lecture14 panel data

7/24/2019 Lecture14 Panel Data

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ECON4150 - Introductory Econometrics

Lecture 14: Panel data

Monique de Haan

([email protected])

Stock and Watson Chapter 10


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OLS: The Least Squares Assumptions

Yi=0+ 1Xi+ui

Assumption 1: conditional mean zero assumption: E[ui|Xi] =0

Assumption 2: (Xi,Yi)are i.i.d. draws from joint distribution

Assumption 3: Large outliers are unlikely

Under these three assumption the OLS estimators are unbiased,consistent and normally distributed in large samples.

Last week we discussed threats to internal validity

In this lecture we discuss a method we can use in case of omittedvariables

Omitted variable is a determinant of the outcomeYi Omitted variable is correlated with regressor of interestXi


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Omitted variables

Multiple regression model was introduced to mitigate omitted variablesproblem of simple regression

Yi=0+ 1X1i+2X2i+3X3i+...+kXki+ui

Even with multiple regression there is threat of omitted variables:

some factors are difficult to measure

sometimes we are simply ignorant about relevant factors

Multiple regression based on panel data may mitigate detrimental effectof omitted variableswithout actually observing them.


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Panel data

Cross-sectional data:

A sample of individuals observed in 1 time period

2010

Panel data: same sample of individuals observed in multiple time periods

2010

2011

2012


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Panel data; notation

Panel data consist of observations onnentities (cross-sectional units)andTtime periods

Particular observation denoted with two subscripts (iandt)

Yit =0+ 1Xit+uit

Yitoutcome variable for individuali in yeart

For balanced panel this results innT observations

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Advantages of panel data

More control over omitted variables.

More observations.

Many research questions typically involve a time component.

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The effect of alcohol taxes on traffic deaths

About 40,000 traffic fatalities each year in the U.S.

Approximately 25% of fatal crashes involve driver who drunk alcohol.

Government wants to reduce traffic fatalities.

One potential policy: increase the tax on alcoholic beverages.

We have data on traffic fatality rate and tax on beer for 48 U.S. states in1982 and 1988.

What is the effect of increasing the tax on beer on the traffic fatality rate?

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Data from 1982

0

1

2

3

4

Trafficfa

tality

rate

0 .5 1 1.5 2 2.5 3

Tax on beer (in real dollars)

Traffic deaths and alcohol taxes in 1982

FatalityRatei,1982 = 2.01 + 0.15 BeerTaxi,1982

(0.

14) (0.

18)

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Data from 1988

0

1

2

3

4

Trafficfa

tality

rate

0 .5 1 1.5 2 2.5

Tax on beer (in real dollars)

Traffic deaths and alcohol taxes in 1988

FatalityRatei,1988 = 1.86 + 0.44 BeerTaxi,1988

(0.

11) (0.

16)

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Panel data: before-after analysis

Both regression using data from 1982 & 1988 likely suffer from omitted

variable bias

We can use data from 1982 and 1988 together as panel data

Panel data withT =2

Observed areYi1,Yi2 andXi1,Xi2

Suppose model is

Yit=0+ 1Xit+2Zi+uit

and we assumeE(uit|Xi1,Xi2,Zi) =0

Zi are (unobserved) variables that vary between states but not over time

(such as local cultural attitude towards drinking and driving)

Parameter of interest is 1

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Panel data

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Panel data: before

Consider cross-sectional regression for first period (t=1):

Yi1 =0+ 1Xi1+ 2Zi+ui1 E[ui|Xi1,Zi] =0

Ziobserved: multiple regression ofYi1 on constant,Xi1 andZileads tounbiased and consistent estimator of1

Zinot observed: regression ofYi1 on constant andXi1 only results inunbiased estimator of1 whenCov(Xi1,Zi) =0

What can we do if we dont observe Zi?

