A COMPARISON .OF ALTERNATIVE PRODUCTION FUNCTION MODELS
USING NON-NESTED HYPOTHESIS TESTS
by
Mary Ellen Embleton
A thesis submitted in partial fulfillment
of the requirements for the degree
of
Master of Science
in
Applied Economics
MONTANA STATE UNIVERSITY
Bozeman, Montana
September 1987
i i
APPROVAL
of a thesis submitted by
Mary Ellen Embleton
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and has been found to be satisfactory regarding content, English usage,
format, citations, bibliographic style, and consistency, and is ready
for submission to the College of Graduate Studies.
Date
Chairperson, Graduate Committee
Approved for the Major Department
Date
Head, Major Department
Approved for the College of Graduate Studies
Date
Graduate Dean
i i i
STATEMENT OF PERMISSION TO USE
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the
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Permission
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Signature _______________________________
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iv
ACKNOWLEDGMENTS
I
Dr.
would
like to express my appreciation to my committee
Bruce Beattie,
members,
Dr. Michael Frank and Dr. John Marsh for their time
and commitment during work on this thesis.
Special
thanks
go
to
understanding over the years.
my
family
for
their
encouragement
and
v
TABLE OF CONTENTS
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STATEMENT OF PERMISSION TO USE ............................ .
i ; i
ACKNOWLEDGMENTS ........................................... .
iv
TABLE OF CONTENTS ......................................... .
v
LIST OF TABLES .................................... , ....... .
vii
LIST OF FIGURES............................................
viii
ABSTRACT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
;X
CHAPTER
1.
INTRODUCTION .................................... .
1
Previous Studies .............................. .
Overview and Objectives of Study ............. .
Study Objectives .......................... .
outline of Thesis ............................ .
2
7
8
9
11
REGULARITY CONDITIONS: MONOTONICITY AND
CONCAVITY ....................................... .
13
Method a 1 ogy .· ..............................
2.
Regularity Conditions ........................
Selected Functional Forms ....................
The a~adratic Production Function .........
The Transcendental Production Function ....
The Translog Production Function ..........
The Spillman Production Function ..........
The von Liebig Production Function ........
3.
.
.
.
.
.
.
.
.
21
22
23
ECONOMETRIC THEORY AND MODEL SELECTION .......... .
. 25
The General Statistical Model ................ .
Model Selection: Nested Models ........... .
26
27
14
17
17
19
vi
TABLE OF CONTENTS-Continued
Page
4.
5.
Model Selection: Non-Nested Models ....... .
Interpretation of Non-Nested Hypothesis Tests.
28
EMPIRICAL RESULTS ............................... .
33
The Empirical Models ......................... .
Empi rfca 1 Results ............................ .
Model Selection Test Results ................. .
33
42
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS ........ .
49
~ecommendations
for Further Study ......
~··
31
37
... .
53
REFERENCES ................................................ .
58
APPENDICES ................................................ .
65
A:
Data Set . ....................................... .
66
8:
Calculation of the Standard Errors for the
von Liebig Model ................................ .
68
Calculation of the Non-Nested Test for the
von Liebig Model ................................ .
72
C:
Vii
Ll ST OF TABLES
Table
1.
Page
Production Function Estimates and Related
Statistics ....................................... .
38
2.
Model Selection Test Results .........•............
43
3.
Profit Maximizing Values for the Various
Functions ......................... ·............... .
51
4.
Values Implied by Translog Function .............. .
52
5.
Experimental Yields of Corn for Varying Levels
of Fertilizer Inputs ............................. .
65
viii
LIST OF FIGURES
Figure
Page
1.
Strict Concavity ................................. .
16
2.
Strict Quasi-Concavity .......................... .
18
3.
Three Examples of an 5-Shaped Production
4.
Surface ......................................... .
20
Estimated Response Surface for the Translog
Function ........................................ .
48
ix
ABSTRACT
The selection of an appropriate mathematical form for a production
process is critical in applied research.
The practical problem is that
little is known, a priori, about the process which generated the sample
data.
Often, the researcher is faced with several alternative or
competing specifications which purport to explain the same phenomena.
The purpose of model selection tests is to "best" describe the
production process under consideration.
If the models belong to the
same parametric family, the problem is one to which standard hypothesis
testing procedures may be applied.
When the models belong to different
parametric families, additional methods for testing model specification,
based on the fundamental work of Cox, can be used.
These tests are
referred to as non-nested hypothesis tests.
In this study, five production function models have been chosen to
represent crop response to fertilizer inputs.
They are the quadratic,
transcendental, translog, Spillman and von Liebig functions. Non-nested
hypothesis tests are carried out to determine which of the competing
models best describes the production process which generated the sample
data. The results of the tests indicate that the translog is the
superior specification.
In particular, the results support input
substitution and a plateau response with respect to phosphorous. With
regard to nitrogen, the results indicate that the res~onse surface
increases at a decreasing rate for low levels of the input and decreases
at an increasing rate at higher levels.
CHAPTER 1
INTRODUCTION
Two fundamental concerns of production economists are the efficient
choice of production processes,
function,
and
Selection of an appropriate mathematical form for
resource allocation.
the
given by the production
production process is critical in applied research.
