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 application of this approach is in the context of complete demand systems where the sum of expenditure shares must equal one. Barten (1969) considered the maximum likelihood estimation of such a system of equations with independent and identical normal disturbance 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, Farley, Lucas, and Ring (1968) and Powell (1969) considered the Generalized Least 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 and/or autocorrelations; see, for instance, Mandy and Martins-Filho (1993) and Berndt and Savin (1975).
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