Panel and Pseudo-Panel Estimation of Cross-Sectional and ...
Data organization. Regression Models Time series Cross-sectional Panel Multi-dimensional panel.
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Transcript of Data organization. Regression Models Time series Cross-sectional Panel Multi-dimensional panel.
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Data organization
Year Sales
2005 $10,2002006 $10,9002007 $11,0002008 $8,5002009 $10,400
Time Series
Location Sales
Virginia $10,400Florida $10,300
Colorado $8,300Maine $10,200
Cross-Sectional
Year Location Sales
2005 Virginia $9,0002005 Florida $9,5002005 Colorado $9,2002005 Maine $8,8002006 Virginia $9,2002006 Florida $10,5002006 Colorado $10,7002006 Maine $9,3002007 Virginia $8,7002007 Florida $8,9002007 Colorado $11,0002007 Maine $9,7002008 Virginia $8,0002008 Florida $8,4002008 Colorado $9,3002008 Maine $9,0002009 Virginia $8,0002009 Florida $9,7002009 Colorado $8,5002009 Maine $9,100
Panel
Year Location Holiday Sales
2005 Virginia Christmas $9,2002005 Virginia July 4 $8,4002005 Virginia Labor Day $8,9002005 Florida Christmas $9,1002005 Florida July 4 $8,4002005 Florida Labor Day $10,5002005 Colorado Christmas $10,3002005 Colorado July 4 $9,4002005 Colorado Labor Day $10,9002005 Maine Christmas $8,9002005 Maine July 4 $9,1002005 Maine Labor Day $8,7002006 Virginia Christmas $8,2002006 Virginia July 4 $8,9002006 Virginia Labor Day $8,9002006 Florida Christmas $10,3002006 Florida July 4 $11,0002006 Florida Labor Day $8,5002006 Colorado Christmas $8,1002006 Colorado July 4 $9,2002006 Colorado Labor Day $10,2002006 Maine Christmas $10,2002006 Maine July 4 $8,1002006 Maine Labor Day $8,6002007 Virginia Christmas $9,6002007 Virginia July 4 $10,4002007 Virginia Labor Day $10,8002007 Florida Christmas $10,3002007 Florida July 4 $9,1002007 Florida Labor Day $10,9002007 Colorado Christmas $10,8002007 Colorado July 4 $9,6002007 Colorado Labor Day $10,2002007 Maine Christmas $10,4002007 Maine July 4 $9,6002007 Maine Labor Day $11,0002008 Virginia Christmas $8,2002008 Virginia July 4 $9,8002008 Virginia Labor Day $8,9002008 Florida Christmas $9,2002008 Florida July 4 $10,4002008 Florida Labor Day $9,0002008 Colorado Christmas $10,7002008 Colorado July 4 $9,6002008 Colorado Labor Day $8,6002008 Maine Christmas $8,1002008 Maine July 4 $8,6002008 Maine Labor Day $8,0002009 Virginia Christmas $9,8002009 Virginia July 4 $8,8002009 Virginia Labor Day $10,4002009 Florida Christmas $10,7002009 Florida July 4 $8,3002009 Florida Labor Day $9,6002009 Colorado Christmas $9,1002009 Colorado July 4 $8,3002009 Colorado Labor Day $9,6002009 Maine Christmas $10,2002009 Maine July 4 $9,6002009 Maine Labor Day $8,200
Multi-Dimensional Panel
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Regression Models
• Time series
• Cross-sectional
• Panel
• Multi-dimensional panel
t t ty x u
i i iy x u
, , ,i t i t i ty x u
, , , , , ,i s t i s t i s ty x u
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Errors in Uni-dimensional Data
In standard time series or cross-sectional data sets, we must adjust for non-independent errors.
Serial correlationErrors correlated across time
Spatial correlationErrors correlated across cross-sections
HeteroskedasticityError variance changes over time or cross-sections
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Errors in Panel Data
Heterogeneous serial correlationErrors correlated across time and differently for different cross-sections.
Heterogeneous spatial correlationErrors correlated across cross-sections but differently for different time periods.
Heterogeneous heteroskedasticityError variance changes over time, but does so differently for different cross-sections.
Serial-spatial correlationPast errors from one cross-section are correlated with future errors from a different cross-section.
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Generalized Least Squares
1 1 2 1 3
1 2 2 2 3
1 3 2 3 3
var cov , cov ,
cov , var cov ,
cov , cov , var
t t t t t
t t t t t
t t t t t
u u u u u
u u u u u
u u u u u
The error covariance matrix shows the covariances of error terms across different observations.
