journal of forest science

JOURNAL O OREST SCIENCE, 47, 2001 (7): 285–293
Simultaneous modelling of stand volume yield, dominant height
and basal area growth models
L.YUANCAI 1 , C. P. MARQUES 2 , J. M. BENTO 2
1 Research Institute of Forest Resource Information Technique, Chinese Academy of Forestry, Beijing, China
2 Department of Forestry, Universidade de Trás-os-Montes e Alto Douro, Vila Real, Portugal
ABSTRACT: Forest growth and yield models are often composed of a system of compatible, interdependent and analytically
related equations rather than of a single equation. Most forest researchers have recognized that the forest stand dynamics
system should be described by a system of simultaneous and interdependent equations. The two- and three-stage
least squares (2SLS and 3SLS) and seemingly unrelated regression (SUR) techniques from econometrics have been widely
used to estimate the coefficients of forest growth and yield model system. This study 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 by initial stand conditions. The coefficients of the system are
estimated from Eucalyptus globulus Labill. stands in the Central inland of Portugal by using 3SLS, SUR and OLS, respectively.
The three methods are evaluated and compared on some statistic indicators. The results indicate that there is a small
difference between the three approaches in the particular case, but the system estimation methods perform better for
a system of simultaneously interdependent equations in theory. Therefore, the appropriate system estimation approaches
are recommended for estimating coefficients in simultaneously interdependent systems of forestry equations.
Keywords: comparison; growth and yield model; simultaneous equation
Modelling methodology of growth models at present
is becoming increasingly sophisticated as forest biometricians
put forward new biological rationale, advanced
statistical techniques, and powerful computing technology
to solve growth and yield prediction problems.
An extensive variety of growth and yield models have
been developed. However, growth and yield models,
which describe forest stand dynamics by using data from
permanent sample plots or inventory sample plots with
real growth series, are often composed of a system of
compatible, interdependent and analytically related equations
(CLUTTER 1963; SULLIVAN, CLUTTER 1972; BOR-
DERS, BAILEY 1986; BURKHART 1986; REED 1986;
DANIELS, BURKHART 1988). Most forest researchers
have recognized that there is a strong correlation and
feedback mechanism between variables that are used to
describe growth and yield relationships. Therefore, forest
stand dynamics should be described by a simultaneous
and interdependent system of equations rather than
by separate and isolated individual equations.
Financial support for this work was partially provided by the NSFC program (30070620).
In forest biometrics, there are two categories of methods
for estimating simultaneous and interdependent systems
of equations.
One method of estimating parameters in a system of
equations is to fit one (or more) of the equations using
Ordinary Least Squares (OLS) techniques and to solve
for the coefficients in the other(s) by invoking the specified
algebraic relationships between parameters (SULLI-
VAN, CLUTTER 1972). This solution has obvious
shortcomings. As the parameter estimates are stochastic,
the final result will depend on the arbitrary choice which
equation(s) is(are) fitted and which is(are) derived. Thus,
BURKHART and SPRINZ (1984), REED and GREEN
(1984), BYRNE and REED (1986), KNOEBEL et al. (1986)
and REED (1986) simultaneously estimated the structural
parameters by minimizing the squared error loss function,
which brought about a substantial improvement for
the total system. However, BORDERS and BAILEY (1986)
and LEMAY (1990) indicated that estimates obtained by
minimizing squared error loss functions may not be con-
J. FOR. SCI., 47, 2001 (7): 285–293 285