Panel Data Regression - s u

Panel Data Regression 090507

1/22

Panel Data Up to now we have analyzed either Cross-sectional data (data collected on several individuals/units at one point in time) or Time series data (data collected on one individual/unit over several 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 there will be space as well as time dimensions.

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


3/22

Panel Data Examples

The individuals/units can for example be workers, …rms, states or countries 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


4/22

Panel Data Examples Some american surveys: The National Longitudinal Survey of Youth (NLSY) tracks labor market outcomes for thousands of individuals, beginning in their teenage years The Panel Survey of Income Dynamics (PSID) since 1968 collects data on 5000 families about various socioeconomic and demographic variables The Survey of Income and Program Participation (SIPP), conducts interviews four times a year about the respondents economic conditions


5/22

Panel Data

Potential gains take heterogenity into account, get individual-speci…c estimates 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.


6/22

Panel Data

Balanced/unbalanced Short panel/long panel


7/22

Panel Data

Two kinds of models:

FIXED EFFECTS MODELS RANDOM EFFECTS MODELS The two types of analyses make conceptually contrasting assumptions about e¤ects as either random or …xed


8/22

Example with 2 explanatory variables: Yit = β1 + β2 X2it + β3 X3it + 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


9/22

Pooled OLS Regression

Treats all observation as equivalent and OLS as usual Yit = β1 + β2 X2it + β3 X3it + uit In this case the error term captures "everything" Naive, ignores time and space


10/22

Fixed E¤ects Models with Dummy Variables

Several kinds of …xed e¤ects models, di¤ers in the asumptions about The intercept The slope coe¢ cients


11/22

Fixed E¤ects Models with Dummy Variables The intercept The slope coe¢ cients

Varies over individuals p

Varies over time

Di¤erent intercepts for di¤erent individuals β1i Yit = β1i + β2 X2it + β3 X3it + uit but each individuals intercept does not vary over time If the number of individuals is N = 4 Yit = α1 + α2 D2i + α3 D3i + α4 D4i + β2 X2it + β3 X3it + uit


12/22

Fixed E¤ects Models with Dummy Variables Varies over individuals The intercept The slope coe¢ cients

Varies over time p

Di¤erent intercepts for di¤erent time periods instead β1t Yit = β1t + β2 X2it + β3 X3it + uit If the number of time periods is T = 20 Yit = λ1 + λ2 D2t + λ3 D3t + ... + λ20 D20t + β2 X2it + β3 X3it + uit


13/22


Varies over individuals p

Varies over time p

Di¤erent intercepts for di¤erent individuals AND time periods β1it Yit = β1it + β2 X2it + β3 X3it + uit For N = 4 and T = 20 Yit

= α1 + α2 D2i + α3 D3i + α4 D4i + λ1 + λ2 D2t + λ3 D3t + ... + λ20 D20t + β2 X2it + β3 X3it + uit


14/22


Varies over individuals p p

Varies over time

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

Yit

= α1 + α2 D2i + α3 D3i + α4 D4i + β2 X2it + β3 X3it +γ1 D2i X2it + γ2 D2i X3it + γ3 D3i X2it + γ4 D3i X3it +γ5 D4i X2it + γ6 D4i X3it + uit

the number of interaction terms is number of dummy variables number of explanatory variables Chapter 16 Panel Data Regression Models

15/22

Fixed E¤ects Models with Dummy Variables

Both intercepts and slopes varies over individuals and time requires even more variables


16/22

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


17/22

Random E¤ects Models Now, in the Random E¤ects Model Yit = β1i + β2 X2it + β3 X3it + uit the intercepts/e¤ects β1i are assumed to be random variables with mean 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


18/22

Random E¤ects Models

each individual in the sample is considered to be a drawing from an in…nite (or "close to") population of individuals which share the common mean value β1 Yit Yit

= β1 + β2 X2it + β3 X3it + εi + uit = β1 + β2 X2it + β3 X3it + wit

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


19/22


Assumptions about the error components

εi

N 0, σ2ε

E ( εi εj ) = 0 uit

for i 6= j

N 0, σ2u

E (uit uis ) = E (uit uit ) = E (uit ujs ) = 0 E (εi uit ) = 0


9 > > > > > > > > > > > =

> > > > > for i 6= j t 6= s > > > > > > ; 20/22


)


8 > > > > < > > > > :

E (wit ) = 0 Var (wit ) = σ2ε + σ2u Corr (wit , wis ) =

σ2ε σ2ε +σ2u

21/22

Random E¤ects vs Fixed E¤ects depends on whether or not the individuals can be viewed as a random sample from a large population E (εi Xi ) = 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 more e¢ cient than …xed e¤ects (if the assumptions hold)


22/22