Regression Extensions - Econometric Analysis of Panel Data - Lecture Slides, Slides of Econometrics and Mathematical Economics

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Econometric Analysis of Panel Data

7. Regression Extensions of Linear

Individual Effects Models

Extensions

 Heteroscedasticity

 Autocorrelation

 Covariance Structures

 Measurement Error

 Spatial Autocorrelation

OLS Estimation

i

i i

1

N N

i 1 i 1

[ ] =

[ ] u + =

Coefficient Estimator, M = F or R

Robust C

−

= =

F F

i i i i F i

R R

i i i R i

M M M M

M i i i i

Fixed Effects : y X D ε Z θ w

Random Effects : y X β i ε Z θ w

Least Squares

θ Z Z Z y

1 1

N N N

i 1 i 1 i 1

ovariance Matrix based on the White Estimator

Est.Asy.Var[ ]

− −

= = =

M M M M M M M M

M i i i i i i i i

θ Z Z Z w w Z Z Z

GLS Estimation

2 2 2 2 2

u i

2

1

N N

i 1 i 1

No natural format (yet)

[I ]=

, I

ε ε ε

−

= =

i

M -1 M M -1 M -

R i i i i i i i

Fixed Effects :

Random Effects : Ω ii + I = ii Φ

(Feasible) Generalized Least Squares

θ Z Φ Z Z Φ y Φ

2

i

1

2 N

i 1

Tˆ

Est.Asy.Var[ ]

−

ε =

M -1 M

R i i i

ii

θ Z Φ Z

Baltagi and Griffin’s Gasoline Data

World Gasoline Demand Data, 18 OECD Countries, 19 years

Variables in the file are

COUNTRY = name of country

YEAR = year, 1960-

LGASPCAR = log of consumption per car

LINCOMEP = log of per capita income

LRPMG = log of real price of gasoline

LCARPCAP = log of per capita number of cars

See Baltagi (2001, p. 24) for analysis of these data. The article on which the

analysis is based is Baltagi, B. and Griffin, J., "Gasoline Demand in the OECD: An

Application of Pooling and Testing Procedures," European Economic Review, 22,

1983, pp. 117-137. The data were downloaded from the website for Baltagi's

text.

Heteroscedastic Gasoline Data

Baltagi Gasoline Data - 18 countries, 19 years

COUNTRY

3.

4.

4.

5.

5.

6.

6.

3.

0 5 10 15 20

LGASPCAR

Evidence of Heteroscedasticity

Country Specific Residual Variances

COUNTRY

.

.

.

.

.

.

0 5 10 15 20

VI

Heteroscedasticity in the FE Model

 Ordinary Least Squares

 Within groups estimation as usual.

 Standard treatment – this is just a (large) linear

regression model.

 White estimator

i

1

N N

i 1 i 1

1 1

N N T 2 N

i 1 i 1 ,it it i it i i 1

1

N N

i 1 i 1

Var[ | ] ( )( )

White Robust Covariance Matrix Estimator

Est.Var[ | ]

−

= =

− −

= = ε =

−

= =

= Σ Σ Σ σ − − Σ

i i

i D i i D i

i i

i D i t=1 i D i

i

i D i

b X M X X M y

b X X M X x x x x X M X

b X X M X

i

1

T 2 N

it it i it i i 1

e ( )( )

−

=

i

t=1 i D i

x x x x X M X

Heteroscedasticity in Gasoline Data

+----------------------------------------------------+

| Least Squares with Group Dummy Variables |

| LHS=LGASPCAR Mean = 4.296242 |

| Fit R-squared = .9733657 |

| Adjusted R-squared = .9717062 |

+----------------------------------------------------+

Least Squares - Within

+---------+--------------+----------------+--------+---------+----------+

|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|

+---------+--------------+----------------+--------+---------+----------+

LINCOMEP .66224966 .07338604 9.024 .0000 -6.

LRPMG -.32170246 .04409925 -7.295 .0000 -.

