Greene W.H. Econometric Analysis
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CHAPTER 9
✦
The Generalized Regression Model
289
a. Compute the ordinary least squares regression of y on a constant, x
1
, and x
2
.Be
sure to compute the conventional estimator of the asymptotic covariance matrix
of the OLS estimator as well.
b. Compute the White estimator of the appropriate asymptotic covariance matrix
for the OLS estimates.
c. Test for the presence of heteroscedasticity using White’s general test. Do your
results suggest the nature of the heteroscedasticity?
d. Use the Breusch–Pagan/Godfrey Lagrange multiplier test to test for hetero-
scedasticity.
e. Reestimate the parameters using a two-step FGLS estimator. Use Harvey’s for-
mulation, Var[ε
i
|x
i1
, x
i2
] = σ
2
exp(γ
1
x
i1
+ γ
2
x
i2
).
2. (We look ahead to our use of maximum likelihood to estimate the models discussed
in this chapter in Chapter 14.) In Example 9.4, we computed an iterated FGLS
estimator using the airline data and the model Var[ε
it
|Loadfactor
i,t
] = exp(γ
1
+
γ
2
Loadfactor
i,t
). The weights computed at each iteration were computed by esti-
mating (γ
1
,γ
2
) by least squares regression of ln ˆε
2
i,t
on a constant and Loadfactor.
The maximum likelihood estimator would proceed along similar lines, however
the weights would be computed by regression of [ˆε
2
i,t
/ ˆσ
2
i,t
− 1] on a constant and
Loadfactor
i,t
instead. Use this alternative procedure to estimate the model. Do you
get different results?
10
SYSTEMS OF EQUATIONS
Q
10.1 INTRODUCTION
There are many settings in which the single equation models of the previous chapters
apply to a group of related variables. In these contexts, it makes sense to consider the
several models jointly. Some examples follow.
1. Munnell’s (1990) model for output by the 48 continental U.S. states is
ln GSP
it
= β
1i
+ β
2i
ln pc
it
+ β
3i
ln hwy
it
+ β
4i
ln water
it
+ β
5i
ln util
it
+ β
6i
ln emp
it
+ β
7i
unemp
it
+ ε
it
.
Taken one state at a time, this provides a set of 48 linear regression models. The
application develops a model in which the observations are correlated across time
within a state. An important question pursued here and in the applications in the
next example is whether it is valid to assume that the coefficient vector is the same
for all states (individuals) in the sample.
2. The capital asset pricing model of finance specifies that for a given security,
r
it
−r
ft
= α
i
+ β
i
(r
mt
−r
ft
) + ε
it
,
where r
it
is the return over period t on security i, r
ft
is the return on a risk-free
security, r
mt
is the market return, and β
i
is the security’s beta coefficient. The distur-
bances are obviously correlated across securities. The knowledge that the return on
security i exceeds the risk-free rate by a given amount gives some information about
the excess return of security j, at least for some j’s. It may be useful to estimate
the equations jointly rather than ignore this connection.
3. Pesaran and Smith (1995) proposed a dynamic model for wage determination in 38
UK industries. The central equation is of the form
y
it
= α
i
+ x
it
β
i
+ γ
i
y
i,t−1
+ ε
it
.
Nair-Reichert and Weinhold’s (2001) cross-country analysis of growth of develop-
ing countries takes the same form. In both cases, each group (industry, country)
could be analyzed separately. However, the connections across groups and the in-
teresting question of “poolability”—that is, whether it is valid to assume identical
coefficients—is a central part of the analysis. The lagged dependent variable in the
model produces a substantial complication.
4. In a model of production, the optimization conditions of economic theory imply that
if a firm faces a set of factor prices p, then its set of cost-minimizing factor demands
for producing output Q will be a set of equations of the form x
m
= f
m
(Q, p).
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CHAPTER 10
✦
Systems of Equations
291
The empirical model is
x
1
= f
1
(Q, p|θ) + ε
1
,
x
2
= f
2
(Q, p|θ) + ε
2
,
···
x
M
= f
M
(Q, p|θ) + ε
M
,
where θ is a vector of parameters that are part of the technology and ε
m
represents
errors in optimization. Once again, the disturbances should be correlated. In addi-
tion, the same parameters of the production technology will enter all the demand
equations, so the set of equations has cross-equation restrictions. Estimating the
equations separately will waste the information that the same set of parameters
appears in all the equations.
