Introduction to Panel Data Analysis

Introduction to Panel Data Analysis
Youngki Shin
Department of Economics
Email: yshin29@uwo.ca
Statistics and Data Series at Western
November 21, 2012
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Motivation
More observations mean more information.
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Motivation
More observations mean more information.
More observations with a certain structure mean much more
information: pooled cross sections and panel data
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Motivation
More observations mean more information.
More observations with a certain structure mean much more
information: pooled cross sections and panel data
How can we extract additional information from pooled cross sections
or panel data?
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Example
Effect of an Incinerator on Housing Prices
With cross-sectional data in 1981, we have
̂rprice = 101, 307.5 − 30, 688.27nearinc
(3, 093.0) (5, 827.71)
n = 142
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Example
Effect of an Incinerator on Housing Prices
With cross-sectional data in 1981, we have
̂rprice = 101, 307.5 − 30, 688.27nearinc
n = 142
(3, 093.0) (5, 827.71)
With another cross-sectional data in 1978 when there were no incinerator,
we have
̂rprice = 82, 517.23 − 18, 824.37nearinc
n = 179
(2, 653.79) (4, 744.59)
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Example
Effect of an Incinerator on Housing Prices
With cross-sectional data in 1981, we have
̂rprice = 101, 307.5 − 30, 688.27nearinc
n = 142
(3, 093.0) (5, 827.71)
With another cross-sectional data in 1978 when there were no incinerator,
we have
̂rprice = 82, 517.23 − 18, 824.37nearinc
n = 179
(2, 653.79) (4, 744.59)
Therefore, the true effect of the incinerator in not −30, 688.27 but
−30, 688.27 − (−18, 824.37) = −11, 863.90.
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Outline
Data Structure
Policy Evaluation with Pooled Cross Sections
Three Approaches in Panel Data Estimation
First Difference (FD) Estimator
Fixed Effect (FE) Estimator
Random Effect (RE) Estimator
Empirical Application: Smoking on Birth Outcomes
Concluding Remarks
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Outline
Data Structure
Policy Evaluation with Pooled Cross Sections
Three Approaches in Panel Data Estimation
First Difference (FD) Estimator
Fixed Effect (FE) Estimator
Random Effect (RE) Estimator
Empirical Application: Smoking on Birth Outcomes
Concluding Remarks
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Data Structure (cont.)
A set of pooled cross sections is obtained by sampling randomly from
a large population at different time points.
A (typical) panel data set follow the same individuals over time.
For example, consider that I sample three individuals from this room
at two time points:
Time Pooled Panel
t=1 John, Jane, Evelyn Eric, Andrew, Rachel
t=2 Kyle, Justin, Lisa Eric, Andrew, Rachel
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Data Structure (cont.)
A Snapshot of Data
Table: Pooled Data
year rprice nearinc y81
1978 60000 0 0
1978 54000 1 0
1978 38000 1 0
. . .
1981 82000 1 1
1981 52000 0 1
1981 97000 0 1
Table: Panel Data
id year inf unem
12 1950 7.3 3.5
12 1951 9.1 2.7
16 1950 5.3 5.4
16 1951 4.6 6.7
. . . .
43 1950 7.1 4.2
43 1951 8.5 3.2
47 1950 6.7 5.4
47 1951 2.6 9.4
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Data Structure (cont.)
There are also very useful panel structures other than the
individual-time combination.
1 Twins data: i is for twins id, and t is for the individual among the
specific twins. Control for unobserved generic factors.
2 School data: students sampled from many schools (or classrooms).
Then, i is for school id, and t is for the student in school i.
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Data Structure (cont.)
Examples of pooled cross sections:
Current Population Survey (CPS), USA
Examples of a panel data:
Labor Market Activity Survey (LMAS), Canada
Panel Study of Income Dynamics (PSID), USA
National Longitudinal Survey of Youth (NLSY), USA
A time series of provincial (or country) level data.
ex) inflation and unemployment rate of 50 countries in 1950–2010.
It is usually easier to collect pooled cross sections than to do panel
data.
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Outline
Data Structure
Policy Evaluation with Pooled Cross Sections
Three Approaches in Panel Data Estimation
First Difference (FD) Estimator
Fixed Effect (FE) Estimator
Random Effect (RE) Estimator
Empirical Application: Smoking on Birth Outcomes
Concluding Remarks
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Policy Evaluation with Pooled Cross Sections
Difference-in-Difference Estimator
Terminology:
Treatment Group: those who are affected by a policy (a treatment)
Control Group: those who are not.
The object of policy evaluation is to measure the (mean) difference of
outcomes between the treatment group and the control group. This
measure is also called the average treatment effect.