This includes
micro (firm level) as well as macro (policy oriented) problems.
The
selection
production
of
process
any specific functional form to
imposes certain constraints
involved and optimum resource use.
on
the
represent
relationships
The practical problem is to select a
mathematical specification which appears,
or is known, to be consistent
with the production phenomena under investigation.
This can
difficult
competing
cations.
simpl~
While
when
there
the
are several alternative
or
be
quite
specifi-
Mathematical specifications of production functions range from
single-equation models to very complex multiple-equation models.
choice between competing specifications is
appropriate
functional
form
is rarely apparent
necessary,
a
the
most
However,
priori.
guides to the appropriate functional form may come from previous experimentation, economic theory, or environmental factors.
Economic theory provides insight in model specification,
choice of functional form.
bit
certain
important
in
general
the
including
An appropriate functional form should exhi--
properties or principles that are thought
economic,
as well as
technical,
analysis
to
of
be
the
2
particular
production process under study.
These technical properties
can be derived in general terms or through formal estimation of
produc-
tion functions.
When
ately
a model(s) has been specified such that it appears to
represent
the
phenomena under consideration and
properties have been derived,
chosen
parameters,
with
technical
it is desirable to determine how well the
mathematical specification(s) performs.
other things,
the
This
includes,
assessment of the statistical precision of the
determination
economic
theory,
accur-
among
estimated
of how well the results of estimation comply
and formal hypothesis tests to
choose
between
competing models.
Previous Studies
Most
nature
soil
of
the early work directed toward specifying
algebraic
of production functions was made by agricultural economists
scientists
attempting
to define the
yields and fertilizer or nutrient inputs.
formulating
(1840,
time
the
crop
It
1855).
that
growth.
"Liebig's
response
relationship
between
Law
crop
One of the first attempts at
to nutrient inputs was made by
von
Liebig
was a widely held belief of soil scientists of
there existed an absence of nutrient substitution
Liebig
and
proposed
a
model,
of the Minimum",
which has come
to
in
be
which maintains that plant
the
plant
known
as
growth
is
directly proportional to the supply of the nutrient present in the ·least
amount.
describes
Another
The
a
model
allows
for
no substitution
between
inputs
response surface which is linear and
reaches
a
early effort to estimate
and
plateau.
agricultural production functions was
3
made
by
Spillman (1923,
empirical
support for the Law of Diminishing Returns.
exponential
ments
Spillman was interested
1924).
on
yield equation based on the results of
marginal
agricultural
He proposed
fertilizer
physical productivity (MPP) but
negative MPP when the input level gets large.
early
production
finding
functions,
does
an
expert-
The Spillman function allows
cotton in North Carolina.
diminishing
in
not
for
allow
In addition, unlike many
the
coefficients
of
the
function were presumed tp vary with environmental conditions.
important
Another
work
in
the
area
specification was by Cobb and Douglas (1928).
a general power function to data for U.S.
the
of
production
function
Cobb and Douglas applied
manufacturing industries
for
years 1899-1922 in an attempt to compute the actual total shares of
product
attributable
selected
a
functional
elasticities
among
to
the two
form
to unity,
inputs,
capital
that restricted the
and
sum
labor.
of
which meant that the division of
the
factor
total
output
capital and labor was such that it just exhausted total
However,
they
specified
general
stated
power function,
creasing
function,
product.
that alternative forms of the function could
which would not require constant
production
They
shares.
The
be
unrestricted
which has come to be known as the Cobb-Douglas
can exhibit either constant,
increasing
marginal physical productivity and returns to scale.
or
de-
That is,
depending on parameter restrictions, the function can either increase at
an increasing, constant, or decreasing rate under single factor or scale
variation.
The Cobb-Douglas function also assumes a constant elasticity
of production.
4
Among
growing
Soil
soil
scientists and agricultural economists,
debate
about the nature of crop response to
there· was
nutrient
a
inputs.
scientists held that crop response was of the plateau type,
while
many agricultural economists supported polynomial specifications due
their
relatively good fit and computational ease.
problem .for
the
applied
researcher
when
to
This often posed
attempting
to
a
choose
an
appropriate model.
The
problem of selection among alternative
studied
by
Johnson
(1953).
production models
He considered the problems
involved
choosing particular algebraic specifications to represent the
ship
between yield and fertilizer inputs.
considered:
a
polynomial,
and
fertility
each
general
an
their
(Cobb-Douglas)
experiments in North Carolina,
with
statistical
precision,
relation-
function,
simple
a
Based
on
corn
parameters were estimated for
Yield estimates implied by each
observed yields.
in
Three functional forms were
exponential (Spillman) function.
functional form.
compared
power
was
function
The results were judged in
how well they complied
with
were
terms
of
biological
logic, and the possibility of extrapolation beyond the sample data.
While
the
polynomial specification fit the actual
more closely than the other equations,
t~onal
observations
Johnson argued that the
form did not conform to accepted biological logic.
func-
The power
function fit the data poorly and appeared to be deficient in terms
biological logic as well.
form
was
of
It was argued that the Spillman exponent1al
was more in accord with biological logic,
the most suitable for extrapolation.
fit the data well and
5
Another
carried
out
applied
They
better,
model
specification
They proposed a
functions of the von Liebig type.
experimental
to
systems.
study in the area of
by Lanzer and Paris (1981).
response
estimating
or
important
data
obtained
on
was
method
for
The model
was
Brazilian
intercropping
found that von Liebig type functions performed as well,
than
traditional polynomial specifications which
did
not
incorporate the agronomic principles of nutrient non-substitution and
a
linear plateau response.