5. The essential form of a model for equilibrium in a market is
Q
Demand
= α
1
+ α
2
Price + α
3
Income + d
α + ε
Demand
,
Q
Supply
= β
1
+ β
2
Price + s
β + ε
Supply
,
Q
Equilibrium
= Q
Demand
= Q
Supply
,
where d and s are other variables that influence the equilibrium through their
impact on the demand and supply curves, respectively. This model differs from
those suggested thus far because the implication of the third equation is that Price
is not exogenous in the equation system. The equations of this model fit more
appropriately in the instrumental variables framework developed in Chapter 8
than in the regression models developed in Chapters 1 to 7. The multiple equations
framework developed in this chapter provides additional results for estimating
“simultaneous equations models” such as this.
The multiple equations regression model developed in this chapter provides a
modeling framework that can be used in many different settings. The models of pro-
duction and cost developed in Section 10.5 provide the platform for a literature on
empirical analysis of firm behavior. At the macroeconomic level, the “vector autore-
gression models” used in Chapter 21 are specific forms of the seemingly unrelated
regressions model of Section 10.2. The simultaneous equations model presented in
Section 10.6 lies behind the specification of the large variety of models considered in
Chapter 8.
This chapter will develop the essential theory for sets of related regression equa-
tions. Section 10.2 examines the general model in which each equation has its own fixed
set of parameters, and examines efficient estimation techniques. Section 10.2.6 examines
the “pooled” model with identical coefficients in all equations. Production and consumer
demand models are a special case of the general model in which the equations of the
model obey an adding-up constraint that has important implications for specification
and estimation. Section 10.3 suggests extensions of the seemingly unrelated regression
model to the generalized regression models with heteroscedasticity and autocorrelation
that are developed in Chapters 9 and 20. Section 10.4 broadens the seemingly unrelated
regressions model to nonlinear systems of equations. In Section 10.5, we examine a
classic application of the seemingly unrelated regressions model that illustrates the
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PART II
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Generalized Regression Model and Equation Systems
interesting features of the current genre of demand studies in the applied literature.
The seemingly unrelated regressions model is then extended to the translog specifica-
tion, which forms the platform for many recent microeconomic studies of production
and cost. Finally, Section 10.6 merges the results of Chapter 8 on models with endoge-
nous variables with the development in this chapter of multiple equation systems. In
Section 10.6, we will develop simultaneous equations models. These are systems of
equations that build on the seemingly unrelated regressions model to produce equation
systems that include interrelationships among the dependent variables. The supply and
demand model suggested in example 5 above, of equilibrium in which price and quantity
in a market are jointly determined, is an application.
10.2 THE SEEMINGLY UNRELATED
REGRESSIONS MODEL
All the examples suggested in the chapter introduction have a common multiple equa-
tion structure, which we may write as
y
1
= X
1
β
1
+ ε
1
,
y
2
= X
2
β
2
+ ε
2
,
··· (10-1)
y
M
= X
M
β
M
+ ε
M
.
There are M equations and T observations in the sample of data used to estimate them.
1
The second and third examples embody different types of constraints across equations
and different structures of the disturbances. A basic set of principles will apply to them
all, however.
2
The seemingly unrelated regressions (SUR) model in (10-1) is
y
i
= X
i
β
i
+ ε
i
, i = 1,...,M. (10-2)
Define the MT × 1 vector of disturbances,
ε = [ε
1
, ε
2
,...,ε
M
]
.
We assume strict exogeneity of X
i
,
E [ε |X
1
, X
2
,...,X
M
] = 0,
and homoscedasticity
E [ε
m
ε
m
|X
1
, X
2
,...,X
M
] = σ
mm
I
T
.
We assume that a total of T observations are used in estimating the parameters of
the M equations.
3
Each equation involves K
i
regressors, for a total of K =
M
i=1
K
i
.
We will require T > K
i
. The data are assumed to be well behaved, as described in
1
The use of T is not meant to imply any necessary connection to time series. For instance, in the fourth
example, above, the data might be cross sectional.
2
See the surveys by Srivastava and Dwivedi (1979), Srivastava and Giles (1987), and Fiebig (2001).
3
There are a few results for unequal numbers of observations, such as Schmidt (1977), Baltagi, Garvin, and
Kerman (1989), Conniffe (1985), Hwang (1990), and Im (1994). But, the case of fixed T is the norm in practice.