Consider that you are testing the effect of a new drug. How can you
design the experiment? Randomization.
Recall the incinerator and housing prices example. Is randomization
possible?
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Policy Evaluation with Pooled Cross Sections
Difference-in-Difference Estimator
Consider an example of a drug test:
blprs i = β 0 + β 1 treat i + u i
If you randomized the control/treatment groups well, i.e.
Cov(treat i , u i ) = 0, then you can estimate the effect of the drug by a
single cross section.
In policy evaluation in social sciences, treat i and u i are easily
correlated:
log(wage i ) = β 0 + β 1 jbtrn i + u i
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Policy Evaluation with Pooled Cross Sections
Difference-in-Difference Estimator
Pooled cross sections help us to evaluate the policy effect correctly by
measuring the difference twice (before and after the policy
implementation.)
Recall the two regressions in the incinerator example:
rprice = γ 0 + γ 1 nearinc + u in years 1978 and 1981
ˆδ 1 = ˆγ 1,81 − ˆγ 1,78
= ( rprice 81,nr − rprice 81,fr
)
−
(
rprice78,nr − rprice 78,fr
)
If perfectly randomized, the second term is 0.
This estimator is called the Difference-in-Difference estimator.
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Policy Evaluation with a Pooled Cross Section
Difference-in-Difference Estimator
The effect can be estimated just by a single regression with some
dummy variable.
rprice = β 0 + δ 0 y81 + β 1 nearinc + δ 1 y81 · nearinc + u
This result is not intuitive. Just follow the logic:
Before (y81 = 0) After (y81 = 1) After-Before
Control (nearinc = 0) β 0 β 0 + δ 0 δ 0
Treatment (nearinc = 1) β 0 + β 1 β 0 + δ 0 + β 1 + δ 1 δ 0 + δ 1
Treatment-Control β 1 β 1 + δ 1 δ 1
Therefore, δ 1 in the above regression gives the same estimate of the
Difference-in-Difference estimator.
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Outline
Data Structure
Policy Evaluation with Pooled Cross Sections
Three Approaches in Panel Data Estimation
First Difference (FD) Estimator
Fixed Effect (FE) Estimator
Random Effect (RE) Estimator
Empirical Application: Smoking on Birth Outcomes
Concluding Remarks
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Panel Data and the First Difference (FD) Estimator
In panel data, we follow the same individual over time. This specific
structure enables us to conduct a better analysis.
Specifically, we can control for certain types of omitted variables
called unobserved heterogeneity.
Let us think about some examples:
log(wage it ) = β 0 + δ 0 d2 t + β 1 educ it + a i + u it
} {{ }
v it
Notation: now we have two subscripts, i and t.
Both a i and u it are unobservables called a fixed effect and an
idiosyncratic error, respectively.
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Panel Data and the First Difference (FD) Estimator
For simplicity, consider two periods model:
y it = β 0 + δ 0 d2 t + β 1 x it + a i + u it t = 1, 2.
The pooled OLS does not work well since a i is usually correlated with
x it , i.e. Cov(v it , x it ) ≠ 0.
A simple solution is the First-Difference (FD) estimator.
y i2 = (β 0 + δ 0 ) + β 1 x i2 + a i + u i2 t = 2
y i1 = β 0 + β 1 x i1 + a i + u i1 t = 1
Taking a difference gives
y i2 − y i1 = δ 0 + β 1 (x i2 − x i1 ) + (u i2 − u i1 )
or
∆y i = δ 0 + β 1 ∆x i + ∆u i .
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Panel Data and the First Difference (FD) Estimator
The (pooled) OLS works in the new regression,
1 ∆u i and ∆x i are uncorrelated;
2 ∆x i has some variation.
∆y i = δ 0 + β 1 ∆x i + ∆u i ,
if
The second condition is violated if x it does not change over time:
ex) gender, race, etc.. Then, ∆x i = 0.
Even in the wage equation example,
log(wage it ) = β 0 + δ 0 d2 t + β 1 educ it + a i + u it ,
Most working population do not increase the years of educ.
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Panel Data and the First Difference (FD) Estimator
More than Two Time Periods
When panel data contain more than two time periods, we can still
apply the FD estimator to control for unobserved heterogeneity.
The sufficient condition for the estimator to be valid is
This condition is violated when
Cov(x it , u is ) = 0 for all t and s.
1 Future regressors react to the past dependent variable (feedback);
2 Regressors contain a lagged dependent variable;
3 An important (i.e. related to x it ) time-varying regressor is omitted.
Take differences with adjacent time periods and run the following
regression when t = 1, 2, and 3:
∆y it = α 0 + α 3 d3 t + β 1 ∆x it + ∆u it for t = 2, 3.
