Post on 24-Oct-2021
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
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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
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CH 17 Panel Data Regression Models 2
Data and MethodologiesData
Pooled Date Panel Date
Individual Effects
Pooled OLSFixed-effect
ModelRandom-effect
Model
?HausmanTest
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Example
itititit hispablackeducwage 3210log
itititit unionmarriedxperexpere 762
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itu
T = 8 (‘80, ’81, …, ’87),
N = 545n N × T 4,360
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7
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CH 17 Panel Data Regression Models 3
▪ The Pooled Data Analysis ▪
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The Pooled OLS
hispablackeducwage 3210log
unionmarriedxperexpere 762
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u
10
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The Pooled OLS
• To account for the possibilities that the distributions are different across time.
hispablackeducwage 3210log
unionmarriedxperexpere 762
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uyear 8
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CH 17 Panel Data Regression Models 4
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The Pooled OLS
• To account for the possibilities that the distributions are different across time.
– Use year dummy
– Test H0 : 2 3 ··· 8 0
hispablackeducwage 3210log
unionmarriedxperexpere 762
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u 82 82
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15
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CH 17 Panel Data Regression Models 5
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▪ The Panel Data Analysis ▪
。The Fixed Effects。The Random Effects
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CH 17 Panel Data Regression Models 6
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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The Fixed Effects Model
• 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 Fixed Effects Model
• 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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The Fixed Effects Model
• 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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CH 17 Panel Data Regression Models 7
Example : Fixed Effects
ituddd 54532 54532
itititit hispablackeducwage 3210log
itititit unionmarriedxperexpere 762
54
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Example : Fixed Effects
itititit hispablackeducwage 3210log
itititit unionmarriedxperexpere 762
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54532 54532 ddd itu
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CH 17 Panel Data Regression Models 8
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The Fixed Effects Model
• 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)
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The Fixed Effects Model
• 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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CH 17 Panel Data Regression Models 9
The Fixed Effects Model
• 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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Example : Fixed Effects
itu 832 832
itititit hispablackeducwage 3210log
itititit unionmarriedxperexpere 762
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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Example : Fixed Effects
itititit hispablackeducwage 3210log
itititit unionmarriedxperexpere 762
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52432 52432 ddd
832 832 itu
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CH 17 Panel Data Regression Models 10
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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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The Random Effects Model
• 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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CH 17 Panel Data Regression Models 11
The Random Effects Model
• 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
itititit hispablackeducwage 3210log
itititit unionmarriedxperexpere 762
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iti ua
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CH 17 Panel Data Regression Models 12
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Example
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itititit unionmarriedxperexpere 762
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iti ua 832 832
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Dependent Variable: LWAGE
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CH 17 Panel Data Regression Models 13
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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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Fixed or Random Effects?
• 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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Fixed or Random Effects?
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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CH 17 Panel Data Regression Models 14
Fixed or Random Effects?
• 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
itititit hispablackeducwage 3210log
itititit unionmarriedxperexpere 762
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iti ua
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CH 17 Panel Data Regression Models 15
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APPENDIX
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The Random Effects Model
• 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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CH 17 Panel Data Regression Models 16
The Random Effects Model
iitit yyy *
iitit xxx *
ititit xy
iitiitiit xxyy 1
Define
iitit *
.10
‥‥‥‥‥ (1)
‥‥‥‥ (2)
(1) – (2)
iii xy
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The Random Effects Model
• 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
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• 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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CH 17 Panel Data Regression Models 17
The Random Effects Model
• 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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The Random Effects Model
• Feasible GLS
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2
ˆˆ
ˆ1ˆ
au
u
T
where
iitiitiit xxyy 1
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