journal of forest science

sistent or most efficient if the equations are simultaneous.
The reason is that the method does not use the information
on correlation between residuals of the remaining
resultant estimators. VAN DEUSEN (1988) discussed the
relationship of the minimization of error loss functions
to the Seemingly Unrelated Regression (SUR) fitting
technique, and showed that estimation based on minimizing
squared error loss functions is very similar to the
SUR techniques from econometrics, and suggested that
if the system of equations belongs to the SUR framework,
estimates of the structural parameters will be consistent
and asymptotically efficient.
The other method of simultaneous fitting techniques
widely used in econometrics is the so-called system method.
or the system of forestry equations, URNIVAL and
WILSON (1971) have considered it as the first application
of econometric fitting techniques to forestry systems
to correct for simultaneity bias and contemporaneous
correlation. Many other researchers have also discussed
and used two and three-stage least squares (2SLS and
3SLS), and SUR techniques from econometric methods
to estimate the structural parameters in a system of forestry
equations (e.g. MURPHY 1983; AMATEIS et al.
1984; BURKHART 1986; BORDERS, BAILEY 1986; REED
1986; VAN DEUSEN 1988; BORDERS 1989; SOMERS,
ARRAR 1991; LYNCH, MURPHY 1995; AMARO 1997;
ZHANG et al. 1997). BORDERS (1989) discussed in detail
three types of equation systems which are: (i) seemingly
unrelated regression equations (SUR); (ii) recursive
systems of equations; (iii) system of simultaneous equations,
and described associated fitting methods in different
types of equation systems, as well as an alternative
parameter estimation procedure that can be used for any
number of analytically related linear or non-linear equation
systems.
It is evident from the given examples that a wide variety
of cases exists where the systems of related equations
are used in forestry. No single model can be expected to
be the “best” for all the purposes that are practical, precise
and common in forest prediction (BURKHART 1997).
Therefore, when a growth and yield model is selected for
forest prediction, one must think of a system of growth
and yield models capable of providing logical and consistent
estimates for varying degrees of stand details.
Based on the idea, this study (i) analyzes a system of
three interdependent equations for predicting the future
stand volume yield, the future stand dominant height and
the future stand basal area for eucalyptus plantations as
an integrated system given initial stand conditions;
(ii) applies some statistical methods to simultaneously
estimate the structural parameters of the system. The
cross-equation correlations and simultaneous nature exist
between the error components of these equations. No
attention is paid to the fact that the error terms from related
equations are almost certain to be correlated, and
the available information on the cross-equation interdependence
is not utilized; (iii) evaluates different system
estimation methods for the system of nonlinear equations
and compares them with the traditional single-equation
technique based on OLS.
SYSTEM O EQUATIONS
VARIABLES O EQUATION SYSTEMS
In systems of interdependent equations, the variables
occurring on the left-hand side (LHS) are referred to as
endogenous variables (or dependent variables). They are
the system outputs and are assumed to be determined by
the joint interaction with other variables within a system.
These endogenous variables may also appear as predictor
variables on the right-hand side (RHS) of equations
within the system. All variables that only appear on the
RHSs of the equations in the system are referred to as
exogenous or predetermined variables that are determined
outside the system.
CLASSIICATIONS O EQUATION SYSTEMS
Pindyck and Rubinfeld in 1981, as referenced in BOR-
DERS (1989), classified the systems of related equations
into three categories: (i) seemingly unrelated equations,
(ii) recursive equations and (iii) simultaneous equations.
The nature of each type of equation system is different
and it is very useful to understand the nature of different
systems of equations for fitting them.
The types of structural models in different systems of
equations are discussed and described below. The position
of the zero elements in the B matrix indicates which
endogenous variables do not appear in different structural
equations. This is used as a criterion for distinguishing
between various types of structures of different equation
systems.
(i) System of seemingly unrelated equations:
If B is diagonal, i.e., if:
0 ......... 0
0 B ...... 0
22
B = .
.
0 0.......... Bii Only one endogenous variable appears in each equation.
This means that the equations are not simultaneous
but seemingly related. A typical form of the seemingly
unrelated equation system was discussed by BORDERS
(1989). Obviously, there are no analytical relationships
between equations if cov(e ,e ) = 0 for all pairs of error
i j
terms i and j. If cov(e ,e ) ≠ 0, then the equations are
i j
related. When such cross-equation correlations are
present, a three-step fitting procedure discussed by REED
(1986) may be desirable for parameter estimation of this
type of equation system.
Seemingly unrelated regression estimates are obtained
by estimating a set of nonlinear equations with crossequation
constraints imposed, but with a diagonal covariance
matrix of the disturbances across equations. These
286 J. FOR. SCI., 47, 2001 (7): 285–293
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