1. Panel Data Models
Sintayehu H.

2. What are panel data?
• Panel (or longitudinal) data combine time-series and cross-
sectional data in a very specific way.
• Panel data include observations on the same variables from
the same cross-sectional sample from two or more different
time periods.
– For example, if you surveyed 200 students when they graduated
from your school and then administered the same questionnaire to
the same individuals five years later, you would have created a
panel data set.
• Not every data set that combines time-series and cross-
sectional data meets this definition. In particular, if different
variables are observed in the different time periods or if the
data are drawn from different samples in the different time
periods, then the data are not considered to be panel data

3. Why use panel data?
• As mentioned earlier, panel data certainly will increase
sample sizes, but a second advantage of panel data is to
provide insight into analytical questions that can’t be
answered by using time-series or cross-sectional data
alone.
• For example, panel data can help policymakers design
programs aimed at reducing unemployment by allowing
researchers to determine whether the same people are
unemployed year after year or whether different
individuals are unemployed in different years.
• A final advantage of using panel data is that it often allows
researchers to avoid omitted variable problems that
otherwise would cause bias in cross-sectional studies. We’ll
come back to this topic soon.

4. Type of variables we use?
• There are four different kinds of variables that we
encounter when we use panel data.
• First, we have variables that can differ between individuals
but don’t change over time, such as gender, ethnicity, and
race.
• Second, we have variables that change over time but are
the same for all individuals in a given time period, such as
the retail price index and the national unemployment rate.
• Third, we have variables that vary both over time and
between individuals, such as income and marital status.
• Fourth, we have trend variables that vary in predictable
ways such as an individual’s age.

5. Data formats and use
• To estimate an equation using panel data, it’s crucial
that the data be in the right format because regression
packages like Stata and Eviews need to identify which
observations belong to which time periods and which
cross-sectional entities.
• Unfortunately different software programs have
different format requirements for panel data.
• Stata, for example, requires that a panel data set
include a date counter and an id number counter, but it
doesn’t require that the data be in any particular order.

6. Data formats and use…
• The use of panel data requires a slight expansion of our notation.
• In the past we’ve used the subscript i to indicate the observation
number in a cross-sectional data set, so Yi indicated Y for the ith
cross-sectional observation.
• Similarly, we’ve used the subscript t to indicate the observation
number in a time-series data set, so Yt indicated Y for the tth time-
series observation.
• In a panel data set, however, variables will have both a cross-
sectional and a time-series component, so we’ll use both subscripts.
• As a result, Yit indicates Y for the ith cross-sectional and tth time-
series observation.
• This notation expansion also applies to independent variables and
error terms.

7. What’s the best way to estimate panel
data equations?
• The two main approaches are the fixed effects
model discussed in this section and the
random effects model featured in the next
section.

8. The Fixed Effects Model
• The fixed effects model estimates panel data
equations by including enough dummy
variables to allow each cross-sectional entity
(like a state or country) and each time period
to have a different intercept:

9. The fixed effects model…
• As you’d expect with a panel data set, Y, X, and e have two
subscripts.
• Although there is only one X in Equation 16.4, the model can be
generalized to any number of independent variables.
• Why do we need something as complicated as Equation 16.4?
• To answer, let’s begin by taking a look at the problems that would
arise if we estimated our model without accounting for the fact that
our observations are from a panel data set.
• Our equation would look like this:

10. The fixed effects model…
• To understand V, remember that because we’re dealing
with panel data, we have observations from several,
maybe many, entities and from several, maybe many,
time periods.
• Just about everyone would agree that no two states
are exactly alike. They have different cultures, histories,
and institutions.
• It’s easy to imagine that those differences might lead to
different outcomes in all sorts of things we might want
to explain.
• Our Yit could be income, health, or crime, for instance.

11. The fixed effects model…
• It’s also easy to see that things like a state’s history and
culture are pretty constant from year to year.
• They might be hard to measure, but we know that they
don’t change, and we know that they make each state
different from all the others.
• It is very likely that these unchanging and unmeasured
differences are correlated with X, but Equation 16.5
doesn’t include them, so they are omitted variables.
• And that’s a problem, right?