LCARPCAP -.64048288 .02967885 -21.580 .0000 -9.

+---------+--------------+----------------+--------+---------+----------+

White Estimator

+---------+--------------+----------------+--------+---------+----------+

LINCOMEP .66224966 .07277408 9.100 .0000 -6.

LRPMG -.32170246 .05381258 -5.978 .0000 -.

LCARPCAP -.64048288 .03876145 -16.524 .0000 -9.

+---------+--------------+----------------+--------+---------+----------+

White Estimator using Grouping

+---------+--------------+----------------+--------+---------+----------+

LINCOMEP .66224966 .06238100 10.616 .0000 -6.

LRPMG -.32170246 .05197389 -6.190 .0000 -.

LCARPCAP -.64048288 .03035538 -21.099 .0000 -9.

Feasible GLS

i

T 2

2

it

,i

i

e

ˆ

T

ε

 

Σ

σ = 



 

t=

2

,it

2 2 2

it i ,i i ,i

1

N N

i 1 i 1

Requires a narrower assumption, estimation of is not feasible.

(Same as in cross section model.)

E[ | ] ; Var[ | ] =

ε

ε ε

−

= =

i

i -1 i i -1 i

i D i D i i D i D i

X ε X I Ω

β X M Ω M X X M Ω M y

1

N N

i 1 i 1 2 2

,i ,i

= weighted within groups LS with constant weights within groups.

−

= =

ε ε

i i

i D i i D i

X M X X M y

i i i

2

,i

y

(Not a function of. Proof left to the reader.)

ε

x β

Modeling the Scedastic Function

2 2

,i i i

2 2

,i i

2 2

,i i

Suppose = a function of , e.g., = f( ).

Any consistent estimator of = f( ) brings

full efficiency to FGLS. E.g.,

= exp ( )

Estimate using ordinary least squares app

ε ε

ε ε

ε ε

′ σ σ

′ σ σ

′

σ σ

z z δ

z δ

z δ

2 2

it i it

lied to

log(e ) log + + w

Second step FGLS using these estimates

ε

′ = σ z δ

Two Step Estimation

i

T 2

2 2 2 it

,iR ,iM i

i

Benefit to modeling the scedastic function vs. using the robust estimator:

e

vs. = exp ( ) ˆ ˆ ˆ

T

Inconsistent; T Consistent in N

is fixed.

ε ε ε

σ = σ σ

t=

z δ

Fixed T is irrelevant

Does it matter? Do the two matrices converge to the same matrix?

N N

i 1 i 1 2 2

,iR ,iM

vs.

N N

It is very unlikely to matter very much.

What if we use Harvey's maximum likelihood estimator instead of LS.

Unlikely t

= =

ε ε

σ σ

i i

i D i i D i

X M X X M X

o matter. In Harvey's model, the OLS estimator is consistent in NT.

Downside: What if the specified function is wrong.

Probably still doesn't matter much.

Ordinary Least Squares

 Standard results for OLS in a GR model

 Consistent

 Unbiased

 Inefficient

 Variance does (we expect) converge to zero;

1 1

N N N

i 1 i 1 i 1

N N N N

i 1 i i 1 i i 1 i i 1 i

1 1

N N N

i 1 i i 1 i i 1 i i N

i 1 i i i i

Var[ | ]

T T T T

f f f , 0 < f < 1.

T T T T

− −

= = =

= = = =

− −

= = =

=

i i i i i i i

ii i i i i ii i

X X X Ω X X X

b X

X X X Ω X X X

Estimating the Variance for OLS

i

i

1 1

N N T 2 N

i 1 i 1 it it it i 1

T 1 N

i=1 t=1 it it t=

White correction?

Est.Var[ | ]= e

Does this work? No. Observations are correlated.

Cluster Estimator

Est.Var[ | ] ( ) ( e )(

− −

= = =

−

i i t=1 i i

b X X X x x X X

b X X'X x

i

T 1

it it

e ) ( )

−

x X'X