bootstrap for panel data (ppt), hounkannounon
DESCRIPTION
Bootstrap for panel data.TRANSCRIPT
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Chapter 1 Chapter 2 Chapter 3
Bootstrap for Panel Data Models with anApplication to the Evaluation of Public
Policies
Bertrand G. B. HounkannounonUniversite de Montreal
Ph.D. defense
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Chapter 1 Chapter 2 Chapter 3
ACKNOWLEDGMENT
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Chapter 1 Chapter 2 Chapter 3
THESIS
The purpose of this thesis is to develop bootstrap methods forpanel data models, to prove their validity and apply them in the
framework of evaluation of public policies.
Chapter 1 : Double resampling bootstrap for the mean of apanel
Chapter 2 : Bootstrap for panel regression models withrandom effects
Chapter 3 : Bootstrapping Differences-in-DifferencesEstimates
-
Chapter 1 Chapter 2 Chapter 3
THESIS
The purpose of this thesis is to develop bootstrap methods forpanel data models, to prove their validity and apply them in the
framework of evaluation of public policies.
Chapter 1 : Double resampling bootstrap for the mean of apanel
Chapter 2 : Bootstrap for panel regression models withrandom effects
Chapter 3 : Bootstrapping Differences-in-DifferencesEstimates
-
Chapter 1 Chapter 2 Chapter 3
THESIS
The purpose of this thesis is to develop bootstrap methods forpanel data models, to prove their validity and apply them in the
framework of evaluation of public policies.
Chapter 1 : Double resampling bootstrap for the mean of apanel
Chapter 2 : Bootstrap for panel regression models withrandom effects
Chapter 3 : Bootstrapping Differences-in-DifferencesEstimates
-
Chapter 1 Chapter 2 Chapter 3
THESIS
The purpose of this thesis is to develop bootstrap methods forpanel data models, to prove their validity and apply them in the
framework of evaluation of public policies.
Chapter 1 : Double resampling bootstrap for the mean of apanel
Chapter 2 : Bootstrap for panel regression models withrandom effects
Chapter 3 : Bootstrapping Differences-in-DifferencesEstimates
-
Chapter 1 Chapter 2 Chapter 3
Chapter 1 : Double resampling bootstrap for themean of a panel
The theoretical results and simulations are provided for the samplemean.
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Chapter 1 Chapter 2 Chapter 3
Panel Data
Panel data refers to data sets where observations on individualunits (such as households, firms or countries) are available overseveral time periods.
The availability of two dimensions (cross section and time series)allows for the identification of effects that could not be accountedfor otherwise.
y11 y12 ... ... y1Ty21 y22 ... ... y2T... ... ... .. ...yN1 yN2 .. ... yNT
yit is the cross-sectional i
s observation at period t.
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Chapter 1 Chapter 2 Chapter 3
Bootstrap Methods
Why do Statisticians and Econometricians use bootstrap ?
The true probability distribution of a test statistic is rarelyknown in finite sample.
Avoid Asymptotic fiction: Asymptotic theory uses thebehavior of the statistic at infinity as an approximation.Bootstrap methods can provide a more accurate inference.
Possibility to make weak structure hypothesis. Simulation of nuisance parameters.
Multiple asymptotic distributions in Large Panels (N and Tare both important) : Multiple asymptotic fictions.
-
Chapter 1 Chapter 2 Chapter 3
Bootstrap Methods
Why do Statisticians and Econometricians use bootstrap ?
The true probability distribution of a test statistic is rarelyknown in finite sample.
Avoid Asymptotic fiction: Asymptotic theory uses thebehavior of the statistic at infinity as an approximation.Bootstrap methods can provide a more accurate inference.
Possibility to make weak structure hypothesis. Simulation of nuisance parameters.
Multiple asymptotic distributions in Large Panels (N and Tare both important) : Multiple asymptotic fictions.
-
Chapter 1 Chapter 2 Chapter 3
Bootstrap Methods
Why do Statisticians and Econometricians use bootstrap ?