12. The fixed effects model…
• In previous lectures we learned that omitting a
relevant variable from a model forces much of its
influence into the error term.
• And that partly explains the problem with the error
term V in Equation 16.5.
• But there’s more. Remember that we’re dealing with
panel data. Not only have we combined several cross
sections, but we’ve also combined some time series!
• That means we have even more potential omitted
variables. Why is that?

13. The fixed effects model…
• Well, it’s entirely possible that during each time period,
certain things affect all the entities, but that those common
influences change from period to period.
• Suppose you’re investigating annual traffic fatalities in
states over a period of many years. If the federal
government raises or lowers the maximum highway speed
limit, it affects traffic fatalities in all states.
• Similarly, changing social norms affect traffic fatalities over
time. Attitudes about seat belts, for instance, could play a
big role. People didn’t always buckle up without thinking!
• If you doubt this, ask your grandparents how many of them
used seatbelts back when they were kids.

14. The fixed effects model…
• With the omitted entity characteristics and the omitted
time characteristics, the error term in Equation 16.5
can be broken down into three components:
• where eit is a classical error term, ai refers to the entity
characteristics omitted from the equation, and zt refers
to the time characteristics omitted from the equation.
• If ai and zt are correlated with Xit, we’re going to have a
problem because we will have violated Classical
Assumption III.
• Our estimate of β1 will be biased.

15. The fixed effects model…
• As we learned in class, the solution in theory is simple.
Just include the omitted variables in the model, and
the omitted variable bias will disappear.
• But the omitted variables often are unobservable. And
even if we could see them, we might not be able to
measure them.
• For instance, if the entities are states, the unobserved
characteristics could be such things as culture or
history.
• How in the world would we ever discover what they
are, much less measure them?

16. The fixed effects model…
• As it happens, we already have something in our econometric
toolbox that can solve the problem—dummy variables!
• By including dummy variables for every entity (EFi) but one, we can
control for those unobservable but unchanging entity effects. We
call them entity fixed effects.
• And by including dummy variables for every time period (TFt) but
one, we can control for time fixed effects.
• These entity and time fixed effects will no longer be omitted
variables because they will be represented by the dummy variables.
• Including the dummies transforms V into e and transforms
Equation 16.5 into the basic fixed effects model, Equation 16.4:

17. The fixed effects model…
• The major advantage of the fixed effects model is that it
avoids bias due to omitted variables that don’t change over
time (like geography) or that change over time equally for
all entities (like the federal speed limit).
• What we’re in essence doing is allowing each entity’s
intercept and each time period’s intercept to vary around
the omitted condition baseline (when all the fixed effect
dummies equal zero).
• And the beauty of it is that we don’t even have to know
exactly what things go into the entity and time fixed effects.
• The dummy variables include them all!

18. The fixed effects model…
• The fixed effects model has some drawbacks, however.
• Degrees of freedom for fixed effects models tend to be low
because we lose one degree of freedom for every dummy
variable (the EFs and the TFs) in the equation.
• For example, if the panel contains 50 states and two years,
we lose 50 degrees of freedom by using 49 state dummies
and one year dummy.
• Another potential pitfall is that no substantive explanatory
variables that vary across entities, but do not vary over
time within each entity, can be used because they would
create perfect multicollinearity.

19. The fixed effects model…
• Luckily, these drawbacks are minor when
compared to the advantages of the fixed
effects model, so it is advisable to benefit
from using the fixed effects model whenever
estimating panel data models.

20. An Example of Fixed Effects Estimation
• Let’s take a look at a simple application of the fixed effects model.
• Suppose that you’re interested in the relationship between the death
penalty and the murder rate, and you collect data on the murder rate in
the 50 states.
• If you were to estimate a cross-sectional model (Table 16.1) of the annual
murder rate as a function of, say, the number of convicted murderers who
were executed in the previous three years, you’d end up with:

24. Example…
• In a cross-sectional model for 1990, the murder rate
appears to increase with the number of executions, quite
probably because of omitted variable bias or because of
simultaneity.
• This result implies that the more executions there are, the
more murders there are!
• Such a result is completely counter to our expectations.
• To make things worse, it’s not a fluke (coincidence).
• If we collect data from another year, 1993, and estimate a
single-time-period regression on the 1993 data set, we also
get a positive slope.