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Panel data: after

We also observeYi2 andXi2, hence model for second period is:

Yi2 =0+ 1Xi2+ 2Zi+ui2

Similar to argument before cross-sectional analysis for period 2 mightfail

Problem is again the unobserved heterogeneity embodied inZi

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Before-after analysis (first differences)

We have

Yi1 =0+ 1Xi1+ 2Zi+ui1

and

Yi2 =0+ 1Xi2+ 2Zi+ui2

Subtracting period 1 from period 2 gives

Yi2 Yi1 = (0+ 1Xi2+ 2Zi+ui2) (0+ 1Xi1+ 2Zi+ui1)

Applying OLS to:

Yi2 Yi1 =1(Xi2 Xi1) + (ui2 ui1)

will produce an unbiased and consistent estimator of1

Advantage of this regression is that we do not need data onZ

By analyzing changes in dependent variable we automatically control fortime-invariant unobserved factors

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Data from 1982 and 1988

1

.4

1.2

1

.8

.6

.4

.2

0

.2

.4

.6

.8

F

atality

rate

1988

Fatatlity

rate

1982

.6 .4 .2 0 .2 .4

Beer tax 1988 Beer tax 1982

Traffic deaths and alcohol taxes: beforeafter

Fatalityi,1988 Fatalityi,1982 = 0.07 1.04 (BeerTaxi,1988 BeerTaxi,1982)(0.06) (0.42)

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Panel data with more than 2 time periods

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Panel data with more than 2 time periods

Panel data withT2

Yit =0+ 1Xit+2Zi+uit, i=1, ...,n; t=1, ...,T

Yit is dependent variable;Xit is explanatory variable;Ziare state

specific, time invariant variables

Equation can be interpreted as model with n specific intercepts (one foreach state)

Yit =1Xit+i+uit, with i=0+ 2Zi

i,i=1, ...,nare called entity fixed effects

imodels impact of omitted time-invariant variables onYit

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State specific intercepts

0

.5

1

1.5

2

2.5

3

3.5

4

Predicted

fatalityrate

0 .5 1 1.5 2 2.5 3

beer tax

Alabama Arizona

Arkansa California

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Fixed effects regression modelLeast squares with dummy variables

Having data on Yit and Xithow to determine1?

Population regression model:Yit=1Xit+i+uit

In order to estimate the model we have to quantifyi

Solution: createndummy variablesD1i, ...,Dni

withD1i=1 ifi=1 and 0 otherwise, withD2i=1 ifi=2 and 0 otherwise,....

Population regression model can be written as:

Yit=1Xit+1D1i+2D2i+...+nDni+uit

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Fi d ff i d l


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Fixed effects regression modelLeast squares with dummy variables

Alternatively, population regression model can be written as:

Yit =0+ 1Xit+2D2i+...+nDni+uit

with0 =1 andi= i 0 fori>1

Interpretation of 1 identical for both representations

Ordinary Least Squares (OLS): choose 0, 1, 2..., nto minimizesquared prediction mistakes (SSR):

n

i=1

T

t=1

Yit 0 1Xit2D2i ...nDni

2

SSRis function of 0,1, 2..., n

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Fi d ff t i d l


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Fixed effect regression modelLeast squares with dummy variables

ni=1

Tt=1

Yit 0 1Xit2D2i ...nDni

2

OLS procedure:

Take partial derivatives ofSSRw.r.t. 0,1, 2...,n

Equal partial derivatives to zero resulting inn+1 equations withn+1unknown coefficients

Solutions are the OLS estimators 0,1, 2...,n

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Fi d ff t i d l


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Fixed effect regression modelLeast squares with dummy variables

Analytical formulas require matrix algebra

Algebraic properties OLS estimators (normal equations, linearity) sameas for simple regression model

Extension to multipleXs straightforward: n+knormal equations

OLS procedure is also labeled Least Squares Dummy Variables (LSDV)method

Dummy variable trap: Never include allndummy variables and theconstant term!