The true probability distribution of a test statistic is rarelyknown in finite sample.
Avoid Asymptotic fiction: Asymptotic theory uses thebehavior of the statistic at infinity as an approximation.Bootstrap methods can provide a more accurate inference.
Possibility to make weak structure hypothesis. Simulation of nuisance parameters.
Multiple asymptotic distributions in Large Panels (N and Tare both important) : Multiple asymptotic fictions.
-
Chapter 1 Chapter 2 Chapter 3
Bootstrap Methods
Why do Statisticians and Econometricians use bootstrap ?
The true probability distribution of a test statistic is rarelyknown in finite sample.
Avoid Asymptotic fiction: Asymptotic theory uses thebehavior of the statistic at infinity as an approximation.Bootstrap methods can provide a more accurate inference.
Possibility to make weak structure hypothesis. Simulation of nuisance parameters.
Multiple asymptotic distributions in Large Panels (N and Tare both important) : Multiple asymptotic fictions.
-
Chapter 1 Chapter 2 Chapter 3
Bootstrap Methods
The method consists in drawing many random samples thatresembles as much as possible and estimating the distribution ofthe object of interest over these random samples.
Resample the original data, to create pseudo data.
Use estimations on these pseudo data to make inference.
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Chapter 1 Chapter 2 Chapter 3
Resampling Methods for Panel Data
How to bootstrap panel data ?
y11 y12 ... ... y1Ty21 y22 ... ... y2T... ... ... .. ...yN1 yN2 .. ... yNT
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Chapter 1 Chapter 2 Chapter 3
Cross-sectional Resampling
Cross - sectional Resampling : Resample cross-sectional units.Application of original i.i.d bootstrap in cross-sectiondimension.
y11 y12 ... ... y1Ty21 y22 ... ... y2T... ... ... .. ...yN1 yN2 .. ... yNT
=
yi11 yi12 ... ... yi1Tyi21 yi22 ... ... yi2T... ... ... ... ...
yiN1 yiN2 ... ... yiNT
(i1, i2, .., iN) by i.i.d. drawing with replacement from (1, 2, ...,N).
A statistical unit can appear 0, 1, 2, 3... times in a pseudo dataset.
-
Chapter 1 Chapter 2 Chapter 3
Cross-sectional Resampling
Cross - sectional Resampling : Resample cross-sectional units.Application of original i.i.d bootstrap in cross-sectiondimension.
y11 y12 ... ... y1Ty21 y22 ... ... y2T... ... ... .. ...yN1 yN2 .. ... yNT
=
yi11 yi12 ... ... yi1T
yi21 yi22 ... ... yi2T... ... ... ... ...
yiN1 yiN2 ... ... yiNT
(i1, i2, .., iN) by i.i.d. drawing with replacement from (1, 2, ...,N).
A statistical unit can appear 0, 1, 2, 3... times in a pseudo dataset.
-
Chapter 1 Chapter 2 Chapter 3
Cross-sectional Resampling
Cross - sectional Resampling : Resample cross-sectional units.Application of original i.i.d bootstrap in cross-sectiondimension.
y11 y12 ... ... y1Ty21 y22 ... ... y2T... ... ... .. ...yN1 yN2 .. ... yNT
=
yi11 yi12 ... ... yi1Tyi21 yi22 ... ... yi2T
... ... ... ... ...
yiN1 yiN2 ... ... yiNT
(i1, i2, .., iN) by i.i.d. drawing with replacement from (1, 2, ...,N).
A statistical unit can appear 0, 1, 2, 3... times in a pseudo dataset.
-
Chapter 1 Chapter 2 Chapter 3
Cross-sectional Resampling
Cross - sectional Resampling : Resample cross-sectional units.Application of original i.i.d bootstrap in cross-sectiondimension.
y11 y12 ... ... y1Ty21 y22 ... ... y2T... ... ... .. ...yN1 yN2 .. ... yNT
=
yi11 yi12 ... ... yi1Tyi21 yi22 ... ... yi2T... ... ... ... ...
yiN1 yiN2 ... ... yiNT
(i1, i2, .., iN) by i.i.d. drawing with replacement from (1, 2, ...,N).