25. Example…
• However, if we combine the two cross-sectional data sets
to create the panel data set in Table 16.1, we can estimate
a fixed effects model, using the fixed effects model of
Equation 16.4,
• adjusted to account for 50 states (with Alabama as the
omitted condition) and two time periods (with 1990 as the
omitted condition):

26. Example…
• As can be seen in Equation 16.8 and Figure 16.3, a fixed
effects model estimated on panel data from 1990 and
1993 results in a significant negative estimated slope
for the relationship between the murder rate and the
number of executions.
• This example illustrates how the omitted variable bias
arising from unobserved heterogeneity can be
mitigated with panel data and the fixed effects model.
• When the dataset is expanded to include another year,
you’re in essence looking at each state and comparing
the state to itself over time.

28. Example…
• Note that we included TF93, a year fixed effect variable,
in Equation 16.8.
• A year fixed effect captures any impact that altered the
level of executions across the country for a given year.
• For example, if the Supreme Court declared a
moratorium (suspension) on a type of execution in that
year, we would see a decline in executions across states
that used that type of execution during the year for
reasons unrelated to the relation between murders
and executions for each state.

29. Example…
• You might have noticed the big increase in R2 between
Equations 16.7 and 16.8 (0.24 and 0.96).
• The increase comes from the addition of all the dummy
variables for state and time fixed effects.
• So why don’t the coefficients of the state dummies appear
in Equation 16.8?
• Unless the entity fixed effects are the main focus of the
research, the coefficients usually are omitted from the
results to save space.
• Some large panel data sets have hundreds or even
thousands of entity fixed effects!

30. Fixed effects…
• In our example, we used only two time
periods, but the fixed effects model can be
extended to many more time periods.
• Fixed effects estimation is a standard
statistical routine in most econometric
software packages, making it particularly
accessible for researchers.

31. The Random Effects Model
• An alternative to the fixed effects model is called the
random effects model.
• While the fixed effects model is based on the
assumption that each cross-sectional unit has its own
intercept, the random effects model is based on the
assumption that the intercept for each cross-sectional
unit is drawn from a distribution that is centered
around a mean intercept.
• Thus each intercept is a random draw from an
“intercept distribution” and therefore is independent
of the error term for any particular observation.

32. Random Effects…
• The random effects model has several clear advantages over the
fixed effects model.
• In particular, a random effects model will have quite a few more
degrees of freedom than a fixed effects model, because rather than
estimating an intercept for virtually every cross-sectional unit, all
we need to do is to estimate the parameters that describe the
distribution of the intercepts.
• Another nice property is that you can estimate coefficients for
explanatory variables that are constant over time (like race or
gender).
• However, the random effects estimator has a major disadvantage in
that it requires us to assume that the unobserved impact of the
omitted variables is uncorrelated with the independent variables,
the Xs, if we’re going to avoid omitted variable bias.

33. Choosing Between Fixed and Random
Effects
• How do researchers decide whether to use the
fixed effects model or the random effects model?
• One key is the nature of the relationship between
ai and the Xs.
• If they’re likely to be correlated, then it makes
sense to use the fixed effects model, as that
sweeps away the ai and the potential omitted
variable bias.

34. Choosing between fixed vs random..
• Many researchers use the Hausman test, which is well
beyond the scope of this text, to see whether there is
correlation between ai and X.
• Essentially, this procedure tests to see whether the
regression coefficients under the fixed effects and random
effects models are statistically different from each other.
• If they are different, then the fixed effects model is preferred
even though it uses up many more degrees of freedom.
• If the coefficients aren’t different, then researchers either
use the random effects model (in order to conserve degrees
of freedom) or provide estimates of both the fixed effects
and random effects models.

35. Thank you!