Panel Data Regression Models Heteroskedasticity Chapter 17

CH 17 Panel Data Regression Models 1

Panel Data Regression Models

Chapter 17

1

The Data

• Pooled Data

– Independently pooled cross sections.

– Obtained by sampling randomly from a population at different points in time.

–No correlation in the errors terms across different observations.

◦ Increase the sample size

2

The Data

• Pooled Data

– Samples from different points in time may not be identically distributed.

◦ Heteroskedasticity

3

The Data

• Panel Data

– Longitudinal data

– Randomly selected observations are followed over time.

– Observations are not independently distributed across time.

◦ Individual effects

4


Data and MethodologiesData

Pooled Date Panel Date

Individual Effects

Pooled OLSFixed-effect

ModelRandom-effect

Model

?HausmanTest

5

Example

itititit hispablackeducwage 3210log

itititit unionmarriedxperexpere 762

54

itu

T = 8 (‘80, ’81, …, ’87),

N = 545n N × T 4,360

6

7

8


▪ The Pooled Data Analysis ▪

9

The Pooled OLS

hispablackeducwage 3210log

unionmarriedxperexpere 762

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u

10

11

The Pooled OLS

• To account for the possibilities that the distributions are different across time.



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uyear 8

12


113

The Pooled OLS

• To account for the possibilities that the distributions are different across time.

– Use year dummy

– Test H0 : 2 3 ··· 8 0



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u 82 82

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15

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17

18

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▪ The Panel Data Analysis ▪

。The Fixed Effects。The Random Effects

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The Fixed Effects Model

• The model:

• Decompose the disturbance term into an individual specific effect, ai , and the “remainder disturbance”, uit, that varies over time and entities.

itiitit uaxy

i 1, 2, …, N and t 1, 2, …, T

Assume cov(xit, ai) 0

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• We can think of ai as encapsulating all of the variables that affect yit cross-sectionallybut do not vary over time, ex.,

• Entity-Fixed Effect Model

。The sector that a firm operates in.。A person’s gender.。The country where a bank has its head-

quarters…etc.

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• The simplest types of fixed effects models allow the intercept in the regression model to differ cross-sectionally but not over time, while all of the slope estimates are fixed both cross-sectionally and over time.

• Using dummy variables,

itiNiiitit udNddxy 21 21

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• Note that the intercept has been removed from the equation to avoid the “dummy variable trap”.

• One may otherwise drop one dummy and keep the intercept.

itiNiiitit udNddxy 21 21

elseif,0

obs 1if,11

st

d

elseif,0

obs 2if,12

nd

d

…

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Example : Fixed Effects

ituddd 54532 54532



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25

26

27




54

54532 54532 ddd itu

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• When the fixed effects are modeled in this way, one may test for whether the panel approach is really necessary at all by the F test or the LR test on

• If the null is not rejected, the data can simply be polled together and OLS can be employed.

H0: 2 3 … N 0

(pooled OLS)

30

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• When it is considered that the average value of yit changes over time but not cross-sectionally, it is also possible to have a time-fixed effects model rather than an entity-fixed effects model.

• An example would be where the regulatory environment or tax rate changes part-way through a sample period.

ittitit uaxy

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• For a time-fixed effects model, a similar dummy variable approach could be estimated,

• Essentially, it is identical to a pooled OLS with time dummy.

ittNttitit uTxy 21 21

elseif,0

period 1if,11

st

elseif,0

period 2if,12

nd

…

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itu 832 832



54

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The Fixed Effects Models

• Models that allow for both entity-fixed and time-fixed effects, which contain both cross-sectional and time dummies.

• Two-way Error Component Model

iNiiitit dNddxy 21 21

ittTtt uT 21 21

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54

52432 52432 ddd

832 832 itu

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The Random Effects Model

• The model:

• The intercepts for each cross-sectional unit are assumed to arise from a common intercept plus a random variable ai , that varies cross-sectionally but is constant over time.

ititiit uxay

Assume cov(xit, ai) = 0

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• ai measures the random deviation of each entity’s intercept term from , which has zero mean, is independent of uit, has constant a

2 (var) and is independent of xit . ititiit uxay

itiitit uaxy

ititx

where it ai uit (composite error).

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• The parameters usually are estimated by a generalized least squares (GLS) procedure.

• The transformation involved in this GLS procedure is to subtract a ‘weighted’ mean of the yit over time.

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Example : Random Effects



54

iti ua

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43

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Example



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iti ua 832 832

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Dependent Variable: LWAGE

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Fixed or Random Effects?

• It is often considered that the RE model is more appropriate when the entities in the sample can be thought of as having been randomly selected from the population.

• But a FE model is more plausible when the entities in the sample effectively constitute the entire population.

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• More technically, the transformation involved in the GLS procedure under the RE approach will not remove the explanatory variables that do not vary over time, hence their impact on yit can be enumerated.

• Also, since there are fewer parameters to be estimated with the RE model and therefore degrees of freedom are saved,

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the RE model should produce more efficient estimation than the FE approach.

• However, the RE approach has a major drawback which arises from the fact that it is valid only when the composite error term it (thus both ai and uit) is uncorrelated with all of the explanatory variables, xit.

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• A test for whether this assumption is valid for the RE estimator is based on the Hausman test.

。Test requires equation estimated with RE.

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The Hausman Test

itiitit uaxy 10

REFEq ˆvarˆvarˆvar

REFEq ˆˆˆ

qqqm ˆˆvarˆ 1

H0: ai are not correlated with xitH1: ai are correlated with xit

Test statistic: ~ 2(k)

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Example : Hausman test



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iti ua

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APPENDIX

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• Define the ‘quasi-demeaned’ data as

• will be a function of the variance of the observation error term, , and of the variance of the entity-specific error term,

iitit yyy *

iitit xxx *

221

ua

u

T

u2

.a2

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iitit yyy *

iitit xxx *

ititit xy

iitiitiit xxyy 1

Define

iitit *

.10

‥‥‥‥‥ (1)

‥‥‥‥ (2)

(1) – (2)

iii xy

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• The model becomes

**0

*ititit xy

iitiitiit xxyy 1

or

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22

2

1au

u

T

By definition,

iit var

Tuua222221

iiiti uaua var

TuuaT

tititi

1

1var

Tu22

Tuua22222 11

TTT uuua222221

2u

,itiit ua

(Proof)

Tu2

63

• Pooled OLS is obtained when 0

• Fixed effect (FE) is obtained when 1

• As T gets larger, tends to 1, which makes RE and FE estimates similar.

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2

1au

u

T

iiitiiit uaua Given

where

). (small 2a

) (large 2a

**0

*ititit xy

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• is never known in practice, but can always be estimated.

• Obtain from

where are pooled OLS residuals, and

22var uait

2ˆa

121

ˆˆˆ

1

1

1 12

kTNT

N

i

T

t

T

ts isit

a

2ˆit

.ˆˆˆ 222aitu then obtain by using2ˆ

u

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• Feasible GLS

22

2

ˆˆ

ˆ1ˆ

au

u

T

where

iitiitiit xxyy 1

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Panel Data Regression Models Heteroskedasticity Chapter 17

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