A statistical unit can appear 0, 1, 2, 3... times in a pseudo dataset.
-
Chapter 1 Chapter 2 Chapter 3
Cross-sectional Resampling
Cross - sectional Resampling : Resample cross-sectional units.Application of original i.i.d bootstrap in cross-sectiondimension.
y11 y12 ... ... y1Ty21 y22 ... ... y2T... ... ... .. ...yN1 yN2 .. ... yNT
=
yi11 yi12 ... ... yi1Tyi21 yi22 ... ... yi2T... ... ... ... ...
yiN1 yiN2 ... ... yiNT
(i1, i2, .., iN) by i.i.d. drawing with replacement from (1, 2, ...,N).
A statistical unit can appear 0, 1, 2, 3... times in a pseudo dataset.
-
Chapter 1 Chapter 2 Chapter 3
Cross-sectional Resampling
Cross - sectional Resampling : Resample cross-sectional units.Application of original i.i.d bootstrap in cross-sectiondimension.
y11 y12 ... ... y1Ty21 y22 ... ... y2T... ... ... .. ...yN1 yN2 .. ... yNT
=
yi11 yi12 ... ... yi1Tyi21 yi22 ... ... yi2T... ... ... ... ...
yiN1 yiN2 ... ... yiNT
(i1, i2, .., iN) by i.i.d. drawing with replacement from (1, 2, ...,N).
A statistical unit can appear 0, 1, 2, 3... times in a pseudo dataset.
-
Chapter 1 Chapter 2 Chapter 3
Block Bootstrap Resampling
Block Bootstrap Resampling : Accommodation of traditionalTime series block bootstrap. Resample blocks of time periodsin order to capture temporal dependence.
Y (N,T )
=
y11 = y1t1 y
12 = y1t2 ... y
1T = y1tT
y21 = y2t1 y22 = y2t2 ... y
2T = y2tT
... ... .. ...yN1 = yNt1 y
N2 = yNt2 ... y
NT = yNtT
(t1, t2, ., tT ) taking the form :1, 1 + 1, ., 1 + l 1
block 1
, 2, 2 + 1, ., 2 + l 1 ,block 2
..,K , K + 1, ., K + l 1 block K
where the vector of indices (1, 2, ..., K ) , K = [T/l ] is obtainedby i.i.d. drawing with replacement from (1, 2, .....,T )
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Chapter 1 Chapter 2 Chapter 3
Double Resampling Bootstrap
Double Resampling Bootstrap : Combination of block andcross-sectional resamplings.
Y =
y11 = yi1t1 y
12 = yi1t2 ... y
1T = yi1tT
y21 = yi2t1 y22 = yi2t2 ... y
2T = yi2tT
... ... .. ...yN1 = yiN t1 y
N2 = yiN t2 ... y
NT = yiN tT
where the indices (i1, i2, ....., iN) and (t1, t2, ., tT ) are chosen asdescribed previously.
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Chapter 1 Chapter 2 Chapter 3
Double Resampling Bootstrap Variance
Var(y)
= Var(z)
+
(1 1
K
)Var
(ycros
)+
(1 1
N
)Var
(ybl
)Finite sample property, holding without any assumption
about yit .
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Chapter 1 Chapter 2 Chapter 3
Double Resampling Bootstrap Variance
Var(y)
>
(1 1
K
)Var
(ycros
)Var
(y)
>
(1 1
N
)Var
(ybl
)The two inequalities mean that the double resampling bootstrapinduces a greater variance than the cross-sectional resamplingbootstrap and the block resampling bootstrap.
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Chapter 1 Chapter 2 Chapter 3
InterpretationFor N and K=T/l large enough
CI cros1 CI 1
CI bl1 CI 1
If the Double Resampling Bootstrap (DRB) CI rejects the NullHypothesis, there is NO CHANCE for one dimensionbootstrap CI to Not Reject it.