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Fi ed effect regression model


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Fixed effect regression modelWithin estimation

Typicallynis large in panel data applications

With largencomputer will face numerical problem when solving system

ofn+1 equations

OLS estimator can be calculated in two steps

First step: demeanYit andXit

Second step: use OLS on demeaned variables

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Fixed effect regression model


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We have

Yit = 1Xit+i+uit

Yi = 1Xi+i+ui

Yi = 1T

Tt=1Yit, etc. is entity mean

Subtracting both expressions leads to

Yit Yi = (1Xit+i+uit) (1Xi+i+ui)

Yit =1

Xit+uit

Yit =Yit Yi,etc. is entity demeaned variable

ihas disappeared; OLS on demeaned variables involves solving onenormal equation only!

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Entity demeaning is often called the Within transformation

Within transformation is generalization of "before-after" analysis to morethanT =2 periods

Before-after: Yi2 Yi1 =1(Xi2 Xi1) + (ui2 ui1)

Within: Yit Yi=1(Xit Xi) + (uitui)

LSDV and Within estimators are identical:

FatalityRateit = 0.66 BeerTaxit + State dummies(0.19)

( FatalityRateit FatalityRate) = 0.66 (BeerTaxit BeerTax)(0.19)

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Fixed effects regression model


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Fixed effects regression modeltime fixed effects

In addition to entity effects we can also include time effects in the model

Time effects control for omitted variables that are common to all entitiesbut vary over time

Typical example of time effects: macroeconomic conditions or federalpolicy measures are common to all entities (e.g. states) but vary over

time

Panel data model with entity and time effects:

Yit=1Xit+i+t+uit

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Fixed effects regression modeltime fixed effects

OLS estimation straightforward extension of LSDV/Within estimators ofmodel with only entity fixed effects

LSDV: createTdummy variablesB1t....BTt

Yit= 0+ 1Xit+2D2i+...+nDni

+2B2t+3B3t+...+TBTt+uit

Within estimation: DeviatingYit andXitfrom their entityand time-periodmeans

The effect of the tax on beer on the traffic fatality rate:

FatalityRateit = 0.64 BeerTaxit + State dummies+Time dummies(0.20)

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Fixed effects regression modelstatistical properties OLS

Yit =1Xit+i+t+uit

statistical assumptions are:

ASS #1: E(uit|Xi1, ...,XiT, i, t) =0

ASS #2: (Xi1, ...,XiT,Yi1, ...,YiT)are i.i.d. over the cross-section

ASS #3: large outliers are unlikely

ASS #4: no perfect multicollinearity

ASS #5: cov(uit,uis|Xi1, ...,XiT, i, t) =0 fort=s

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Fixed effects regression modelstatistical properties OLS

ASS #1 to ASS #5 imply that:

OLS estimator 1 isunbiasedandconsistentestimator of1

OLS estimators approximately have a normal distribution

remarks:

ASS #1 is most important

extension to multipleXs straightforward

Yit=1X1it+2X2it+...+kXkit+i+t+uit

additional assumption ASS #5 implies that error terms are uncorrelatedover time (no autocorrelation)

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Fixed effects regression modelClustered standard errors

Violation of assumption #5: error terms are correlated over time:(Cov(uit, uis)=0)

uitcontains time-varying factors that affect the traffic fatality rate (but

that are uncorrelated with the beer tax)

These omitted factors might for a given entity be correlated over time

Examples: downturn in local economy, road improvement project

Not correcting for autocorrelation leads to standard errors which areoften too low

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Fixed effects regression modelClustered standard errors

Solution: compute HAC-standard errors (clustered ses)

robust to arbitrary correlation within clusters (entities) robust to heteroskedasticity assume no correlation across entities

Clustered standard errors valid whether or not there isheteroskedasticity and/or autocorrelation

Use of clustered standard errors problematic when number of entities isbelow 50 (or 42)

In stata: command, cluster(entity)

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The effect of a tax on beer on traffic fatalities


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Dependent variable: traffic fatality rate (number of deaths per 10 000)

Beer tax 0.36*** -0.66*** -0.64*** -0.59*** -0.59*(0.06) (0.19) (0.20) (0.18) (0.33)

State fixed effects - yes yes yes yesTime fixed effects - - yes yes yes

Additional control variables - - - yes yes

Clustered standard errors - - - - yes

N 336 336 336 336 336

Note:* significant at 10% level, ** significant at 5% level, *** significant at 1% level. Control variables: Unemployment rate, per capitaincome, minimum legal drinking age.