CHAPTER 10
✦
Systems of Equations
293
Section 4.4.1, and we shall not treat the issue separately here. For the present, we also
assume that disturbances are uncorrelated across observations but correlated across
equations. Therefore,
E [ε
it
ε
js
|X
1
, X
2
,...,X
M
] = σ
ij
, if t = s and 0 otherwise.
The disturbance formulation is, therefore,
E [ε
i
ε
j
|X
1
, X
2
,...,X
M
] = σ
ij
I
T
,
or
E [εε
|X
1
, X
2
,...,X
M
] = =
⎡
⎢
⎢
⎢
⎢
⎣
σ
11
I σ
12
I ··· σ
1M
I
σ
21
I σ
22
I ··· σ
2M
I
.
.
.
σ
M1
I σ
M2
I ··· σ
MM
I
⎤
⎥
⎥
⎥
⎥
⎦
. (10-3)
It will be convenient in the discussion below to have a term for the particular
kind of model in which the data matrices are group specific data sets on the same set
of variables. Munnell’s model noted in the introduction is such a case. This special
case of the seemingly unrelated regressions model is a multivariate regression model.
In contrast, the cost function model examined in Section 10.4.1 is not of this type—it
consists of a cost function that involves output and prices and a set of cost share equations
that have only a set of constant terms. We emphasize, this is merely a convenient term
for a specific form of the SUR model, not a modification of the model itself.
10.2.1 GENERALIZED LEAST SQUARES
Each equation is, by itself, a classical regression. Therefore, the parameters could be
estimated consistently, if not efficiently, one equation at a time by ordinary least squares.
The generalized regression model applies to the stacked model,
⎡
⎢
⎢
⎢
⎢
⎣
y
1
y
2
.
.
.
y
M
⎤
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎣
X
1
0 ··· 0
0X
2
··· 0
.
.
.
00··· X
M
⎤
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎣
β
1
β
2
.
.
.
β
M
⎤
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎢
⎣
ε
1
ε
2
.
.
.
ε
M
⎤
⎥
⎥
⎥
⎥
⎦
= Xβ + ε. (10-4)
Therefore, the efficient estimator is generalized least squares.
4
The model has a partic-
ularly convenient form. For the tth observation, the M × M covariance matrix of the
disturbances is
=
⎡
⎢
⎢
⎢
⎢
⎣
σ
11
σ
12
··· σ
1M
σ
21
σ
22
··· σ
2M
.
.
.
σ
M1
σ
M2
··· σ
MM
⎤
⎥
⎥
⎥
⎥
⎦
, (10-5)
4
See Zellner (1962) and Telser (1964).
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Generalized Regression Model and Equation Systems
so, in (10-3),
= ⊗ I (10-6)
and
−1
=
−1
⊗ I.
5
Denoting the ijth element of
−1
by σ
ij
, we find that the GLS estimator is
ˆ
β = [X
−1
X]
−1
X
−1
y = [X
(
−1
⊗ I)X]
−1
X
(
−1
⊗ I)y. (10-7)
Expanding the Kronecker products produces
ˆ
β =
⎡
⎢
⎢
⎢
⎢
⎣
σ
11
X
1
X
1
σ
12
X
1
X
2
··· σ
1M
X
1
X
M
σ
21
X
2
X
1
σ
22
X
2
X
2
··· σ
2M
X
2
X
M
.
.
.
σ
M1
X
M
X
1
σ
M2
X
M
X
2
··· σ
MM
X
M
X
M
⎤
⎥
⎥
⎥
⎥
⎦
−1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
M
j=1
σ
1 j
X
1
y
j
M
j=1
σ
2 j
X
2
y
j
.
.
.
M
j=1
σ
Mj
X
M
y
j
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
The asymptotic covariance matrix for the GLS estimator is the bracketed inverse matrix
in (10-7). All the results of Chapter 9 for the generalized regression model extend to
this model (which has both heteroscedasticity and autocorrelation).
This estimator is obviously different from ordinary least squares. At this point,
however, the equations are linked only by their disturbances—hence the name seem-
ingly unrelated regressions model—so it is interesting to ask just how much efficiency
is gained by using generalized least squares instead of ordinary least squares. Zellner
(1962) and Dwivedi and Srivastava (1978) have analyzed some special cases in detail.
1. If the equations are actually unrelated—that is, if σ
ij
= 0 for i = j—then there is
obviously no payoff to GLS estimation of the full set of equations. Indeed, full GLS
is equation by equation OLS.