Numerous
appropriate
comparing
other studies have addressed the problem of selecting
mathematical specification for the production function
the
considerable
performance
of
alternative
with
analysis
of
ducts,
authors
was
Also,
work has been devoted to developing the technical
proper-
the
Among the first authors to deal,
efficient choice of production
functions
and
Heady
(1952).
In a later text with
Dillon
applications of production function studies,
pro-
(1961),
the basic concepts of function selection
to
data
in
the
the mathematical relationships between resources and
expanded
economic
and
specifications.
ties of various functional forms.
general,
an
the
inclu~e
collection
and finalysis for function estimation, and some of the empirical problems
associated
with
considerable
properties
estimation.
detail,
of
the
production
isoquant patterns,
Beattie and Taylor
mathematical
functions,
derivation
such as
(1985)
of
stages
present,
the
of
factor elasticities and returns to scale.
in
technical
production,
They also
provide intuitive and theoretical motivation for the comparative static,
economic
optimization
principles and results that
alternative production function specifications.
are
implied
given
6
The
selection
production
ships
conventional
of
represent
relation-
such
as
the
and
Cobb-Douglas
constant
production
elasticities
of
This realization has motivated substantial research in the
production process.
production
forms
assumes a single stage
specifying
limitations
the
Recognition has been given to the limitations of many
functional
which
production.
area
to
process imposes certain assumptions regarding the
involved.
function
of any specific functional form
more
general,
Halter,
flexible forms
represent
the
aware of
the
Carter and Hocking (1957),
of the Cobb-Douglas function,
function.
to
proposed the
transcendental
The transcendental function allows for
variable
production elasticities and three stages of production, yet remains easy
to estimate from agricultural data.
be
obtained
In addition,
the Cobb-Douglas can
from the transcendental function by
parameter . restrictions.
Christensen,
imposing
appropriate
Jorgenson and Lau (1971,
1973)
proposed an alternative specification that permits disaggregation of the
factor indices and a variety of substitution possibilities.
posed
They
pro-
the transcendental logarithmic production function (translog),
member
of the generalized power production functions,
linear and quadratic terms.
the C.E.S.
Cobb-Douglas
As
Which
has
a
both
special cases, the translog reduces to
(constant elasticity of substitution) and the multiple-input
production
functions.
The flexibility of
the
translog
function has made it a popular specification for representing production
processes.
Berndt
imposes
no
and Christensen (1973) used the translog function,
separability
restrictions
a
priori,
possibilities of substitution between equipment,
to
explore
which
the
structures and labor
7
in
the
u.s.
Separability
manufacturing
can
be
imposed
industry
on the
for
the
translog
years
function
1929~1968.
by
making
appropriate restrictions on the parameters.
Guilkey and Lovell (1980) used Monte Carlo experiments to
the
usefulness of two different translog estimating models,
evaluate
a
single-
equation model and multiple-equation model, to represent technologies of
varying degrees of complexity, substitution possibilities and returns to
The translog function can be used to model a variety of produc-
scale.
since
processes
tion
it
leaves
separability
and
substitution
possibilities as hypotheses to be tested rather than accepted a
priori.
Accordingly, Guilkey and Lovell also sought to test whether the translog
function
performed better than more conventional specifications.
They
found that both models provided dependable (in the sense that the models
were
able
to
reject false hypotheses) estimates
substitution properties,
of
both
scale
and
and that the accuracy of the estimates held up
well under increasing complexity of the functional form being estimated.
Because
lower
of its relative simplicity,
cost,
marginally better performance
and
use of the single-equation translog model was recommended.
Guilkey and Lovell note the inability of the translog function
However,
(a second order approximation) to capture wide departures from unity
of
returns
of
to
scale
and substitution elasticities and
the
tendency
estimates of mean returns to scale to be biased upwards.
Overview and Objectives of Study
There
tion
are countless possible algebraic specifications for
functions,
produc-
exhibiting a wide range of technical coefficients
and
8
In
conditions.
addition
satisfaction
of
concavity),
each
to the parameter restrictions
given
regularity
conditions
required
for
(monotonicity
imposes particular properties with respect
and
to
the
technical characteristics of the function, for example, production elasticities and returns to scale.
In this study,
represent
alternative
differences
specifications
more
in
each
of
observed
case.
However,
specification
Therefore,
assumptions.
and
maintained
inputs.
A two-factor
most
mathematical
three
production
studies.
implicitly assumes specific relationships
it
or
The chosen models represent a broad cross-
of functional forms found in agricultural
factors.
The models
and results can be generalized to models with
explanatory variables.
section
Each
specifications
in plant yield response to nutrient
presumed
is
model
several functional forms are evaluated.
between
is important to examine the validity of
these
When functions are members of the same parametric family,
referred to as nested models,
statistical testing procedures.
this can be accomplished through standard
However, other functional forms are not
members
of the same parametric family and classical hypothesis
methods
cannot be used.
testing
In such cases an alternative approach
non-nested hypothesis testing,
first developed by Cox (1961,
called
1962), is
applied.
