panel data regression - s ugauss.stat.su.se/gu/e/slides/f17.pdfchapter 16 panel data regression...

Panel Data Regression090507

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Panel Data

Up to now we have analyzed either

� Cross-sectional data (data collected on severalindividuals/units at one point in time)

or

� Time series data (data collected on one individual/unit overseveral time periods)

What if we have a combination of these two types of data?

Chapter 16 Panel Data Regression Models 2/22

Panel data are repeated cross-sections over time, in essence therewill be space as well as time dimensions.

Other names are pooled data, micropanel data, longitudinal data,event history analysis and cohort analysis


Panel Data Examples

The individuals/units can for example be workers, �rms, states orcountries

� Annual unemployment rates of each state over several years

� Quarterly sales of individual stores over several quarters

� Wages for the same worker, working at several di¤erent jobs


Panel Data Examples

Some american surveys:

� The National Longitudinal Survey of Youth (NLSY) trackslabor market outcomes for thousands of individuals, beginningin their teenage years

� The Panel Survey of Income Dynamics (PSID) since 1968collects data on 5000 families about various socioeconomicand demographic variables

� The Survey of Income and Program Participation (SIPP),conducts interviews four times a year about the respondentseconomic conditions


Panel Data

Potential gains

� take heterogenity into account, get individual-speci�cestimates

� especially suitable to study dynamics of change� study more sophisticated behavioral models� minimize bias due to aggregation

However, panel data also increases the complexity of the analysis.


Panel Data

� Balanced/unbalanced

� Short panel/long panel


Panel Data

Two kinds of models:

FIXED EFFECTS MODELS

RANDOM EFFECTS MODELS

The two types of analyses make conceptually contrastingassumptions about e¤ects as either random or �xed


Example with 2 explanatory variables:

Yit = β1 + β2X2it + β3X3it + uit

Notice the subscript index it

� i stands for the i : th cross-sectional unit, i = 1, ...,N

� t stands for the t : th time period, i = 1, ...,T


Pooled OLS Regression

Treats all observation as equivalent and OLS as usual

Yit = β1 + β2X2it + β3X3it + uit

In this case the error term captures "everything"

Naive, ignores time and space


Fixed E¤ects Models with DummyVariables

Several kinds of �xed e¤ects models, di¤ers in the asumptionsabout�The interceptThe slope coe¢ cients



Varies over individuals Varies over timeThe intercept

p �The slope coe¢ cients � �

Di¤erent intercepts for di¤erent individuals β1i

Yit = β1i + β2X2it + β3X3it + uit

but each individuals intercept does not vary over time

If the number of individuals is N = 4

Yit = α1 + α2D2i + α3D3i + α4D4i + β2X2it + β3X3it + uit



Varies over individuals Varies over timeThe intercept � p

The slope coe¢ cients � �

Di¤erent intercepts for di¤erent time periods instead β1t

Yit = β1t + β2X2it + β3X3it + uit

If the number of time periods is T = 20

Yit = λ1 + λ2D2t + λ3D3t + ...+ λ20D20t + β2X2it + β3X3it + uit




p p

The slope coe¢ cients � �

Di¤erent intercepts for di¤erent individuals AND time periods β1it

Yit = β1it + β2X2it + β3X3it + uit

For N = 4 and T = 20

Yit = α1 + α2D2i + α3D3i + α4D4i + λ1 + λ2D2t +

λ3D3t + ...+ λ20D20t + β2X2it + β3X3it + uit




p �The slope coe¢ cients

p �

Both intercepts and slopes varies over individuals, introduces a lotof dummy variables

Yit = α1 + α2D2i + α3D3i + α4D4i + β2X2it + β3X3it+γ1D2iX2it + γ2D2iX3it + γ3D3iX2it + γ4D3iX3it+γ5D4iX2it + γ6D4iX3it + uit

the number of interaction terms is number of dummy variables �number of explanatory variables



Both intercepts and slopes varies over individuals and time

requires even more variables


Fixed E¤ects Models

Cautions

� a lot of dummy variables) less df) increased risk of multicollinearity

� have to re�ect on the assumptions about the error term uit- heteroscedasticity?- autocorrelation?

easily gets complicated when both time and space dimensions


Random E¤ects Models

Now, in the Random E¤ects Model

Yit = β1i + β2X2it + β3X3it + uit

the intercepts/e¤ects β1i are assumed to be random variables withmean value

E (β1i ) = β1

and the intercept value for individual i can be expressed as

β1i = β1 + εi i = 1, ...,N

where E (εi ) = 0 and Var (εi ) = σ2ε



each individual in the sample is considered to be a drawing from anin�nite (or "close to") population of individuals which share thecommon mean value β1

Yit = β1 + β2X2it + β3X3it + εi + uitYit = β1 + β2X2it + β3X3it + wit

The error term wit consists of two components, random e¤ectsmodels are sometimes called error components models



Assumptions about the error components

εi � N�0, σ2ε

�E (εi εj ) = 0 for i 6= j

uit � N�0, σ2u

�E (uituis ) = E (uituit ) = E (uitujs ) = 0 for i 6= j t 6= s

E (εiuit ) = 0

9>>>>>>>>>>>=>>>>>>>>>>>;



)

8>>>><>>>>:E (wit ) = 0

Var (wit ) = σ2ε + σ2u

Corr (wit ,wis ) =σ2ε

σ2ε+σ2u


Random E¤ects vs Fixed E¤ects

depends on

� whether or not the individuals can be viewed as a randomsample from a large population

�E (εiXi ) = 0?

If yes: random e¤ects, if no: �xed e¤ects

� the relation between T and N

- for large T and small N not a big di¤erence- for small T and large N random e¤ects estimators are moree¢ cient than �xed e¤ects (if the assumptions hold)


panel data regression - s ugauss.stat.su.se/gu/e/slides/f17.pdfchapter 16 panel data regression...

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