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Panel data: an example


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preturns to schooling

Yit =1Xit+i+uit

Yitis logarithm of individual earnings;Xitis years of completededucation

iunobserved ability

Likely to be cross-sectional correlation between Xit andi, hencestandard cross-sectional analysis with OLS fails

However, in this case panel data does not solve the problem becauseXit

typically lacks time series variation (Xit=Xi)

We have to resort to cross-sectional methods (instrumental variables) toidentify returns to schooling

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Panel data: Cigarette taxes and smoking


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g g

Is there an effect of cigarette taxes on smoking behavior?

Yit =1Xit+i+uit

Yitnumber of packages per capita in state i in yeart,Xitis real tax oncigarettes in statei in yeart

i is a state specific effect which includes state characteristics which areconstant over time

Data for 48 U.S. states in 2 time periods: 1985 and 1995

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g g

Lpackpc = log number of packages per capita in state iin yeart

rtax = real avr cigarette specific tax during fiscal year in statei

Lperinc = log per capita real income

. r egr ess l packpc r t ax l per i nc

Source | SS df MS Number of obs = 96

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - F( 2, 93) = 21. 25Model | 1. 76908655 2 . 884543277 Pr ob > F = 0. 0000

Resi dual | 3. 87049389 93 . 041618214 R- squared = 0. 3137

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squared = 0. 2989

Tot al | 5. 63958045 95 . 059364005 Root MSE = . 20401

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

l packpc | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt er val ]

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

r t ax | - . 0156393 . 0027975 - 5. 59 0. 000 - . 0211946 - . 0100839

l per i nc | - . 0139092 . 158696 - 0. 09 0. 930 - . 3290481 . 3012296

_cons | 5. 206614 . 3781071 13. 77 0. 000 4. 455769 5. 95746

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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Before-After estimation

. gen di f f _r t ax= r t ax1995- r t ax1985

. gen di f f _l packpc= l packpc1995- l packpc1985

. gen di f f _l per i nc= l per i nc1995- l per i nc1985

. r egr ess di f f _ l packpc di f f _r t ax di f f _ l per i nc, nocons


- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - F( 2, 46) = 145. 66

Model | 3. 33475011 2 1. 66737506 Pr ob > F = 0. 0000

Resi dual | . 526571782 46 . 011447213 R- squar ed = 0. 8636

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squared = 0. 8577

Tot al | 3. 86132189 48 . 080444206 Root MSE = . 10699

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

di f f _ l packpc | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt er val ]

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

di f f _r t ax | - . 0169369 . 0020119 - 8. 42 0. 000 - . 0209865 - . 0128872

di f f _l per i nc | - 1. 011625 . 1325691 - 7. 63 0. 000 - 1. 278473 - . 7447771

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Least squares with dummy variables (no constant term)

. r egr ess l packpc r t ax l per i nc st at eB*, nocons


- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - F( 50, 46) = 7317. 61

Model | 2094. 15728 50 41. 8831457 Pr ob > F = 0. 0000

Resi dual | . 263285891 46 . 005723606 R- squared = 0. 9999

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squar ed = 0. 9997

Tot al | 2094. 42057 96 21. 8168809 Root MSE = . 07565

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

l packpc | Coef . St d. Err . t P>| t | [ 95% Conf . I nt er val ]

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

r t ax | - . 0169369 . 0020119 - 8. 42 0. 000 - . 0209865 - . 0128872

l per i nc | - 1. 011625 . 1325691 - 7. 63 0. 000 - 1. 278473 - . 7447771

st at eB1 | 7. 663688 . 3037711 25. 23 0. 000 7. 052229 8. 275148

st at eB2 | 7. 834448 . 2926539 26. 77 0. 000 7. 245367 8. 42353

st at eB3 | 7. 678433 . 3121525 24. 60 0. 000 7. 050103 8. 306763

st at eB4 | 7. 66627 . 3392221 22. 60 0. 000 6. 983451 8. 349088

......