Study Objectives
The
mathematical specifications analyzed in this study include the
quadratic, transcendental, translog, Spillman, and von Liebig production
functions.
cation,
The quadratic function is a second order polynomial specifi-
while the translog function is a polynomial in logarithms.
The
9
transcendental function is a general power function with an
term.
exponential
The Spillman is an exponential function and the von Liebig repre-
sents a linear spline.
Specific study objectives are to:
1.
Establish
faction of
2.
the
parameter restrictions implied by
satis-
monotonicity and concavity restrictions.
Estimate
the parameters of each functional form using experi-
mental data on corn yield
and
the
vs.
plant nutrients (Heady, Pesek and Brown)
evaluate the consistency of the parameter estimates
with
accepted
economic theory.
3.
Statistically
describes
the
test
which
production
model
func~ion
process which generated the sample data
"best"
using
testing
procedures first developed by Cox for non-nested models.
Methodology
It
is
functional
ciently
desirable,
forms,
native
the problem under investigation.
most
common
in agricultural production studies for choosing among
alter-
treated
significance
studies
as
a
The
has been to select the model with the
among
the regression
highest multiple correlation coefficient (R 2 ).
previous
highest
coefficients
in that the choice among alternative models
problem
of
model
evidence
apparent
and
the
This study differs from
specification.
There
questions that may be important when comparing models:
there
among
suffi-
models
statistical
choosing
to determine which of the alternative models
describes
procedure
when faced with the problem of
are
will
be
several
for example, is
provided by the data to suggest that one model
is
the
true model or do the models give significantly different results for the
10
data?
the models belong to the same family of
If
problem
is
However,
one
distributions,
to which standard hypothesis testing can
be
the
applied.
if the models belong to separate families of distributions, an
Such a method for testing separate
alternative approach must be taken.
families of hypotheses was first developed by Cox (1961, 1962).
proposed
Cox
other
hypptheses
general
procedures
for
testing
non-nested
One is based on the Neyman-Pearson likelihood ratio, while
hypotheses.
the
two
is
based
have
on an artificial compound
been
included.
Atkinson
model
(1970)
in
which
elaborated
on
both
the
procedure by which two models are combined into one comprehensive model.
Pesaran (1974) and Pesaran and Deaton (1978) extended Cox's approach
linear
of
regression models under classical assumptions,
first-order
autocorrelation,
Davidson
MacKinnon
to
nonlinear
(1981,
for the presence
and
multivariate
1983)
provided simplified
procedures,
both conceptually and computationally,
based on artificial
nesting
of
the competing hypotheses.
McAleer
(1981) derived a test which is asymptotically equivalent to the
models.
and
and
to·
McAleer (1981)
and
Fisher
and
earlier tests and is valid for small. samples.
Although
procedures
apply.
to
equivalent
tests
developed
MacKinnon
nonlinear
useful
later
models,
were
by Cox,
(1983)
complicated
than
the
they could still be quite
outlined
techniques~
which are easy·to employ
the applied researcher.
to
less
the tests developed
earlier.
difficult
both for
Fisher
to
linear
and
sense
and
tn a practical
The tests are also
initial
asymptotically
(1983)
provided
11
simplified criteria for non-nested tests and emphasized the unity underlying the various procedures.
Specific
applications
of these tests have been made
by
Aneuryn-
Evans and Deaton (1980) to the choice between logarithmic and linear regression models,
gard
and by Ackello-Ogutu, Paris and Williams (1985) in re-
to specifying a crop response function as a polynomial or
plateau
(von Liebig) type model.
In
forms
this
study,
the empirical fit and testing of
the
functional
is carried out using data from an agronomic experiment
involving
the yield response of corn to two variable nutrient inputs, nitrogen (N)
and
potassium ( P2 o5 ) (Heady,
Pesek and Brown).
The. experiments
were
conducted in 1952 with corn on calcareous Ida silt loam in western
Iowa
and
designed
to
The
relationships.
replication.
allow
All
for
the
derivation
of
relevant
production
experimental design included randomized plots
resources
or
inputs
were
held
constant,
and
except
fertilizer and the variable quantities of labor and machine services for
application
and
harvesting.
The
sample data include a total
of
114
observations obtained from an incomplete factorial experimental design.
Outline of Thesis
This
thesis
mathematical
satisfaction
concavity).
considers
the
specification and the parameter restrictions required
for
of
is organized as follows.
given
Chapter
regularity
Chapter
conditions
2
(monotonicity
and
3 outlines the general statistical procedures for
estimation and hypothesis testing.
The procedure for testing non-nested
12
models
is
discussed
Chapter 5.
included.
in
Empirical results of the statistical
Chapter 4.
A summary and conclusions are
tests .are
presented
in
13
CHAPTER 2
REGULARITY CONDITIONS:
MONOTONICITV AND CONCAVITY
The aspects of production function research, including economic and
statistical
specification,
Theoretical
other.
production
tionships
derivation
as
well as
empirical
important
the
estimation
functions provides useful information on input-output
in understanding resource allocation
of
rela-
problems
in
A production function represents a physical relationship
agriculture.
between
are not independent -- each influences
resources and output.
