Abstract
The ith structural equation of the SEM can be written as
where 
is the dependent vector of the ith structural
equation, 
is the full column rank exogenous 
matrix, 
is the observation 
matrix of other
endogenous variables included in the ith equation, 
and
are the structural parameters vector and
are the disturbances. For 
and 
, the stacked
system of the structural equations can be written as
where 
, X is a 
matrix of all predetermined
variables, 
, 
with
be a selector matrix such that 
( 
),
and
.
The disturbance vector 
satisfies 
and 
, where 
is
a non-negative definite matrix.
The 3SLS estimator, denoted by 
, is the GLS estimator of
the transformed SEM
where is replaced by its consistent estimator 
based on
2SLS residuals. The QR decomposition of X is given by
is orthogonal, 
is upper triangular
non-singular matrix, 
and 
.
Computing the Cholesky decomposition 
, the
estimator derives from the solution of the normal equations
where C is a non-singular 
upper triangular matrix,
and 
[Belsley1992, Jennings1980].
For singular or ill-conditioned the above estimation procedure
fails, since 
does not exist. However, this can be
overcomed by rewriting the transformed SEM (3) in the
equivalent form
where the rank of 
is 
, 
has a full column rank and V is a random gK element
vector with zero mean and variance-covariance matrix 
,
defined as 
. Under this
formulation the 3SLS estimator of 
is the solution of the
generalized linear least squares problem
The latter does not require the variance-covariance matrix to be non-singular [Kontoghiorghes and Clarke1995, Kontoghiorghes and Dinenis1996, Kourouklis and Paige1981].
Using the QR decomposition as a basic tool, firstly the numerical
solution of (7) for computing the 3SLS estimator
and its dispersion matrix is presented; the computation of and
redundancies in the SEM are discussed [Golub and Van Loan1983, Kontoghiorghes1995].
Secondly, computational formulae for modifying the SEM after adding or
deleting observations or variables are investigated. Thirdly, the
solution of the SEM with linear equality constraints is considered.
The first approach uses the constraints as additional precise
observations, while the other two approaches reparametarize the
constraints to solve a reduce unconstraint SEM. Finally, sequential
and parallel strategies are proposed for computing the main
factorisations employed to solve (7), followed by
the conclusions and future work, where inconsistencies in the SEM and
implementation issues of the various algorithms are discussed.
