Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
Systems of linear regression equations or SURE
(Seemingly Unrelated Regression)
Ragnar Nymoen
Department of Economics, UiO
29 January 2009
ECON 4610: Lecture 4
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
Overview
This is topic 4 in the lecture plan
Systems of regression equations
SURE as GLS
Relationship to OLS.
SURE on panel data
Main reference is G Ch 10.1 and 10.2;. B Ch 4, 5.4, 5.5, 6.3a,
6.4a; 6N: B; K: Ch 10.1
ECON 4610: Lecture 4
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
Let Yt denote GDP in period t D 1, 2, ..., T .
Ct is “endogenous expenditure” and let Xt denote “exogenous
expenditure”.
Assume that Ct depends on GDP, then our example model is
Yt
D Ct C Xt
(1)
D b1 C b2 Yt C "t , b2
Ct
0
(2)
The Reduced is form:
Yt
Ct
D
11
C
D
21
C
12 Xt
22 Xt
C
1t
(3)
C
2t
(4)
Assume that the disturbances " t are independenty normally
distruted with zero means and constant variances, and that
our parameters of interest are the multipliers 12 and 22 .
ECON 4610: Lecture 4
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
Macro example, cont’d
How can we estimate our parameters of interest?
OLS on (3) and (4) separately? OLS is unbiased.
Something to gain in e¢ ciency, from joint estimation of the
two equations?
After all, they are only seemingly unrelated:
1t and 2t are correlated by construction.
There are restrictions on the coe¢ cients. For example,
11 D 21 . Is there something to gain from incorporating
coe¢ cient restrictions into the estimated procedure?
We concentrate on the correlated disturbances— at the some
remarks about how to incorporate coe¢ cient restrictions.
ECON 4610: Lecture 4
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
Matrix notation
Use the same notation as before, but for two regression models:
y1 D X1
y2 D X2
1
2
C "1 , and
C "2
each X matrix has n (or T ) rows. The number of explanatory
variables may be di¤erent, hence K1 and K2 for the number of
regressors, and coe¢ cients in 1 and 2 . The two disturbance
vectors have classical properties
E ["i ] D 0
Var ["i ] D
2
i I,
i D 1, 2
The conditioning on the respective X matrices is implicit.
ECON 4610: Lecture 4
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
Stacking
We can write the two regression models as one equation by …rst
0
stacking y1 and y2 into a 2n column vector y D y1 y2 , and
then stacking the right hand sides of the two regressions in a
similar way:
y1
y2
X1 0
0 X2
y D X C"
D
1
2
C
"1
"2
(5)
(6)
(6) looks like a usual regression model. But care must be taken,
because the variance of " is obtained by taking expectations of
" "0 D
"1 "01 "1 "02
"2 "01 "2 "02
If E ["1i "2i ] D 12 0 while E ["1i "2j ] D 0 for i 6D j, we have the
analogue to a heteroscedastic covariance matrix.
ECON 4610: Lecture 4
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
The structure of the variance matrix
E ["1i "2i ] D
12
0 written as
E ["1 "02 ] D
12 I
together with
E ["1 "01 ] D
2
1I
and E ["2 "02 ] D
2
2I
gives
Var ."/ D
D
2
1I
12 I
2
1
12
12 I
2
2I
12
2
2
(7)
ID6
I
•.
for the (stacked ) disturbance vector in the regression model
y D X C ".
ECON 4610: Lecture 4
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
We know from Lecture 3 that in this case GLS is e¢ cient
bGLS
1
.X0 • 1 X/
X0 • 1 y
h
i 1
1
X0 [ 6
D .X0 [6 I] X/
D
(8)
1
I] y
We see that the SURE estimator involves the elements of 6
Write these as ij (see Greene p 256)
bSURE D
11 X0 X
1 1
21 X0 X
2 1
12 X0 X
1 2
22 X0 X
2 2
1
11 X0 y
1 1
21 X0 y
2 1
ECON 4610: Lecture 4
C
C
1.
12 X0 y
1 2
22 X0 y
2 2
(9)
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
If
b
12
SURE
D
D
21
D 0 then
11 X0 X
1 1
01
0
22 X0 X
2 2
1
11 X0 y
1 1
22 X0 y
2 2
D
.X01 X1 / 1 X01 y1
.X02 X2 / 1 X02 y2
If the models are unrelated, then the GLS/SURE estimator is
identical to OLS on each of the two models.
No e¢ ciency gain, or improved inference, from using
SURE/GLS.
ECON 4610: Lecture 4
.
