In a regression model with an arbitrary number of error
components, the covariance matrix of the
disturbances has three equivalent representations as
linear combinations of matrices. Furthermore, this
property is invariant with respect to powers, matrix
addition, and matrix multiplication. This result is
applied to the derivation and interpretation of the
inconsistency of the estimated coefficient variances
when the error components structure is improperly
restricted. This inconsistency is defined as the
difference between the asymptotic variance obtained
when the restricted model is correctly specified,
and the asymptotic variance obtained when the
restricted model is incorrectly specified; when some
error components are improperly omitted, and the
remaining variance components are consistently
estimated, it is always negative. In the case where
the time component is improperly omitted from the
two-way model, we show that the difference between
the true and estimated coefficient variances is of
order greater than N–1
in probability.