...st at eB45 | 7. 844359 . 3193189 24. 57 0. 000 7. 201603 8. 487114

st at eB46 | 7. 92666 . 3154175 25. 13 0. 000 7. 291758 8. 561563

st at eB47 | 7. 644741 . 2936826 26. 03 0. 000 7. 053589 8. 235894

st at eB48 | 7. 825943 . 3275694 23. 89 0. 000 7. 16658 8. 485306

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Least squares with dummy variables with constant term

. r egr ess l packpc r t ax l per i nc st at eB*


- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - F( 49, 46) = 19. 17

Model | 5. 37629455 49 . 109720297 Pr ob > F = 0. 0000

Resi dual | . 263285891 46 . 005723606 R- squared = 0. 9533

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squar ed = 0. 9036

Tot al | 5. 63958045 95 . 059364005 Root MSE = . 07565

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -l packpc | Coef . St d. Er r. t P>| t | [ 95% Conf . I nt erval ]

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

r t ax | - . 0169369 . 0020119 - 8. 42 0. 000 - . 0209865 - . 0128872

l per i nc | - 1. 011625 . 1325691 - 7. 63 0. 000 - 1. 278473 - . 7447771

st ateB1 | - . 1530275 . 0900694 - 1. 70 0. 096 - . 3343279 . 0282728

st at eB2 | . 0177322 . 1005272 0. 18 0. 861 - . 1846185 . 220083

..

.

..

.

..

.st ateB42 | - . 771239 . 0918679 - 8. 40 0. 000 - . 9561594 - . 5863186st at eB43 | ( dr opped)

st at eB44 | . 1757536 . 0854144 2. 06 0. 045 . 0038233 . 347684

st at eB45 | . 0276429 . 0948094 0. 29 0. 772 - . 1631985 . 2184843

st at eB46 | . 1099444 . 0918156 1. 20 0. 237 - . 0748708 . 2947597

st ateB47 | - . 1719747 . 0959042 - 1. 79 0. 080 - . 3650198 . 0210705

st at eB48 | . 0092272 . 0787188 0. 12 0. 907 - . 1492255 . 16768

_cons | 7. 816716 . 3458507 22. 60 0. 000 7. 120554 8. 512877

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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Within estimation

. xt r eg l packpc rt ax l per i nc, f e i ( STATE)

Fi xed- ef f ect s ( wi t hi n) r egr essi on Number of obs = 96

Gr oup var i abl e: STATE Number of groups = 48

R- sq: wi t hi n = 0. 8636 Obs per group: mi n = 2

bet ween = 0. 0896 avg = 2. 0

overal l = 0. 2354 max = 2

F( 2, 46) = 145. 66cor r ( u_i , Xb) = - 0. 5687 Prob > F = 0. 0000

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

l packpc | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt er val ]

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

r t ax | - . 0169369 . 0020119 - 8. 42 0. 000 - . 0209865 - . 0128872

l per i nc | - 1. 011625 . 1325691 - 7. 63 0. 000 - 1. 278473 - . 7447771

_cons | 7. 856714 . 3150362 24. 94 0. 000 7. 222579 8. 490849

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

si gma_u | . 25232518

si gma_e | . 07565452

r ho | . 91751731 ( f r act i on of var i ance due t o u_i )

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

F t est t hat al l u_i =0: F( 47, 46) = 13. 41 Prob > F = 0. 0000

lecture14 panel data

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