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Online Econometrics Textbook - Regression Extensions - Seemingly unrelated regression (SUR)

[Home] [Up] [Assumption Violations] [Simultaneity] [Multicollinearity] [Restricted LS] [Distributed Lags] [SUR]

[Pooling I] [Pooling II]

Consider a system of M equations without any feedback mechanisms ordered as

(III.II-1)

or simply as

(III.II-2)

The error-covariance matrix is assumed to be of the form

(III.II-3)

Above (III.II-3), it is also assumed that the error terms of each equation have a zero mean.

The GLS estimator is given by

(III.II-4)

which is a BLUE.

Under one of the following two conditions, OLS applied to each equation is equivalent with GLS (in the context of SUR)

The proof of this last condition is quite simple. Since it is assumed that

(III.II-5)

we can write the GLS estimator of (III.II-4) as

(III.II-6)

(III.II-7)

In general (III.II-7) is biased since different equations may have a different number of regressors. Therefore we might use a formula which attempts to correct for this bias:

(III.II-8)

Of course it also possible to apply an iterative MLE algorithm to compute SUR if the random errors are normally distributed.
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Assumption Violations
Simultaneity
Multicollinearity
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Distributed Lags
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Pooling I
Pooling II
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