One dimension bootstrap methods can reject the Nullhypothesis, and the DRB CI not reject it.
The Double Resampling Bootstrap dominates the resamplingmethods in one dimension, in the sense that It is valid formore processes.
-
Chapter 1 Chapter 2 Chapter 3
InterpretationFor N and K=T/l large enough
CI cros1 CI 1
CI bl1 CI 1
If the Double Resampling Bootstrap (DRB) CI rejects the NullHypothesis, there is NO CHANCE for one dimensionbootstrap CI to Not Reject it.
One dimension bootstrap methods can reject the Nullhypothesis, and the DRB CI not reject it.
The Double Resampling Bootstrap dominates the resamplingmethods in one dimension, in the sense that It is valid formore processes.
-
Chapter 1 Chapter 2 Chapter 3
InterpretationFor N and K=T/l large enough
CI cros1 CI 1
CI bl1 CI 1
If the Double Resampling Bootstrap (DRB) CI rejects the NullHypothesis, there is NO CHANCE for one dimensionbootstrap CI to Not Reject it.
One dimension bootstrap methods can reject the Nullhypothesis, and the DRB CI not reject it.
The Double Resampling Bootstrap dominates the resamplingmethods in one dimension, in the sense that It is valid formore processes.
-
Chapter 1 Chapter 2 Chapter 3
Panel Data Models
IID panel yit = + it
Cross. one-way ECM yit = + i + it
Temp. one-way ECM yit = + ft + it
Two-way ECM yit = + i + ft + it
Factor model yit = + iFt + ityit = + i + iFt + it
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Chapter 1 Chapter 2 Chapter 3
Consistency
A bootstrap method is consistent if :
supxR
P (M (y y) x) P (M (y ) x) PNT
0
with M {N,T ,NT} .
Intuition : The behavior of(y y) is similar to the behavior of(
y ) when the sample size increases.
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Chapter 1 Chapter 2 Chapter 3
Consistency
Y = +
1 ... 12 ... 2... .. ...N ... N
+
f1 ... fTf1 ... fT... ... ...f1 ... fT
+
12...N
( F1 ... FT )+
11 ... 1T21 ... 2T... .. ...N1 ... NT
The cross-sectional resampling is also equivalent to i.i.d.resampling on (1, .., N) . and treats (f1, ..., fT ) and (F1, ....,FT )as constants
yit,cros = + i + ft +
i Ft +
it,cros
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Chapter 1 Chapter 2 Chapter 3
Consistency
Y = +
1 ... 12 ... 2... .. ...N ... N
+
f1 ... fTf1 ... fT... ... ...f1 ... fT
+
12...N
( F1 ... FT )+
11 ... 1T21 ... 2T... .. ...N1 ... NT
The block resampling, is equivalent to block resampling on(f1, .., fT ) and (F1, ...,FT )and treats (1, .., N) and (1, .., N) asconstants.
yit,bl = + i + ft,bl + iF
t,bl +
it,bl
-
Chapter 1 Chapter 2 Chapter 3
Consistency
Y = +
1 ... 12 ... 2... .. ...N ... N
+
f1 ... fTf1 ... fT... ... ...f1 ... fT
+
12...N
( F1 ... FT )+
11 ... 1T21 ... 2T... .. ...N1 ... NT
The double resampling is equivalent to i.i.d. resampling on(1, ...., N) and (1, ...., N) and block resampling on (f1, ...., fT )and (F1, ....,FT ) .