This relationship describes the
formation of resources into a given product.
trans-
Symbolically, a production
function can be written as
t 2 '1 )
where
Y is the product or output and
x
1
,~
•.
~xm
are
the
resources
or
services required to produce V.
Economic
certain
regularity
assumption
with
firms.
theory
requires
that the
production
function
These
conditions
result
conditions.
of a convex technology.
profit-maximizing
That is,
a technology
or cost-minimizing behavior of
satisfy
from
the
consistent
individuals
or
In particular, the function f(o) is assumed to be monotonic and
strictly concave for profit maximization
for cost minimization.
or monotonic and quasi-concave
14
Physical
production is generally a function of many resources
may result in multiple outputs.
This study presumes production function
models where only two resources,
and a single output results.
and
or factors of production, are employed
However,
and results can be generalized to models
the mathematical specifications
~ith
three or more factors.
In
addition, it is assumed that production takes place in a single period.
This chapter presents the mathematical specification of the production
functions considered in this study and the parameter
implied
restrictions
for satisfaction of monotonicity and concavity conditions.
Regularity Conditions
Economic
theory
requires
that each production
function
satisfy
certain regularity conditions to be consistent with an interior solution
for
a
perfectly
equilibrium
competitive, profit-maximizing
for a firm.
or
A function which meets these
cost-minimizing
requirements
is
said to be well-behaved.
The production function should have continuous
first
partial derivatives and be increasing
and
second-order
variable factors at the profit maximizing levels of inputs and
in
the
outputs,
( 2. 2)
However, for large enough values of the inputs, the function can exhibit
negative marginal productivities.
In addition, the Hessian matrix,
formed from the second-order partial derivatives of the function,
( 2. 3)
H
=
H,
15
must
be
negative definite for strict concavity.
That is,
for
all
that the function f(X 1 ,X 2 ) must lie everywhere below
the
(2.4)
This
means
tangent plane defined by the gradient [af(Xl,X2)/aX1,
af(X 1 ,X2)/ax2).
This is illustrated in Figure 1.
At
the cost-minimizing level of input use, the production function
should have continuous first and second-order partial derivatives and be
increasing in the variable factors,
(2.5)
Again,
there
enough
values of the inputs.
from
the
can be a region of negative first derivatives
The bordered Hessian matrix,
Hessian matrix augmented in the last row and
for
B,
column
large
formed
by
the
first partial derivatives of the function,
a 2 ftax 1 ax 2
(2.6)
must
8
=
a 2 ftax 1 ax 2
a 2 ftax~
aftax 1
aftax 2
0
be negative semi-definite for strict quasi-concavity.
(2.7)
+
<
o.
That is,
16
y
Xl
Figure 1.
Strict Concavity
17
Thus,
for changes in the variable factors in a direction tangent to the
contour
that
lines of the function,
(af!X
1
,x~J/aX
1
function decreases.
J
i.e.
perpendicular to the gradient
dX 1 + (af<X 1 ,x 2 Jtax 2 J dX 2
= o,
the value of
so
the
This is illustrated in Figure 2.
Selected Functional Forms
The
choice
of
a
specific
functional form
technical characteristics of the function;
substitution
shape
the
diminishing
marginal
The researcher may choose a von Liebig
response
surface
can
be
represented
by
this
study
maintained
were chosen because they represent
differences
mathematical
in crop response
specifications
to
chosen
both
fertilizer
include
inputs
a
relationship which reaches a plateau at some maximum yield.
in
on
the absence of
if he/she believes that no substitution exists between
that
and
based
The choice can also be made on the basis of the expected
of the response surface.
function
often
for example,
between factors or the existence of
productivity.
is
linear
The models
observed
and
inputs.
The
quadratic,
the
transcendental, translog, Spillman and von Liebig production functions.
This
section
restrictions
includes the algebraic specification
required
for
satisfaction
of
the
and
parameter
given
regularity
conditions for each of the selected models.
The Quadratic Production Function
The
quadratic
specification
allows
ties,
production
function is a
second-order
frequently used in agricultural production
for declining as well as negative marginal physical
is
quite simple computationally,
polynomial
studies.
It
productivi-
and usually provides
a
good
18
\
'/..2
Xl
Figure 2.
'•'
'·
ouasi-cancavitY
I
19
statistical
In
fit.
function is hill-shaped.
reaches
a maximum,
the response surface of the
general~
The function increases at a decreasing
The
then decreases at an increasing rate.
shape of the response
quadratic
surfac~,
as
.i~
rate,
actual
true for .the other models as well,
will depend on the particular values of the parameters involved.
The quadratic function has the. form:
(2.8)
Monotonicity
and
restrictions;
13 0
strict-concavity
>
o.,
>
imply
the
following
o,
parameter
and
Monotonicity and quasi-concavity require only that
>
13 3 , 13 4 <
0.
The Transcendental Production Function
The
transcendental production function is an exponential
function
which allows for variable production elasticities and is usually easy to
estimate. The transcendental function has often been used because of its
flexibility
and
three-stage
because it can .be used to represent
production
function.
