effects regression line relating the
beer tax to the fatality rate is
\
FatalityRate =
0.66BeerTax + StateFixedEffects
(0.20)
where the estimated state xed intercepts are not listed to save
space.