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Econometrica, Vol. 68, No. 3 May, 2000 , 715᎐719
ROBUST WALD TESTS IN SUR SYSTEMS WITH
ADDING-UP RESTRICTIONS
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BY B. RAVIKUMAR, SURAJIT RAY, AND N. EUGENE SAVIN
1. INTRODUCTION
FOR SUR SYSTEMS WITH ADDING-UP RESTRICTIONS, it is well known that the covariance
matrix of disturbances is singular. The usual approach to hypothesis testing in such cases
is to construct the relevant test statistics after deleting an equation. A common applica-
tion of this approach is in the context of complete demand systems where the sum of
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expenditure shares must equal one. Barten 1969 considered the maximum likelihood
estimation of such a system of equations with independent and identical normal distur-
bance vectors. He proved that the value of the likelihood function, and hence, the
maximum likelihood estimates of the parameters are invariant to the equation deleted.
This, in turn, implies that the value of the likelihood ratio statistic for testing linear
restrictions on the coefficients is invariant to the equation deleted. Similarly, McGuire,
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Farley, Lucas, and Ring 1968 and Powell 1969 considered the Generalized Least
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Squares GLS estimation of a system of demand equations. Under the assumption that
the covariance matrix of the stacked disturbance vector is known, they showed that the
GLS estimator and the corresponding quadratic form are invariant to the equation
deleted. Estimation and testing have been extended to SUR systems with specific forms
of heteroskedasticity andror autocorrelations; see, for instance, Mandy and Martins-Filho
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1993 and Berndt and Savin 1975 .
In practice, the likelihood function andror the covariance matrix of the stacked
disturbance vector are usually unknown. Similarly, the functional form of heteroskedas-
ticity andror autocorrelations is also unknown. In this paper, we consider SUR systems
with adding-up restrictions where the same explanatory variables are present in all
equations and where heteroskedasticity andror autocorrelation of unknown forms may
be present. For this case, the coefficients are usually estimated by least squares, equation
by equation. For testing the typical hypotheses of interest, we show that the robust Wald
statistic, i.e., the statistic based on the heteroskedasticity and autocorrelation consistent
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HAC covariance matrix estimator, is invariant to the equation deleted. Our proof of
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invariance does not rely on parametric assumptions as in Barten 1969 or on knowledge
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of the covariance matrix as in Powell 1969 . Furthermore, the adding-up restrictions we
consider are more general than Barten’s. As in Powell, the weighted sum of the
dependent variables in this paper adds up to one of the explanatory variables, not
necessarily a constant. Our proof exploits the properties of generalized inverses and
depends only on the existence of first and second moments. It should be noted that even
though our robust Wald test is invariant to the equation deleted, it is not invariant to
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nonlinear transformations of the null hypothesis see Gregory and Veall 1985 .
1 We gratefully acknowledge the helpful comments of a co-editor and the referees. In particular,
we thank an anonymous referee for suggesting a shorter and more elegant proof of our main
theorem.
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