6
2. If the equations have identical explanatory variables—that is, if X
i
= X
j
—then
OLS and GLS are identical. We will turn to this case in Section 10.2.2.
7
3. If the regressors in one block of equations are a subset of those in another, then GLS
brings no efficiency gain over OLS in estimation of the smaller set of equations;
thus, GLS and OLS are once again identical.
8
In the more general case, with unrestricted correlation of the disturbances and
different regressors in the equations, the results are complicated and dependent on
5
See Appendix Section A.5.5.
6
See also Baltagi (1989) and Bartels and Fiebig (1992) for other cases in which OLS = GLS.
7
An intriguing result, albeit probably of negligible practical significance, is that the result also applies if the
X
s are all nonsingular, and not necessarily identical, linear combinations of the same set of variables. The
formal result which is a corollary of Kruskal’s theorem [see Davidson and MacKinnon (1993, p. 294)] is that
OLS and GLS will be the same if the K columns of X are a linear combination of exactly K characteristic
vectors of . By showing the equality of OLS and GLS here, we have verified the conditions of the corollary.
The general result is pursued in the exercises. The intriguing result cited is now an obvious case.
8
The result was analyzed by Goldberger (1970) and later by Revankar (1974) and Conniffe (1982a, b).
CHAPTER 10
✦
Systems of Equations
295
the data. Two propositions that apply generally are as follows:
1. The greater is the correlation of the disturbances, the greater is the efficiency gain
accruing to GLS.
2. The less correlation there is between the X matrices, the greater is the gain in
efficiency in using GLS.
9
10.2.2 SEEMINGLY UNRELATED REGRESSIONS
WITH IDENTICAL REGRESSORS
The case of identical regressors is quite common, notably in the capital asset pricing
model in empirical finance—see the chapter introduction. In this special case, general-
ized least squares is equivalent to equation by equation ordinary least squares. Impose
the assumption that X
i
= X
j
= X, so that X
i
X
j
= X
X for all i and j in (10-7). The
inverse matrix on the right-hand side now becomes [
−1
⊗X
X]
−1
, which, using (A-76),
equals [ ⊗ (X
X)
−1
]. Also on the right-hand side, each term X
i
y
j
equals X
y
j
, which,
in turn equals X
Xb
j
. With these results, after moving the common X
X out of the
summations on the right-hand side, we obtain
ˆ
β =
⎡
⎢
⎢
⎢
⎢
⎣
σ
11
(X
X)
−1
σ
12
(X
X)
−1
··· σ
1M
(X
X)
−1
σ
21
(X
X)
−1
σ
22
(X
X)
−1
··· σ
2M
(X
X)
−1
.
.
.
σ
M1
(X
X)
−1
σ
M2
(X
X)
−1
··· σ
MM
(X
X)
−1
⎤
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎣
(X
X)
M
l=1
σ
1l
b
l
(X
X)
M
l=1
σ
2l
b
l
.
.
.
(X
X)
M
l=1
σ
Ml
b
l
⎤
⎥
⎥
⎥
⎥
⎥
⎦
. (10-8)
Now, we isolate one of the subvectors, say the first, from
ˆ
β. After multiplication, the
moment matrices cancel, and we are left with
ˆ
β
1
=
M
j=1
σ
1 j
M
l=1
σ
jl
b
l
=b
1
M
j=1
σ
1 j
σ
j1
+b
2
M
j=1
σ
1 j
σ
j2
+···+b
M
M
j=1
σ
1 j
σ
jM
.
The terms in parentheses are the elements of the first row of
−1
= I, so the end result
is
ˆ
β
1
=b
1
. For the remaining subvectors, which are obtained the same way,
ˆ
β
i
=b
i
, which
is the result we sought.
10
To reiterate, the important result we have here is that in the SUR model, when all
equations have the same regressors, the efficient estimator is single-equation ordinary
least squares; OLS is the same as GLS. Also, the asymptotic covariance matrix of
ˆ
β
for this case is given by the large inverse matrix in brackets in (10-8), which would be
estimated by
Est. Asy. Cov[
ˆ
β
i
,
ˆ
β
j
] = ˆσ
ij
(X
X)
−1
, i, j = 1,...,M, where
ˆ
ij
= ˆσ
ij
=
1
T
e
i
e
j
.
Except in some special cases, this general result is lost if there are any restrictions on β,
either within or across equations.
9
See also Binkley (1982) and Binkley and Nelson (1988).
