Data organization. Regression Models Time series Cross-sectional Panel Multi-dimensional panel. Errors in Uni -dimensional Data In standard time series or cross-sectional data sets, we must adjust for non-independent errors. Serial correlation Errors correlated across time - PowerPoint PPT Presentation
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Data organization Year Sales 2005 $10,200 2006 $10,900 2007 $11,000 2008 $8,500 2009 $10,400 Tim e Series Location Sales Virginia $10,400 Florida $10,300 Colorado $8,300 Maine $10,200 C ross-Sectional Year Location Sales 2005 Virginia $9,000 2005 Florida $9,500 2005 Colorado $9,200 2005 Maine $8,800 2006 Virginia $9,200 2006 Florida $10,500 2006 Colorado $10,700 2006 Maine $9,300 2007 Virginia $8,700 2007 Florida $8,900 2007 Colorado $11,000 2007 Maine $9,700 2008 Virginia $8,000 2008 Florida $8,400 2008 Colorado $9,300 2008 Maine $9,000 2009 Virginia $8,000 2009 Florida $9,700 2009 Colorado $8,500 2009 Maine $9,100 Panel Year Location H oliday Sales 2005 Virginia Christmas $9,200 2005 Virginia July 4 $8,400 2005 Virginia LaborD ay $8,900 2005 Florida Christmas $9,100 2005 Florida July 4 $8,400 2005 Florida LaborD ay $10,500 2005 Colorado Christmas $10,300 2005 Colorado July 4 $9,400 2005 Colorado LaborD ay $10,900 2005 Maine Christmas $8,900 2005 Maine July 4 $9,100 2005 Maine LaborD ay $8,700 2006 Virginia Christmas $8,200 2006 Virginia July 4 $8,900 2006 Virginia LaborD ay $8,900 2006 Florida Christmas $10,300 2006 Florida July 4 $11,000 2006 Florida LaborD ay $8,500 2006 Colorado Christmas $8,100 2006 Colorado July 4 $9,200 2006 Colorado LaborD ay $10,200 2006 Maine Christmas $10,200 2006 Maine July 4 $8,100 2006 Maine LaborD ay $8,600 2007 Virginia Christmas $9,600 2007 Virginia July 4 $10,400 2007 Virginia LaborD ay $10,800 2007 Florida Christmas $10,300 2007 Florida July 4 $9,100 2007 Florida LaborD ay $10,900 2007 Colorado Christmas $10,800 2007 Colorado July 4 $9,600 2007 Colorado LaborD ay $10,200 2007 Maine Christmas $10,400 2007 Maine July 4 $9,600 2007 Maine LaborD ay $11,000 2008 Virginia Christmas $8,200 2008 Virginia July 4 $9,800 2008 Virginia LaborD ay $8,900 2008 Florida Christmas $9,200 2008 Florida July 4 $10,400 2008 Florida LaborD ay $9,000 2008 Colorado Christmas $10,700 2008 Colorado July 4 $9,600 2008 Colorado LaborD ay $8,600 2008 Maine Christmas $8,100 2008 Maine July 4 $8,600 2008 Maine LaborD ay $8,000 2009 Virginia Christmas $9,800 2009 Virginia July 4 $8,800 2009 Virginia LaborD ay $10,400 2009 Florida Christmas $10,700 2009 Florida July 4 $8,300 2009 Florida LaborD ay $9,600 2009 Colorado Christmas $9,100 2009 Colorado July 4 $8,300 2009 Colorado LaborD ay $9,600 2009 Maine Christmas $10,200 2009 Maine July 4 $9,600 2009 Maine LaborD ay $8,200 M ulti-DimensionalPanel
Transcript
Data organization
Year Sales2005 $10,2002006 $10,9002007 $11,0002008 $8,5002009 $10,400
Time Series
Location SalesVirginia $10,400Florida $10,300
Colorado $8,300Maine $10,200
Cross-Sectional
Year Location Sales2005 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 Sales2005 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
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
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
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.
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 uu u u u uu 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
cov ,
0 t s
u t su u
t s
Ordinary Least Squares Assumptions
0 00 00 0
uuu
11 1
For the regression model
ˆ ' '
t t ty x u
X X X Y
Ordinary Least Squares (Heteroskedasticity)
cov ,
0 t
t s
u t su u
t s
1
2
3
0 00 00 0
t
t
t
uu
u
11 1
For the regression model
ˆ ' '
t t ty x u
X X X Y
Ordinary Least Squares (Serial Correlation)
| |cov , t st su u u
2
2
u u uu u uu u u
11 1
For the regression model
ˆ ' '
t t ty x u
X X X Y
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 st s
, ,
and cov ,
0 otherwisei t j s
u i j t su u
0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0
0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0
0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 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
2
2
2
2
2
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0 00 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
21 1 1
1 1 12
1 1 1
22 2 2
2 2 22
2 2 2
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
23 3 3
3 3 32
3 3 3
Two-Dimensional Panel Data: OLS (heterogeneous serial correlation)
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
Random vs. Fixed Effects
Random Effects AssumptionPro: Estimators are more efficientCon:Estimators are inconsistent if any of the three errors are not
IIN(0,σ2) across all dimensions.
Fixed Effects AssumptionPro: 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.
Random vs. Fixed Cross-Sectional Effects
2Estimation Procedure Estimate Standard Error Regression ROLS 0.595 0.004 0.63
Random Effects 0.588 0.004 0.59Fixed Effects 0.518 0.009 0.65
, , ,
2 2 2, ,~ 0, , ~ 0, , ~ 0,
35, 400.5
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
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
DrawbacksLess efficient estimators due to lost information about cross-sectional error covariance.Remove the ability to restrict parameter values across cross-sections.
Alternatives to Panel Techniques
, , ,
Run standard OLS on
i t i t i ty x u
Pooled Regression
DrawbacksLess efficient estimators due to lost information about cross-sectional error covariance.Restricts parameter values to be equal across cross-sections.
Alternatives to Panel Techniques
, , ,
Run standard OLS on
i t i i t i ty x u
Pooled Regression with Cross-Sectional Dummies
DrawbacksThis 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.
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.
Extra stuff
Panel data reveals information that is unattainable with non-panel data.
Three-Dimensional Structure of the ASA-NBER Data Set
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.
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 shocksDiscrete shocks
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.