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
Identical regressors
Consider the case that the two models have the same regressors:
X1 D X2
bSURE
D
D [6
D [6
11 .X0 X /
1 1
21 .X0 X /
1 1
1
.X01 X1 /]
.X01 X1 / 1 ]
1
12 .X0 X /
11 X0 y
1 1
1 1
22 .X0 X /
21 X0 y
1 1
1 1
11 X0 y C 12 X0 y
1
1 1
1 2
21 X0 y C 22 X0 y
1 1
1 2
11 X0 y C 12 X0 y
1 1
1 2
21 X0 y C 22 X0 y
1 1
1 2
ECON 4610: Lecture 4
C
C
12 X0 y
1 2
22 X0 y
1 2
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
If OLS is applied on each model then
bj D .X01 X1 / 1 X01 yj , j D 1, 2
since X2 D X2 , implying
.X01 X1 /bj D X0j yj
bS
11 b C 12 X0 X b
1
1 1 2
21 b C 22 X0 X b
1
1 1 2
0
2
0
0
1
1
X1 X1 11 b1
12 .X1 X1 /
1 .X1 X1 /
0
2
0
1
1
X01 X1 21 b1
21 .X1 X1 /
2 .X1 X1 /
2 11
b1 C 21 12 b2 C 12 21 b1 C 12 22 b2
1
11 b C
12 b C 2 21 b C 2 22 b
21
1
21
2
1
2
2
2
2 11
2 12
21
22
b1 . 1
C 12
/ C b2 . 1
C 12
/
2 21
2 22
11
12
b1 . 21
C 2
/ C b2 . 21
C 2
/
D [6
D
D
D
.X01 X1 / 1 ]
X01 X1
X01 X1
ECON 4610: Lecture 4
C
C
12 X0 X b
1 1 2
22 X0 X b
1 1 2
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
Write out
66
1
DI
as
2
1
21
12
2
2
.
2
1
11
2
1
.
21
21
11
12
21
22
C
12
12
C
11
12
C
C
12
2
2
2
2
D I
21
/ D 1
22
21
22
D 0
D 0
D 1
From this we see that the SURE estimator becomes
bSURE D
b1
b1
in the case of identical regressors X1 D X2 .
ECON 4610: Lecture 4
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
SURE vs OLS
These result generalize to systems of any number of regression
equations.
Because of it GLS interpretation, the SURE estimator (for
known 6) is more e¢ cient (lower variance) and gives more
correct inference than OLS
The result that bSURE D bOLS when disturbances are
correlated between equations is intuitive
But the result that bSURE D bOLS in the case of identical
regressors, even though disturbances are correlated, is truly
mind boggelig!
It also has practical importance. Often, from theory, the
regressors are the same for all equations, vf the macro
example above.
ECON 4610: Lecture 4
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
Engle functions
Let y1 and y2 contain n variables (observations) of family
expenditures on two commodities. Let X1 and X2 contain
only one explanatory variable: total consumption expenditure
in the N families in the sample, all families face the same
relative price, which is then subsumed in the constant term of
the model.
Known as Engle functions, since the main parameter of
interest is the Engle derivative or the Engle elasticity.
Since X1 D X2 , OLS is e¢ cient (with a caveat) even though
there is good reason to believe that the disturbances are
correlated.
ECON 4610: Lecture 4
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
Caveat
The expenditure on the two goods sum to total expenditure,
this should be incorporated into the estimation to ensure both
e¢ ciency and logical consistency.
This can be attained by working from an explicit functional
form of the utility function, and/or by restrictions on 1 , 2 .
This caveat extends to system of demand functuions, where
we observe variation in goods prices, and therefore have both
price and income elasticities as parameters of interest.
ECON 4610: Lecture 4
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
Factor demand
Assume a 2 factor Cobb-Douglas function. Cost minimization
for a given output level leads to the model
y1 D X1
y2 D X2
1
2
C "1 , and
C "2
where y1 and y2 contain the n observations of the logs of the
volumes of two factors, and X1 D X2 contain intercept, and
the logs of real output and the two factor prices.
OLS will be an e¢ cient estimator, and the more e¢ cient the
more of the restrictions on 1 , 2 implied by theory that we
are able on impose.
Some of the restrictions are linear and are therefore easy to
impose. For example we nominal factor prices by relative
factor prices (formally: the restricted least square estimator
with R D q as constraint).
ECON 4610: Lecture 4
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
Feasible SURE
As with GLS we will in practice need to estimate 6.
OLS residuals (from each equation separately) give a
consistent estimate of 6, and like in the GLS case this
delivers a feasible SURE that is asymptotically e¢ cient
In small samples, there may be a trade-o¤ between the gains
taking into account the cross equation residual correlation,
and the cost of estimating it.
ECON 4610: Lecture 4
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
Panel data: the pooled estimator
Assume that yi (i D 1, 2, ...n/ is T 1. Hence we collect data
from T time periods in n vectors for the individuals in a cross
section. Consistently, Xi (i D 1, 2, ...n/ is T K , and we have the
same variables in all Xi s.
2
3
2
32
3 2
3
y1
X1
"1
1
6 y2 7
6 X2 7 6 2 7 6 "2 7
6
7
6
76
7 6
7
4 : 5 D 4 : 54 : 5C4
5
yn
Xn
"n
k
D X C"
If we assume that all non-zero cross-section residual correlations
occur in the same time period, then
E ["i "0j ] D
ij IT T
for all i 6D j
and E ["i "0i ] D
ECON 4610: Lecture 4
2
i IT T
Macro example
Seemingly unrelated regressions (SURE)
Economic examples
SURE on panel data
Pooled estimator: structure of variance matrix
2
6
Var ."/ D 6
4
D 6
2
1I
12 I
2
2I
12 I
:
1n I
IT
T
..
..
:
..
2n I ..
•.
1n I
12 I
:
2I
n
3
7
7
5
Hence the GLS estimator is this case is equivalent to (8)
h
bGLS D .X0 [6
i
1
I] X/
1
X0 [ 6
1
I] y
ECON 4610: Lecture 4
(10)