yit = + i + f
t,bl +
i Ft,bl +
it
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Chapter 1 Chapter 2 Chapter 3
Consistency
yit,cros = + i + ft +
i Ft +
it,cros
yit,bl = + i + ft,bl + iF
t,bl +
it,bl
yit = + i + f
t,bl +
i Ft,bl +
it
(ycros y
)= ( ) +
(F F
)+([inter ]
)(ybl y
)=
(fbl f
)+(Fbl F
)+([inter ]
bl
)(y y) = ( ) + (f bl f )+ (F bl F)+ ( )
-
Chapter 1 Chapter 2 Chapter 3
Summary of Bootstrap Consistency
Cross-sect. Block DoubleResampling Resampling Resampling
Cross. one-way ECM Consistent Consistentyit = + i + it
Temp. one-way ECM Consistent Consistentyit = + ft + it
Two-way ECM Consistentyit = + i + ft + it
Factor model Consistent Consistentyit = + i + iFt + it
-
Chapter 1 Chapter 2 Chapter 3
Simulations
(N,T ) (10, 10) Cross Bl(1) Bl(2) 2Res(1) 2Res(2)
yit = + it 4.5 4.3 4.7 1.0 2.0yit = + i + it 5.2 50.1 40.9 5.0 5.1
Temp ECM 0.00 49.1 5.3 5.2 5.0 6.5yit = + 0.25 66.8 10.1 6.5 11.3 9.4ft + it 0.50 63.1 22.2 12.8 24.1 16.7
0.00 5.4 5.2 5.5 1.0 1.3Factor 0.25 4.7 7.5 5.4 1.2 1.6yit = + 0.50 5.0 11.3 7.5 2.3 2.0iFt + it 0.95 5.0 29.3 24.3 4.2 4.3
1.00 4.8 34.0 29.5 4.2 4.92-ECM 0.00 13.8 14.0 9.9 5.6 5.2
yit = + i 0.25 17.2 16.9 12.9 7.1 7.5ft + it 0.50 24.2 28.4 17.3 14.1 12.7
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Chapter 1 Chapter 2 Chapter 3
Simulations(N,T ) (30, 30) Cross Bl(2) Bl(3) 2Res(2) 2Res(3)
yit = + it 5.0 4.8 5.3 1.1 1.3yit = + i + it 4.8 71.3 68.7 4.7 4.9
Temp ECM 0.00 71.6 69.4 5.3 5.2 5.2yit = + 0.25 77.0 9.3 6.9 9.9 7.5ft + it 0.50 83.6 15.3 13.2 15.4 14.3
0.00 4.6 4.7 5.0 0.8 1.2Factor 0.25 4.4 6.0 5.6 1.3 1.1yit = + 0.50 5.7 9.2 8.3 1.3 1.2iFt + it 0.95 5.0 38.8 39.0 5.4 4.1
1.00 4.6 65.0 57.9 5.0 5.52-ECM 0.00 13.1 13.6 14.0 4.6 5.0
yit = + i 0.25 23.0 18.0 12.6 7.1 6.9ft + it 0.50 30.3 23.0 19.2 12.1 10.8
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Chapter 1 Chapter 2 Chapter 3
Simulations
(N,T ) (60, 60) Cross Bl(3) Bl(5) 2Res(3) 2Res(5)
yit = + it 5.6 4.5 5.2 0.8 1.0yit = + i + it 4.4 79.7 77.7 4.2 4.8
Temp ECM 0.00 78.3 5.7 5.9 5.8 6.1yit = + 0.25 83.7 7.8 6.1 7.8 6.3ft + it 0.50 88.6 12.5 8.5 12.8 8.7
0.00 4.8 5.1 4.5 0.5 0.9Factor 0.25 5.0 6.0 5.4 1.3 0.9yit = + 0.50 4.7 7.2 5.6 1.0 1.4iFt + it 0.95 5.2 40.3 33.2 3.7 3.9
1.00 5.2 73.1 67.3 5.4 4.92-ECM 0.00 15.7 14.6 14.8 4.7 5.4
yit = + i 0.25 22.8 15.1 12.8 7.4 5.3ft + it 0.50 30.8 20.0 12.7 11.7 8.0
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Chapter 1 Chapter 2 Chapter 3
Conclusion
The Double Resampling Bootstrap (DRB) method dominatesresampling methods in one dimension, in the sense that theset of DGP for which DRB is valid is greater.
The double resampling is valid under general conditions oncross-sectional and temporal heterogeneity as well ascross-sectional dependence.