11 1
For the regression model
ˆ ' '
t t ty x u
X X X Y
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cov ,
0 t s
u t su u
t s
Ordinary Least Squares Assumptions
0 0
0 0
0 0
u
u
u
11 1
For the regression model
ˆ ' '
t t ty x u
X X X Y
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Ordinary Least Squares (Heteroskedasticity)
cov ,
0 t
t s
u t su u
t s
1
2
3
0 0
0 0
0 0
t
t
t
u
u
u
11 1
For the regression model
ˆ ' '
t t ty x u
X X X Y
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Ordinary Least Squares (Serial Correlation)
| |cov , t st su u u
2
2
u u u
u u u
u u u
11 1
For the regression model
ˆ ' '
t t ty x u
X X X Y
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Two-Dimensional Panel Data: OLS Assumptions
, , ,
11 1
For the regression model
ˆ ' '
i t i t i t i ty x v u
X X X Y
cov ,
0 otherwisei j
v i jv v
cov ,
0 otherwiset s
t s
, ,
and cov ,
0 otherwisei t j s
u i j t su u
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0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Two-Dimensional Panel Data: OLS Assumptions
, , ,
, , i t i t i t i t
i t i t
y x v u
x
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2
2
2
2
2
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
2
Two-Dimensional Panel Data: OLS (homogeneous serial correlation)
, , ,
, , i t i t i t i t
i t i t
y x v u
x
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21 1 1
1 1 12
1 1 1
22 2 2
2 2 22
2 2 2
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
23 3 3
3 3 32
3 3 3
Two-Dimensional Panel Data: OLS (heterogeneous serial correlation)
, , ,
, , i t i t i t i t
i t i t
y x v u
x
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2 2 21 1 1 1,2 1,2 1,2 1,3 1,3 1,3
1 1 1 1,2 1,2 1,2 1,3 1,3 1,32 2 2
1 1 1 1,2 1,2 1,2 1,3 1,3 1,3
21,2 1,2 1,2
1,2 1
2 22 2 2 2,3 2,3 2,3
,2 1,2 2 2 2 2,3 2,3 2,32 2 2
1,2 1,2 1,2 2 2 2 2,3 2,3 2,3
21,3 1,3 1,3
1,3 1,3 1,32
1,3 1,3 1,3
2 22,3 2,3 2,3 3 3 3
2,3 2,3 2,3 3 3 32 2
2,3 2,3 2,3 3 3 3
Two-Dimensional Panel Data: OLS (serial-spatial correlation)
, , ,
, , i t i t i t i t
i t i t
y x v u
x
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OLS vs. Panel Estimation
2Estimation Procedure Estimate Standard Error Regression R
OLS 0.482 0.017 0.37
Cross-Sectional Effects 0.499 0.014 0.46
Time Effects 0.486 0.013 0.48
Both Effects 0.505 0.009 0.67
, , ,
2 2 2 2, ,~ 0, , ~ 0, , ~ 0, , ~ 0,
35, 40
0.5
i t i t i t i t
i v t i t i u i t t u
y x v u
v IIN IIN u IIN u IIN
N T
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Fixed versus Random Effects
Under the random effects assumption, and are treated as stochastic.
Under the fixed effects assumption, they are treated as fixed in repeated samples.
iv t
, , ,i t i t i t i ty x v u
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Random vs. Fixed Effects
Random Effects Assumption
Pro: Estimators are more efficient
Con:Estimators are inconsistent if any of the three errors are not IIN(0,σ2) across all dimensions.
Fixed Effects Assumption
Pro: Estimators are consistent regardless of and .
Con:Estimators are less efficient.
iv t
, , ,i t i t i t i ty x v u
See Hausman test for endogeneity.
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Random vs. Fixed Cross-Sectional Effects
2Estimation Procedure Estimate Standard Error Regression R
OLS 0.595 0.004 0.63
Random Effects 0.588 0.004 0.59
Fixed Effects 0.518 0.009 0.65
, , ,
2 2 2, ,~ 0, , ~ 0, , ~ 0,
35, 40
0.5
i t i t i t i t
t i t i u i t t u
y x v u
IIN u IIN u IIN
N T
Test statistic = 22
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Alternatives to Panel Techniques
1, 1 1 1, 1,
2, 2 2 2, 2,
For cross-section 1
For cross-section 2
etc.
t t t
t t t
y x u
y x u
Separate Regressions
Drawbacks
Less efficient estimators due to lost information about cross-sectional error covariance.
Remove the ability to restrict parameter values across cross-sections.
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Alternatives to Panel Techniques
, , ,
Run standard OLS on
i t i t i ty x u
Pooled Regression
Drawbacks
Less efficient estimators due to lost information about cross-sectional error covariance.
Restricts parameter values to be equal across cross-sections.
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Alternatives to Panel Techniques
, , ,
Run standard OLS on
i t i i t i ty x u
Pooled Regression with Cross-Sectional Dummies
Drawbacks
This is the fixed effects panel technique.
If the cross-sectional dummies are IIN, then parameter estimates are less efficient than under the random effects panel technique.
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Procedures to use with panel data
Generalized least squares (GLS)Generalized method of moments (GMM)
OLS with “automated” corrections for serial correlation, etc. is GLS.
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Extra stuff
Panel data reveals information that is unattainable with non-panel data.
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Three-Dimensional Structure of the ASA-NBER Data Set
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Shock Occurrence vs. Shock Impact
These shocks all occur in quarter 6 but impact inflation in different quarters.
These shocks all impact inflation in quarter 9 but occur in different quarters.
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Shock Occurrence vs. Shock Impact
, 1ˆ ˆˆth th t hu
1, 1ˆ ˆ ˆth th t hv u u
, , 11
1ˆN
th ith i t hi
F FN
Cumulative shocks
Cross-sectional shocks
Discrete shocks
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Shock Occurrence vs. Shock Impact
Shock Measure Shocks Occur From Shocks Impact Inflation From
Cumulative shocks
th
Beginning of quarter t – h to the end of quarter t.
Beginning of quarter t – h to the end of quarter t.
Cross-sectional shocks
uth
Beginning of quarter t – h to the end of quarter t – h.
Beginning of quarter t – h to the end of quarter t.
Discrete shocks
vth
Beginning of quarter t – h to the end of quarter t – h.
Beginning of quarter t to the end of quarter t.