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Additional Remarks on FD Estimator
Due to the expansion over the time dimension, serial correlation may
arise.
Also, we cannot exclude the heteroskedasticity problem.
Since we use the OLS estimator, we can apply the White correction or
the HAC estimation method as before.
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Fixed Effect Estimator
Consider a simple error component model again:
y it = β 1 x it + a i + u it , t = 1, . . . , T and i = 1, . . . , n.
We assume that the idiosyncratic error u it is ‘innocuous’ in the sense:
E(u it |X i ) = 0 or E(u it |x it ) = 0.
However, the individual fixed effect a i could be arbitrarily correlated
with x it .
We have already known that the FD estimator cancels out the
unobserved heterogeneity a i .
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Fixed Effect Estimator
There is a different way to cancel out unobserved heterogeneity.
First, fix the individual i and take an average over time:
ȳ i = β 1 ¯x i + a i + ū i .
where
ȳ i = 1 T
T∑
y it , ¯x i = 1 T
t=1
T∑
x it , and ū i = 1 T
t=1
T∑
u it .
t=1
The point is
ā i = 1 T
T∑
a i = 1 T Ta i = a i .
t=1
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Fixed Effect Estimator
Now, take a difference between two equations:
y it = β 1 x it + a i + u it , t = 1, 2, . . . , T .
ȳ i = β 1 ¯x i + a i + ū i .
Then, what we have is
y it − ȳ i = β 1 (x it − ¯x i ) + (u it − ū i ),
t = 1, 2, . . . , T
or
ÿ it = β 1 ẍ it + ü it , t = 1, 2, . . . , T .
We may apply the pooled OLS on the last equation.
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Fixed Effect Estimator
The FE estimator uses information from within group (i) variation:
ÿ i1 = y i1 − ȳ i
ÿ i2 = y i2 − ȳ i
.
ÿ iT = y iT − ȳ i
For this reason, the FE estimator is also called within estimator.
This can be readily extended to a multiple regression model:
ÿ it = β 1 ẍ 1it + β 2 ẍ 2it + . . . + β k ẍ kit + ü it
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Fixed Effect Estimator
FD vs. FE
If T = 2, the FD estimator and the FE estimator are identical:
( )
yi1 + y i2
ÿ i2 ≡ y i2 − ȳ i = y i2 −
= y 1 − y 2
≡ 1 2
2 2 ∆y i2.
Therefore,
ÿ i2 = β 1 ẍ it + ü it
⇐⇒ 1 2 ∆y i2 = β 1
1
2 ∆x i2 + 1 2 ∆u i2
⇐⇒ ∆y i2 = β 1 ∆x i2 + ∆u i2
However, they are different in a finite sample if T > 2. Unless there is
a unit root (or severe serial correlation) problem, you would better use
the FE estimator.
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Random Effect Estimator
In the random effect model:
we assume that
y it = β 0 + β 1 x it + a i + u it ,
Cov(x it , a i ) = 0.
Then, we come back to the ‘nice’ world where we don’t need to
cancel out a i . Just use the pooled OLS?
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Random Effect Estimator
In the random effect model:
we assume that
y it = β 0 + β 1 x it + a i + u it ,
Cov(x it , a i ) = 0.
Then, we come back to the ‘nice’ world where we don’t need to
cancel out a i . Just use the pooled OLS?
No. There is a serial correlation problem.
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Random Effect Estimator
Serial Correlation in the RE model
We have two components in the error term:
v it = a i + u it
Suppose that u it is totally innocuous again:
Cov(a i , u it ) = Cov(u it , u is ) = 0 for t ≠ s.
Now, we calculate Corr(v it , v is ) and show that it is not zero:
Var(v it ) = Var(a i + u it ) = σ 2 a + σ 2 u
Cov(v it , v is ) = E ((a i + u it )(a i + u is ))
= E(a 2 i + a i u is + a i u it + u it u is )
= E(a 2 i ) = σ 2 a
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Random Effect Estimator
Serial Correlation in the RE model
Therefore,
Corr(v it , v is ) =
σ2 a
σ 2 a + σ 2 u
≠ 0
Any inference based on the pooled OLS would be incorrect.
However, we know how to fix this problem. Do GLS!
We want to transform the original model into
ỹ it = β 0 + β 1˜x it + ṽ it
where ṽ it does not have the serial correlation anymore.
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Random Effect Estimator
We multiplied ρ and took a difference when there is a AR(1) serial
correlation. In this case, we multiply
and take a difference as
[
σu
2 λ = 1 −
σu 2 + T σa
2
] (1/2)
y it − λȳ i = β 0 (1 − λ) + β 1 (x it − λ¯x i ) + v it − λ¯v i
We can show that ṽ it (= v it − λ¯v i ) is not serially correlated.