10
See Hashimoto and Ohtani (1990) for discussion of hypothesis testing in this case.
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Generalized Regression Model and Equation Systems
10.2.3 FEASIBLE GENERALIZED LEAST SQUARES
The preceding discussion assumes that is known, which, as usual, is unlikely to be the
case. FGLS estimators have been devised, however.
11
. The least squares residuals may
be used (of course) to estimate consistently the elements of with
ˆσ
ij
= s
ij
=
e
i
e
j
T
. (10-9)
The consistency of s
ij
follows from that of b
i
and b
j
.
12
A degrees of freedom correction
in the divisor is occasionally suggested. Two possibilities that are unbiased when i = j
are
s
∗
ij
=
e
i
e
j
[(T − K
i
)(T − K
j
)]
1/2
and s
∗∗
ij
=
e
i
e
j
T − max(K
i
, K
j
)
.
13
(10-10)
Whether unbiasedness of the estimator of used for FGLS is a virtue here is uncertain.
The asymptotic properties of the feasible GLS estimator,
ˆ
ˆ
β do not rely on an unbiased
estimator of ; only consistency is required. All our results from Chapters 8 and 9 for
FGLS estimators extend to this model, with no modification. We shall use (10-9) in what
follows. With
S =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
s
11
s
12
··· s
1M
s
21
s
22
··· s
2M
.
.
.
s
M1
s
M2
··· s
MM
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(10-11)
in hand, FGLS can proceed as usual.
10.2.4 TESTING HYPOTHESES
For testing a hypothesis about β, a statistic analogous to the F ratio in multiple regression
analysis is
F[J, MT − K] =
(R
ˆ
β − q)
[R(X
ˆ
−1
X)
−1
R
]
−1
(R
ˆ
β − q)/J
ˆε
−1
ˆε/(MT − K)
. (10-12)
The computation requires the unknown . If we insert the FGLS estimate
ˆ
based on
(10-9) and use the result that the denominator converges to one, then, in large samples,
the statistic will behave the same as
ˆ
F =
1
J
(R
ˆ
ˆ
β − q)
[R
(
Var[
ˆ
ˆ
β]R
]
−1
(R
ˆ
ˆ
β − q). (10-13)
This can be referred to the standard F table. Because it uses the estimated , even
with normally distributed disturbances, the F distribution is only valid approximately.
In general, the statistic F[J, n] converges to 1/ J times a chi-squared [J ]asn →∞.
11
See Zellner (1962) and Zellner and Huang (1962). The FGLS estimator for this model is also labeled
Zellner’s efficient estimator, or ZEF, in reference to Zellner (1962) where it was introduced
12
Perhaps surprisingly, if it is assumed that the density of ε is symmetric, as it would be with normality, then
b
i
is also unbiased. See Kakwani (1967).
13
See, as well, Judge et al. (1985), Theil (1971), and Srivastava and Giles (1987).
CHAPTER 10
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297
Therefore, an alternative test statistic that has a limiting chi-squared distribution with
J degrees of freedom when the null hypothesis is true is
J
ˆ
F = (R
ˆ
ˆ
β − q)
[R
(
Var[
ˆ
ˆ
β]R
]
−1
(R
ˆ
ˆ
β − q). (10-14)
This can be recognized as a Wald statistic that measures the distance between R
ˆ
ˆ
β and q.
Both statistics are valid asymptotically, but (10-13) may perform better in a small or
moderately sized sample.
14
Once again, the divisor used in computing ˆσ
ij
may make a
difference, but there is no general rule.
A hypothesis of particular interest is the homogeneity restriction of equal coeffi-
cient vectors in the multivariate regression model. That case is fairly common in this
setting. The homogeneity restriction is that β
i
= β
M
, i = 1,...,M −1. Consistent with
(10-13)–(10-14), we would form the hypothesis as
Rβ =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
I0··· 0 −I
0I··· 0 −I
···
00··· I −I
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎛
⎜
⎜
⎜
⎜
⎝
β
1
β
2
···
β
M
⎞
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎝
β
1
− β
M
β
2
− β
M
···
β
M−1
− β
M
⎞
⎟
⎟
⎟
⎟
⎠
= 0. (10-15)
This specifies a total of (M −1)K restrictions on the KM ×1 parameter vector. Denote
the estimated asymptotic covariance for (
ˆ
ˆ
β
i
,
ˆ
ˆ
β
j
) as
ˆ
V
ij
. The bracketed matrix in (10-13)
would have typical block
[R
(
Var[
ˆ
ˆ
β]R
]
ij
=
ˆ
V
ij
−
ˆ
V
iM
−
ˆ
V
Mj
+
ˆ
V
MM
This may be a considerable amount of computation. The test will be simpler if the
model has been fit by maximum likelihood, as we examine in Section 14.9.3. Pesaran
and Yamagata (2008) provide an alternative test that can be used when M is large and
T is relatively small.