Resampling only in the cross section dimension is not valid inthe presence of temporal heterogeneity
Block resampling only in the time series dimension is not validin the presence of cross section heterogeneity.
The bootstrap does not require the researcher to choose oneof several asymptotic approximations available for panelmodels.
-
Chapter 1 Chapter 2 Chapter 3
Conclusion
The Double Resampling Bootstrap (DRB) method dominatesresampling methods in one dimension, in the sense that theset of DGP for which DRB is valid is greater.
The double resampling is valid under general conditions oncross-sectional and temporal heterogeneity as well ascross-sectional dependence.
Resampling only in the cross section dimension is not valid inthe presence of temporal heterogeneity
Block resampling only in the time series dimension is not validin the presence of cross section heterogeneity.
The bootstrap does not require the researcher to choose oneof several asymptotic approximations available for panelmodels.
-
Chapter 1 Chapter 2 Chapter 3
Conclusion
The Double Resampling Bootstrap (DRB) method dominatesresampling methods in one dimension, in the sense that theset of DGP for which DRB is valid is greater.
The double resampling is valid under general conditions oncross-sectional and temporal heterogeneity as well ascross-sectional dependence.
Resampling only in the cross section dimension is not valid inthe presence of temporal heterogeneity
Block resampling only in the time series dimension is not validin the presence of cross section heterogeneity.
The bootstrap does not require the researcher to choose oneof several asymptotic approximations available for panelmodels.
-
Chapter 1 Chapter 2 Chapter 3
Conclusion
The Double Resampling Bootstrap (DRB) method dominatesresampling methods in one dimension, in the sense that theset of DGP for which DRB is valid is greater.
The double resampling is valid under general conditions oncross-sectional and temporal heterogeneity as well ascross-sectional dependence.
Resampling only in the cross section dimension is not valid inthe presence of temporal heterogeneity
Block resampling only in the time series dimension is not validin the presence of cross section heterogeneity.
The bootstrap does not require the researcher to choose oneof several asymptotic approximations available for panelmodels.
-
Chapter 1 Chapter 2 Chapter 3
Conclusion
The Double Resampling Bootstrap (DRB) method dominatesresampling methods in one dimension, in the sense that theset of DGP for which DRB is valid is greater.
The double resampling is valid under general conditions oncross-sectional and temporal heterogeneity as well ascross-sectional dependence.
Resampling only in the cross section dimension is not valid inthe presence of temporal heterogeneity
Block resampling only in the time series dimension is not validin the presence of cross section heterogeneity.
The bootstrap does not require the researcher to choose oneof several asymptotic approximations available for panelmodels.
-
Chapter 1 Chapter 2 Chapter 3
Chap 2 : Bootstrap for panel regression models withrandom effects
Extension to previous results to panel linear regression model.
yit = + Vi + Wt + Xit + it = Zit + it
it = i + ft + iFt + uit
-
Chapter 1 Chapter 2 Chapter 3
Residuals based bootstrap
yit = Zit + it
Use OLS estimator of to get the residuals.
uit = yit Zit
Resample the residuals to create pseudo data.
yit = Zit + uit
Repeat in other to have many realizations of {Y ,Z} and and use them to make inference.
-
Chapter 1 Chapter 2 Chapter 3
Pairs bootstrap
yit = Zit + it
Resample directly {Y ,Z} to create pseudo data {Y ,Z }.
Run OLS regression with {Y ,Z } to have
Repeat to have many realizations of and use them to makeinference
-
Chapter 1 Chapter 2 Chapter 3
Bootstrap Validity
supxRK
P (M ( ) x) P (M ( ) x) PNT
0
M {
N,T ,NT}
Intuition : The behavior of(
)is similar to the behavior of(
)
when the sample size increases.