The
response
the· neoclassiCal·
surface
of
the
transcendental function is very flexible. Depending on the values of the
parameters,
the function can exhibit a wide range of
example, it can be as simple as the Cobb-Douglas function, which
increases
three
at a decreasing rate,
stage,
s-shaped
this
thesis
in
function surface.
always
or it can exhibit the classical two or
production surface with respect
proportional factor variation.
For
curvat~res.
to
single
or
The term, "s-shaped", is used throughout
describing a classical two or three
stage
production
The term is used in a generic sense to represent one
of either three cases as demonstrated. in Figure 3.
y
y
y
I
I
II
I
I
I
III·
I
I
I
I
I
I
""'------.....:...----.:..'--J"l/X2
Case A.
Two Stages Without Plateau
. Figure 3.
Case B.
Two Stages With Plateau
Three Examples of an 5-Shaped Production Surface
Case C.
Three Stages With Maximum
N
o.
21
The transcendental function, chosen for this study, is given by
(2. 9)
Monotonicity
and
restrictions;
Po
strict~concavity
o, 0
>
<
p1
~·
·imply
1,
o
<
the
P2
~
following·
1,
> 0.
The
tr~nscendental
<
o.
Po> 0 and p1 ,
p2
and
Monotonicity and quasi-concavity requires simply that
parameter
~
p4
3•
function reduces to the Cobb-Douglas when p3
=
The Translog Production Function
The
translog production function was first developed to be a
representation
flexibl~
It
is
of technology than the Cobb-Douglas
frequently used fn production studies because it is
approximation
means),
of
any
(such
without imposing restrictions on other technical
function.
The
flexible,
depending
transcendental,
Douglas,
technology about a base point
more
function.
a
flexible
as
aspe~ts
sample
of. the
response surface ot the translog function is also
on
the
the
translog
values
of
function
the
parameters.
can be reduced
can exhibit the classical s-shape;
to
very
Like
the
the
Cobb-
or exhibit a wide range of
curvatures.
The translog function is a polynomial in logarithms of the form
The translog function cannot globally satisfy monotonicity and concavity
conditions.
over'
a
~owever,
region
of
the function is locally regular or "well-behaved"
the positive orthant
if
the
following
parameter
22
restrictions
the
hold;
[~ 1 +
<
p4 1n<X 1 )
following range of values for
-~~IPe
[ exp(-~ 1 tp 5 Jx 1
and
~3
0
~
3,
~
4 < 0.
x2
<
< exp((1 -
the
~
1
Jt~
+ ~ 3 1nCX 2 J] < 1,
0 <. [~ 2
+
variables:
-~~~~~]
.
5 >x 1
,
The translog function reduces to the Cobb-Douglas when
= P4 = Ps = 0 ·
The Spillman Production Function
The
which
Spillman production function is an
allows
agricultural
diminishing
surface
yield
exponential-type
for diminishing marginal returns.
function
It has been
used
studies to represent crop response to nutrient inputs
returns
in
the fattening of
livestock.
The
can be represented by a smooth s-shaped curve which
plateau,
given
appropriate .parameter
values.
in
and
production
reaches
The
a
function
increases at an increasing rate, reaches an inflection point after which
it increases at a
rate until it reaches a plateau, indicating
decrea~ing
the maximum value of the function.
The Spillman function has the form
(2.11)
The Spillman function is not globally regular.
regular when
0
<
since
~5
< 1.
.•:
1 >
o,
0 <
~
2 < 1,
0 < p3 < 1,
0 <
Notice that p1 represents the maximum
the 1 imit as
well behaved.
·..
~
However,
x1
and
x2
~
it is 1oca 11 y
4 <1, and
obtainable
output
go to infinity is 13 1 when the function
is
23
The von Liebig Production Function
The
which
von
does
surface
Liebig
not
which
pro~uction
spline
implies
linear and reaches a yield plateau.
is
crop response to
economists
linear
allow for input substitution and
widely used by soil scientists,
represent
function is a
a
It
agricultural
economics
sp~cifieations
specification
against
fertili~er/nutrient
is
to represent crop response.
von
has
been
Agricultural
inputs.
literature has centered on
a
response
rather than agricultural economists, to
have traditionally preferred polynomial models,
quadratic and square root,
function
Liebig
function
such as
However, recent
testing
to
th~
polynomial
determine
a better representation of crop response to
which
nutrient
inputs (Ackello-Ogutu, Paris and Williams, Lanzer and Paris).
The von Liebig function
~as
the form
(2.12)
where Y* represents maximum yield.
~
4
~1'
>
o.
The function is quasi-concave if
2,
A positive level of output for zero levels of inputs requires
133 > o.
This chapter has described the nature of the regularity
required
for
satisfaction of second-order conditions
economic optimization,
the
~
five
functions
restrictions
required
conditions.
Given
conditions
associated
with
presented the algebraic specification of each of
chosen
for
study,
and
discussed
for satisfaction of monotonicity
that
profit-maximization
is
and
parameter
concavity
maintained,
regularity conditions give some insight into which functional forms
such
and
.24
parameter
econometric
values are appropriate.
theory
The following chapter presents
the
and estimation procedures required to· conduct
the
actual statistical tests.
