Chapter 11_Seemingly Unrelated Regressions (SUR)

Chapter 11 SEEMINGLY UNRELATED REGRESSIONS Seemingly unrelated regressions SUR are often a set of equations with distinct dependent and independent variables, as well as different coeff

Chapter 11

SEEMINGLY UNRELATED REGRESSIONS

Seemingly unrelated regressions (SUR) are often a set of equations with distinct dependent and independent variables, as well as different coefficients, are linked together

by some common immeasurable factor

Consider the following set of equations: there are 1, 2, … M, such that

1 1 1 1

(T 1) (T k) (k1) (T 1)

2 2 2 2

(T 1) (T k) (k1) (T 1)

× = × × + × country 2

…

( 1) ( ) ( 1) ( 1)

T T k k T

• Assume each (i = 1, 2, …, M) meets classical assumptions so OLS on each equation

separately in fine

• Although each of M equations may seem unrelated, the system of equations may be linked through their mean – zero error structure

• We use cross-equation error covariance to improve the efficiency of OLS M equations are estimated as a system

' 2

( i i) i T ii T

'

( i j) ij T

Where ij : contemporaneous covariance between errors of equations i and j

1

( 1)

2

( 1)

( 1)

T

M

T

MT

Y

Y

Y

×

×

×



1 ( )

2 ( )

( ) ( )

T k

T k

M

T k

TM kM

X

X

X

×

×

×

×







( 1) ( 1)

( 1) ( 1) ( 1) ( 1)

kM NT

Assumption: there is a such that:

(1) ↔ Y X= β ε+

( )

( )

MT MT

E εε

×

′

M M

I







Where:

M M

Σ =







The equation (1) can be estimated by GLS if E(εε’) is known:

ˆ [ '( ( ')) ] [ '( ( ')) ]

ˆ [ '( ) ] [ '( ) ]

GLS is the best linear unbiased estimator:

SUR

Advantages of SUR over single-equation OLS

1 Gain in efficiency:

Because β will have smaller varriance than ˆSUR ˆ

OLS

β

1( ) ( 1) 2( )

( ) ( 1) ( 1)

ˆ

ˆ

OLS k OLS OLS

M OLS k TM

β β β

β

×

×

×

=





Note that βˆi OLS( ) is efficient estimator for i, but β is not efficient estimator for , and ˆOLS

ˆ

SUR

β is efficient estimator for

2 Test or impose cross-section restriction (Allowing to test or impose)

Usually E(εε’) unknown

Feasible GLS estimation

1 Estimate each equation by OLS, save residuals

( 1)

i T

e

×

, i = 1, 2, …, M

2 Compute sample variances and covariances

1

ˆ

T

it jt t

ij

e e

σ = =

−

∑

all ij pairs

M M

Σ =







1

M M

=

−







( )

( )

MT MT

E εε

×

′

( ) ( )

ˆ

M M T T I

3 βˆFGLS =[X'(Σ ⊗ˆ I)−1X] [−1Σ ⊗X'(ˆ I)−1Y]

→Σˆ is a consistent estimator of

It is also possible to interate 2 & 3 until convergence which will produce the maximum likelihood estimator under multivariate normal errors In other words, βˆFGLS and β will ˆML have the same limiting distribution such that:

asy

Where is consistently estimated by

1 1

ˆ

ˆ [X'( I) X]

Definition: For any two matrices A,B

A⊗ is defined by the matrix consisting of each element of A time the entire second B

matrix B

Propositions:

(1) (A⊗B C)( ⊗D)=AC⊗BD

11 12

21 22





11 12

21 22





A⊗B − =A− ⊗B− if inverses are defined

Because: ( ) ( 1 1) ( 1 1)

A⊗B A− ⊗B− = AA− ⊗BB− =I

A⊗B − =A− ⊗B−

(3) ( )/ / /

A⊗B = A ⊗B (you show)

1 When ij = 0 for all i≠j: the equations are not linked in any fashion and GLS does not

provide any efficiency gains → we can show that ˆ

OLS

β = βˆSUR

SUR

1 1

(Σ ⊗ )− = Σ ⊗ =− I

11

22

1

1

1

MM

I

I

I

σ

σ

σ







SUR

1 /

1

11 ( )

1 /

( )

22

/ ( )

1

1

T k

T k

M M

T k

MM

I X

X

X X

I

σ

σ

σ

−

×

×

×













1 /

1 1

11

/

2 2 22

/ 1

MM

X X

X X

X X

σ

σ

σ

−







/ 1

1 1 11

/ 1

2 2 22

X X

X X

X X

σ

σ

σ

−

−

−







iOLS i i ii

SUR i

i VarCov β → no efficiency gains at all

Exercise: Show:

1 2

ˆ ˆ ˆ

OLS OLS SUR

β β β

  in this case

Note:

1 The greater is the correlation of disturbance, the greater is the gain in efficiency in using SUR & GLS

2 The less correlation then is in between the X matrices, the greater is gain in using

GLS

3 When X1=X2 = = X M = X

SUR

[(I X) ( − I I)( X)]−

[ − (X X)]− (X X)−

/ 1

X X

σ

−

=







→ no efficiency gain

iOLS ii

X =

0 0

X X

X







1 Contemporaneous correlation (spatial correlation):

/

( )

ij ij

i j

ij E

σ σ

ε ε

σ

=





 H0: σij =0 for all i≠j

HA: H0 false

LM test statistic:

1

2 2 ( 1)

M i

ij M M

i j

= =

Where r ij is calculated using OLS residuals:

/ / / ( )( )

i j ij

i i j j

e e r

e e e e

Under H0 → 2

( 1) 2

M M

λ χ − If accept H0 → no efficiency gain

2 Restrictions on coefficients:

H0: Rβ =0

HA: H0 false

The general F test can be extended to the SUR system However, since the statistic requires usingΣˆ, the test will only be valid asymptotically Where β =(β β1, 2, ,βM) Within SUR framework, it is possible to test coefficient restriction across equations One possible test statistic is:

( ) ( 1) ( 1) ( ) ( ) ( )

( 1)

( FGLS ) [ ( FGLS) ] ( FGLS )

m k k m m k k k k m

m

×



2

asy

m

W  χ under H0

Heteroscedasticity and autocorrelation are possibilities within SUR framework I will focus on autocorrelation because SUR systems are often comprised of time series observations for each equation Assume the errors follow:

, , 1

i t i i t u it

ε =ρ ε − +

Where u it is white noise The overall error structure will now be:

(E εε′)

11 11 12 12 1 1

21 21 22 22 2

M M

M MM

M M M M MM MM MT MT

=







Where:

1 1

1 1

1

T

T

ij

−

−







1 2 /

1 2 ( )

i i

iT E

ε ε

ε

 

 

 

 

 



Estimation:

1 Run OLS equation by equation by equation Compute consistent estimate of ρi:

1 2 2 1

ˆ

T

it it t

i T

it t

e e e

=

∑

Transform the data, using Cochrane-Orcutt, to remove the autocorrelation

2 Calculate FGLS estimates using the transformed data

• Estimate Σ using the transformed data as in GLS

• Use Σˆ to calculate FGLS