-
Chapter 1 Chapter 2 Chapter 3
Theoretical Related Literature
Kapetanios (2008) A bootstrap procedure for panel datasetswith many cross-sectional units : N-asymptotic theoreticalresults with iid cross-sectional vector.
yit = + Vi + Xit + it
Goncalves (2010) The Moving Blocks Bootstrap for PanelRegression Models with Individual Fixed Effects:Accommodation of Moving Blocks Bootstrap to linear panelmodels.
yit = Vi + Wt + Xit + it
-
Chapter 1 Chapter 2 Chapter 3
Theoretical Related Literature
Kapetanios (2008) A bootstrap procedure for panel datasetswith many cross-sectional units : N-asymptotic theoreticalresults with iid cross-sectional vector.
yit = + Vi + Xit + it
Goncalves (2010) The Moving Blocks Bootstrap for PanelRegression Models with Individual Fixed Effects:Accommodation of Moving Blocks Bootstrap to linear panelmodels.
yit = Vi + Wt + Xit + it
-
Chapter 1 Chapter 2 Chapter 3
Theoretical contribution
We prove that of the Cross-section resampling bootstrap isvalid only for parameters associated with cross-section varyingregressors in the presence of random effects.
yit = + Vi + Wt + Xit + it
-
Chapter 1 Chapter 2 Chapter 3
Theoretical contribution
We prove that the block resampling bootstrap is valid only forparameters associated with time varying regressors thepresence of random effects.
yit = + Vi + Wt + Xit + it
-
Chapter 1 Chapter 2 Chapter 3
Theoretical contribution
We prove that the double resampling bootstrap induces acorrect inference for all the vector of the parameters in thepresence of random effects.
yit = + Vi + Wt + Xit + it
-
Chapter 1 Chapter 2 Chapter 3
Simulations
(N;T ) = (10; 10)
Cros. Bloc. D-Res
1 31.4 34.4 9.42-way Vi 12.6 59.4 6.0ECM Wt 58.5 12.2 9.9
i + ft + it Xit 26.3 28.7 7.02-way ECM 1 27.0 35.8 9.9with spatial Vi 12.7 53.8 9.5dependence Wt 45.9 11.0 6.8
i + ft + iFt + it Xit 18.4 24.1 5.5
-
Chapter 1 Chapter 2 Chapter 3
Simulations
(N;T ) = (20; 20)
Cros. Bloc. D-Res
1 25.2 24.4 8.92-way Vi 8.1 67.5 6.9ECM Wt 67.3 7.8 7.2
i + ft + it Xit 26.4 27.1 5.52-way ECM 1 23.8 25.4 8.5with spatial Vi 7.8 59.8 6.4dependence Wt 60.8 8.6 6.7
i + ft + iFt + it Xit 21.4 19.8 5.7
-
Chapter 1 Chapter 2 Chapter 3
Simulations
(N;T ) = (30; 30)
Cros. Bloc. D-Res
1 26.0 23.8 6.52-way Vi 8.7 73.8 5.2ECM Wt 73.8 7.3 4.7
i + ft + it Xit 24.4 28.2 5.82-way ECM 1 24.2 22.5 6.7with spatial Vi 6.8 65.2 6.0dependence Wt 68.6 7.4 5.9
i + ft + iFt + it Xit 20.5 21.2 5.5
-
Chapter 1 Chapter 2 Chapter 3
Simulations
(N;T ) = (50; 50)
Cros. Bloc. D-Res
1 24.3 20.5 6.02-way Vi 5.5 81.6 5.5ECM Wt 78.2 5.2 5.6
i + ft + it Xit 24.8 25.3 5.42-way ECM 1 22.6 20.9 6.0with spatial Vi 6.0 73.2 5.8dependence Wt 77.3 5.2 4.7
i + ft + iFt + it Xit 19.5 20.4 4.9
-
Chapter 1 Chapter 2 Chapter 3
Chapter 3: BootstrappingDifferences-in-Differences Estimates
How bootstrap method can help to avoid spurious findings in theevaluation of public policies using panel data.
Double Resampling Bootstrap avoids size distortions and givesmore reliable evaluation of public policies
-
Chapter 1 Chapter 2 Chapter 3
Differences-in-Differences Estimation
Basic setup : Y outcome of interest
Two groups : Treatment group, Control group of statistical units,Two periods before and after a public intervention.