10.2.5 A SPECIFICATION TEST FOR THE SUR MODEL
It is of interest to assess statistically whether the off diagonal elements of are zero. If
so, then the efficient estimator for the full parameter vector, absent heteroscedasticity or
autocorrelation, is equation by equation ordinary least squares. There is no standard test
for the general case of the SUR model unless the additional assumption of normality of
the disturbances is imposed in (10-2) and (10-3). With normally distributed disturbances,
the standard trio of tests, Wald, likelihood ratio, and Lagrange multiplier, can be used.
For reasons we will turn to shortly, the Wald test is likely to be too cumbersome to apply.
With normally distributed disturbances, the likelihood ratio statistic for testing the null
hypothesis that the matrix in (10-5) is a diagonal matrix against the alternative that
is simply an unrestricted positive definite matrix would be
λ
LR
= T[ln |S
0
|−ln |S
1
|], (10-16)
14
See Judge et al. (1985, p. 476). The Wald statistic often performs poorly in the small sample sizes typical in
this area. Fiebig (2001, pp. 108–110) surveys a recent literature on methods of improving the power of testing
procedures in SUR models.
298
PART II
✦
Generalized Regression Model and Equation Systems
where S
1
is the residual covariance matrix defined in (10-9) (without a degrees of free-
dom correction). The residuals are computed using maximum likelihood estimates of
the parameters, not FGLS.
15
Under the null hypothesis, the model would be efficiently
estimated by individual equation OLS, so
ln |S
0
|=
M
i=1
ln (e
i
e
i
/T ),
where e
i
= y
i
−X
i
b
i
. The limiting distribution of the likelihood ratio statistic under the
null hypothesis would be chi-squared with M(M − 1)/2 degrees of freedom.
The likelihood ratio statistic requires the unrestricted MLE to compute the residual
covariance matrix under the alternative, so it is can be cumbersome to compute. A
simpler alternative is the Lagrange multiplier statistic developed by Breusch and Pagan
(1980) which is
λ
LM
= T
M
i=2
i−1
j=1
r
2
ij
(10-17)
= (T/2)[trace(R
R) − M],
where R is the sample correlation matrix of the M sets of T OLS residuals. This has the
same large sample distribution under the null hypothesis as the likelihood ratio statistic,
but is obviously easier to compute, as it only requires the OLS residuals.
The third test statistic in the trio is the Wald statistic. In principle, the Wald statistic
for the SUR model would be computed using
W = ˆσ
[Asy. Var( ˆσ )]
−1
ˆσ ,
where ˆσ is the M(M − 1)/2 length vector containing the estimates of the off-diagonal
(lower triangle) elements of , and the asymptotic covariance matrix of the estimator
appears in the brackets. Under normality, the asymptotic covariance matrix contains the
corresponding elements of 2 ⊗ /T. It would be possible to estimate the covariance
term more generally using a moment-based estimator. Because
ˆσ
ij
=
1
T
T
t=1
e
it
e
jt
is a mean of T observations, one might use the conventional estimator of its variance
and its covariance with ˆσ
lm
, which would be
f
ij,lm
=
1
T
1
T − 1
T
t=1
(e
it
e
jt
− ˆσ
ij
)(e
lt
e
mt
− ˆσ
lm
). (10-18)
The modified Wald statistic would then be
W
= ˆσ
[F]
−1
ˆσ
15
In the SUR model of this chapter, the MLE for normally distributed disturbances can be computed by
iterating the FGLS procedure, back and forth between (10-7) and (10-9) until the estimates are no longer
changing. We note, this procedure produces the MLE when it converges, but it is not guaranteed to converge,
nor is it assured that there is a unique MLE. For our regional data set, the iterated FGLS procedure does not
converge after 1,000 iterations. The Oberhofer–Kmenta (1974) result implies that if the iteration converges,
it reaches the MLE. It does not guarantee that the iteration will converge, however. The problem with this
application may be the very small sample size, 17 observations. One would not normally use the technique
of maximum likelihood with a sample this small.