25
CHAPTER 3
ECONOMETRIC THEORY AND MODEL SELECTION
The
problem
algebraic
form
of model specification is complex.
for
the
production
function
to
In
be
selecting
an
estimated,
the
researcher is often faced with the· problem of choosing among many alternative
may
or competing models.
come from previous experimentation and/or
allow
the
Once
the
common
researcher to narrow
and
highest apparent statistical
coefficients
regression
or the model with
the
Both
estimated.
estimated,
the
most
significance
highest
the
among
the
R2-statistic.
study differs somewhat in that the choice among alternative models
will be treated as a prciblem of model selection.
provide
are
theory.
form
of discriminating between models has been to select
the
with
economic
the set of functions to be
algebraic forms have been chosen
method
model
This
Guides to.the appropriate functional
The objective is
to
evidence that one or more of the models developed in Chapter
misspecified,
for
the
given data set.
Traditional
methods
2
of
testing rely largely on the requirement that models are nested by linear
restrictions
study
are
applied.
on
the parameters.
non-nested,
The
remainder
Because the models included in
methods for testing non-nested models
of this chapter will discuss the
procedures for carrying out these tests.
must
this
be
theory· and
26
The General Statistical Model
The objective of statistical estimation and inference is to provide
~nowledge
about the unknown parameters of some production model.
simplest case,
In
priori.
serves
as
the functional form of the model is known,
this
case,
or assumed a
the specification of the production
a maintained hypothesis.
In addition to
In the
function
functional
form,
there are a number of basic assumptions underlying the model which allow
the
researcher
to derive parameter estimates
with
certain
desirable
statistical properties (Kmenta).
In general, the linear regression model is given by
( 3. 1 )
where
X/3 +
y -
Y is
an
variable, X is
variables,
[3
€
I
(n x 1) vector of observed
an (n x k)
is
an
each X; is nonstochastic,
(3.3)
£i -N(O, cr 2 ), and
(3.4)
E(£;£j)
= 1,
the
(n x 1) vector of unknown parameters
( 3. 2)
where (i .• j)
on
(i
~
assumed
an
Further it is assumed that
j),
. , n.
can be shown to be best,
are
is
and~
Given these assumptions, the least squares estimators of
terms
dependent
matrix of observed values of the independent
(n x 1) vector of random disturbances.
=0
values
linear and unbiased.
to· be
normally
[3
in (3.1)
Further, since the error
distributed,
the
least
estimators are equivalent to maximum likelihood estimators.
squares
Thus, they
have all the desirable asymptotic properties-- asymptotic unbiasedness,
27
and asymptotic efficiency.
consistency
If the regression model given
in (3.1.) is nonlinear with respect to the parameters, maximum likelihood
estimation
or
the
method
of nonlinear
parameter estimators which have the
least
squares
will
provide
asymptotic properties.
de~irable
Given the assumptions about the random errors, (3.3) and (3.4), the
least squares estimators can be shown to be normally distributed.
forms
the
instance,
This
basis of hypothesis testing within the general
the hypothesis that
~
=0
can be tested using a
statistic is valid in small samples.
include
the
t.agrange
the
actual
multiplier,
model.
Wald
For
t-statistic.
Asymptotically valid
and
This
likelihood
ratio
tests
tests
(Kmenta).
If
functional form of the
relationship
is
unknown,
choice between competing models becomes a practical problem. The problem
of
model
since
selection is particularly prevalent in
the
known.
(a
However,
there
selection
exist a number of ad hoc
of Akaike (1973),
are
criterion
In
on
and information criteribn based on the
additional
general,
these
methods exist
ever)
R2 , adjusted R2 ,
These include:
Amemiya (1980) and Sawyer (1980).
non-nested,
spe.c if i cation.
selection
criterion developed by Mallows (1973) based
square prediction error),
models
economics
actual process which generates the data is seldom (if
which allow such choice~ to be made.
Cp
prpduction
mean
works
When the competing
for
testing
non-nested tests derive
from
model
the
fundamental work of Cox (1961, 1962).
Model Selection:
Nested Models
Standard hypothesis testing procedures, in general, involve testing
restrictions on a general model.
The null,
or maintained
hypothesis,
28
H0 ,
is tested against an alternative,
HA.
The
test
usually more general hypo.thesis,
is a rule which allows the maintained hypothesis to
"accepted" or rejected at a specified significance level.
the same parametric family,
between
referred to as nested
F6r models of
models,
comparisons
functional forms can be made using standard testing procedures.
For example,
the Cobb-Douglas production function is nested within both
the transcendental (2.9) and translog ( 2.10) production
imposing
appropriate
translog
functions can be reduced to the Cobb-Douglas.
parameter restrictions,
the
functions.
transcendental
If the
are non-nested, such linear restrictions are not defined.
is
necess~ry
Therefore, it
Non-Nested Models
by
Cox
testing
non-nested
ratio.
The
However,
·Deaton
an
(1978)
are
He outlined a general
1962).
hypotheses based on the
method
Neyman-Pe~rson
quite
first
alternative model.
Pesaran (1974) and
extended Cox's earlier work
regression
models.
asymptotically
to
linear,
The tests developed by
for
likelihood
complicated.
does allow for detecting departures from one model in
of
multivariate
(1961,
resulting test is often operationally
it
direction
and
models
A procedure for testing separate families of hypotheses was
introduced
By
to develop an alternative procedure.