The λ should be estimated by ˆλ.
This specific GLS estimator is called the Random Effect (RE)
estimator.
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Random Effect Estimator
The RE estimator is something between the pooled OLS and the FE
estimator. Note that in Equation:
y it − λȳ i = β 0 (1 − λ) + β 1 (x it − λ¯x i ) + v it − λ¯v i ,
it becomes the pooled OLS when λ = 0, and does the FE estimator
when λ = 1.
The λ is always between 0 and 1 in the RE model.
As T → ∞, the FE and RE estimators are equivalent since λ → 1.
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Random Effect Estimator
RE vs. FE
If you believe that there is obvious endogenous fixed factor, a i , in
your model, you should use the FE estimator.
Otherwise, the RE estimator will tell you more: non time-varying
regressors, efficiency etc.
Keep in mind that the RE estimator is not even consistent if
Cov(x it , a i ) ≠ 0.
We can test whether Cov(x it , a i ) = 0 or not.
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Random Effect Estimator
Hausman Test
The idea of the Hausman test is simple. The null hypothesis is
H 0 :Cov(x it , a i ) = 0
H 1 :Cov(x it , a i ) ≠ 0
Under H 0 , both RE and FE are consistent:
p p
̂β RE → β, ̂βFE → β.
Thus, we can expect that ̂β RE ≈ ̂β FE .
However, under H 1 , only ̂β FE is consistent. Therefore, we reject H 0 if
the difference between ̂β RE and ̂β FE is large enough.
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Outline
Data Structure
Policy Evaluation with Pooled Cross Sections
Three Approaches in Panel Data Estimation
First Difference (FD) Estimator
Fixed Effect (FE) Estimator
Random Effect (RE) Estimator
Empirical Application: Smoking on Birth Outcomes
Concluding Remarks
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Empirical Application: Smoking on Birth Outcomes
“Infants born to women who smoke during pregnancy have a lower average
birthweight... Low birthweight is associated with increased risk for
neonatal, perinatal, and infant morbidity and mortality.”
(Women and Smoking: A Report of the Surgeon General, 2001, requoted from
Abrevaya (2006))
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Empirical Application: Smoking on Birth Outcomes
The direct medical costs: According to the estimates of Lewit et al.
(1995), the low-birthweight (LBW) infants (less than 10% of births)
account for more than 1/3 of health care costs during the first year of life.
The long-term costs:
“Hack et al. (1995) find that LBW babies have developmental problems in
cognition, attention and neuromotor functioning that persist until
adolescence.” (Abrevaya (2006))
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Empirical Application: Smoking on Birth Outcomes
How to Estimate
The OLS estimates would be biased into the negative direction due to
endogeneity.
IV estimation?
492 Comparison between J. ABREVAYA OLS and IV estimates
from Abrevaya (2006)
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Empirical Application: Smoking on Birth Outcomes
The fixed-effect (FE) estimation can be used if panel data are
available.
Abrevaya (2006) constructed a pseudo panel data set and showed
that the FE estimate is smaller than that of the OLS.
y ib = x ′ ib β + γs ib + c i + u ib
where i is Mom’s id and b is the order of a baby from Mom i.
The estimation results for γ by OLS and FE are −243.27(3.20)
−144.04(4.75), respectively.
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Concluding Remarks
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Concluding Remarks
Pooled cross sections are very similar to a single cross section, but
observations across different time points help evaluate the correct
policy effect.
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Concluding Remarks
Pooled cross sections are very similar to a single cross section, but
observations across different time points help evaluate the correct
policy effect.
Extra information contained in panel data enables us to control for
the individual fixed effect by FD and FE estimators.
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Concluding Remarks
Pooled cross sections are very similar to a single cross section, but
observations across different time points help evaluate the correct
policy effect.
Extra information contained in panel data enables us to control for
the individual fixed effect by FD and FE estimators.
If the fixed effect is not correlated with regressors, we can apply RE
estimator, which is a GLS estimator.
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Concluding Remarks
Pooled cross sections are very similar to a single cross section, but
observations across different time points help evaluate the correct
policy effect.
Extra information contained in panel data enables us to control for
the individual fixed effect by FD and FE estimators.
If the fixed effect is not correlated with regressors, we can apply RE
estimator, which is a GLS estimator.
Panel data are not restricted to the individual-time structure.
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Stata Commands
Load the data set filename.dta.
First, we need to set an id variable and a time variable. Check the
relevant variable names.
xtset id time.
Now type xtsum.
The command for the FE estimator is
xtreg dep x1 x2 x3, . . ., fe
The command for the RE estimator is
xtreg dep x1 x2 x3, . . ., re
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