The Differences-in-Differences (DD) estimator is :
DD = (yT ,2 yT ,1) (yU,2 yU,1) =
y = 0 + 1I2 + I + u
I2 is a time dummy variable, I is a binary program indicator.
By analogy, OLS estimator is called Differences-in-Differences(DD) estimator, even in a more complex linear regression model.
-
Chapter 1 Chapter 2 Chapter 3
Impact Evaluation Using Panel Data
General setup :Introduction of Control Variables X to avoidselection bias, Several periods. The model becomes :
yit = Xit + Iit + uit
i = 1, 2, ....N; t = 1, 2, ....T
Typically its a linear panel data model : several statistical unitsduring several time periods.
Advantages : Robustness in time dimension, Possibility todistinguish short term impact and long term impact.
Difficulties : Heterogeneities, temporal correlation, moderatesample size (specially in time dimension).
-
Chapter 1 Chapter 2 Chapter 3
BDM Exercise
Bertrand, Duflo and Mullainathan (QJE,2004) examines thedifferences-in-differences estimator commonly used with panel datato evaluate the impact of public policies.
Their empirical application uses panel data constructed from theCurrent Population Survey (CPS) on wages of women in the 50states, from 1979 to 1999.
-
Chapter 1 Chapter 2 Chapter 3
BDM Exercise
Formally, consider the next model :
Yist = As + Bt + cXist + Ist + ist
Yist : outcome (wage), As : state effects, Bt : time effects
Ist : dummy intervention variable : Randomly generated
-
Chapter 1 Chapter 2 Chapter 3
BDM Exercise
Yist = As + Bt + cXist + Ist + ist
First regression on individual controls Xist (education and age)
Panel construction with mean of residuals by state and year.
Y st = s + t + Ist + st
-
Chapter 1 Chapter 2 Chapter 3
BDM EXERCISE
Y st = s + t + Ist + st
Placebo public interventions are randomly generated across Statesand Periods its impact measured on wages. By construction, noimpact should be found : = 0.
Intuition of BDM Exercise: Several Researchers evaluateindependently a public policy without real impact, using a correctinference method, only 5% of the Researchers should conclude thatthe public policy has a significant impact (Wrong answer).
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Chapter 1 Chapter 2 Chapter 3
BDM Exercise
Y st = s + t + Ist + st
States BDM-OLS FGLS BDM-BSP
06 48.0 . 43.5
10 38.5 . 22.5
20 38.5 . 13.5
50 43.0 24.0 6.5Table 1 : BDM Simulations Results (Theoretical level 5%)
Several evaluations conclude to a significant impact when there isno impact.Dummy variables not enough to remove all the correlationstructure.Parametric Assumptions for FGLS fail to correct the problem.BDM bootstrap method (without rigorous theoretical justification).
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Chapter 1 Chapter 2 Chapter 3
BDM Revisited
States BDM FGLS BDM-BSP Pair-BSP D.Res.1 D.Res.2
06 48.0 - 43.5 17.1 15.0 4.9
10 38.5 - 22.5 13.3 9.6 5.3
20 38.5 - 13.5 8.1 6.3 5.1
50 43.0 24.0 6.5 6.5 5.1 5.1Table 2 : Simulations Results(Theoretical level 5%)
BDM : BDM Fixed effects OLS
FGLS : Assume AR1 process for Error term
BDM-BSP : BDM Bootstrap
Pair-BSP : Correct Version of BDM Bootstrap (correct bootstrapvariance)
D.Res.1 : Double Resampling , Residuals based bootstrap
D.Res.2 : Double Resampling, Pairs bootstrap
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Chapter 1 Chapter 2 Chapter 3
THANKS !
Chapter 1Bootstrap MethodsTheoretical Results
Chapter 2Chapter 2.1Chapter 2.2
Chapter 3Empirical MotivationBDM Revisited