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
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Data organization. Regression Models Time series Cross-sectional Panel Multi-dimensional panel.
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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
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 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
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
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
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
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
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
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
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)