Model Selection:
Deaton
be
the
Pesaran
and
nonlinear
and
Pesaran
and
equivalent to Cox's tests and can
also
be
difficult to implement.
Davidson and MacKinnon (1981,
provide
Deaton
· ... ·.
simplifications
based
on
1983) and Fisher and McAleer
(1981)
of the tests developed by Cox and Pesaran
artificial
nesting
of
the
comp~ting
and
hypotheses.
29
Finally,
MacKinnon (1983) presented non-nested tests,
based on artifi-
cial regressions, which are much simpler conceptually, easier to employ,
and
also asymptotically equivalent to the tests developed earlier.
It
is the tests presented by MacKinnon that will be employed in this study.
Given
two models to be tested,
M0 and M1 ,
Cox's initial test was
based on a comparison of the observed difference of likelihoods with
estimate of the expected value of this difference if Mo were true.·
an
The
statistic developed by Cox for the test of the null model was
(3.5)
where Lo and L1 denote the log likelihood functions under M0 and
and
are
~
th~
parameters of the two models,
maximum likelihood estimates of rand p,
and r and p denote
respectively.
represents the expected value under the null model.
given
that
M0 was the true model,
would
with
with
test was in defining the small
1
the
The operator Er
Cox showed that T0 ,
asymptotically
mean zero and variance v0 cT 0 ).
distributed
Cox's
be
M1 ,
The main
normally
difficulty
of
T0 ,
Pesaran (1974) developed a comparable statistic which replaced
the
which depended on certain
unknown
parameters
sa~ple
distribution
unknown parameters.
with
their consistent
estimators.
asymptotically equivalent to the one developed by Cox.
method
Pesaran
to
test
and
n~n-nested
linear models in a
Monte
The
test
is
Pesaran used the
Carlo
Deaton (1978) .extended the earlier work to
framework.
nonlinear· and
multivariate models.
Davidson
for
testing
and MacKinnon (1981) proposed several related
non-nested
models which are simpler·
pro~edures
com~utationally
and
30
accommodate
both
linear and nonlinear non-nested models.
procedures,
which
One ·Of
the
is
based .on
the
is the null model,. considered linear,
and g 1 is
they
refer
to
as the
J-test,
artificial regression,
y i = ( 1 - a ) f i ( Xi ,
( 3. 6)
where
f;<Xi,
alternative
~)
model,
A
r.
estimate r of
the
gicz 1 ,
of
A
) + ag i + Ei ,
A
evaluated
1),
In other words,
alternative model.
truth
~
at the
A
maximum
the
likelihood
gi is the best prediction of Yi under
the value of a is
If M0 is true,
Mo can be tested using an asymptotic t-test or
zero.
a
The
likelihood
ratio test of the null hypothesis a= 0.
If
the null model is nonlinear,
first
two
cause
serious computational problems.
this
expressions of (3.6) may be highly correlated.
problem by taking a Taylor's
point
the explanatory variables of
(a= 0,
= f1).
~
Davidson and
This
MacKinnon
the
could
solved
expansion of (3.6) around
~aries
The resulting regression equation,
the
which they
refer to as the P-test, ·is
A
Yi- fi
( 3. 7)
A
where
F
A
A
~) +
A
A
a(gi- f;) + Ei,
The
respect to
~'
Notice, when M0 is linear, F is just the matrix of
fi is equal to
equivalent.
~)with
A
= ~.
A
asymptotic
a)([J-
is the matrix of derivatives of fi(X 1 ,
evaluated at f1
X's,
A
= F(1-
A
X~
test
t-statistic
and the two ·regressions,
of
the hypothesis a
on
a
and
tests
asymptotically uncorrelated with (gi- fi).
=0
(3.6)
is
whether
and
(3.7),
comprised
(y L
-
oi
f i)
are
an
is
31
Since
there
is
the
no
true form of the relationship being tested is
maint~ined
Each model is equally
hypothesis.
unKnow.n,
I ikely
unlikely to be the correct specification.
Therefore,
tion tests are made on a pairwise basis.
The models are
time,
and tested against the
assuming each in
native
models
hypothesis
to determine
against
tained hypothesis.
to be true,
tur~
~hether
model specificataken one at a
the performance of the
the data is consistent with the
or
alter-
alternative
temporarily
main-
The procedure to test the truth of M1 is to reverse
the roles of Mo and M1 and carry out the tests again.
The tests provide
evidence that one, or both models, are misspecified.
Interpretation of Non-Nested Hypothesis Tests
An
objective
of fitting agricultural production functions
However,
describe crop response to nutrient inputs.
many models to explain
any given relationship.
is
to
there are usually
Therefore, the purpose
of hypothesis testing is to provide evidence as to which function represents
a
~esting
better specification of the crop
has
been
completed
response.
(on a pairwise
basis),
possibl& outcomes (for a given significance level).
rejected,
one
Once
there
formal
are
four
Both models may be
in which case the data does not support either
may be rej ectad whi I e model two is not,
the
model;
model
or vice versa, · or neither
model may be rejected, in which case the sample data is insufficient for
choosing among the competing models.
It should be stressed that the test of a= 0 in (3.6) or (3.7) is a
test
of the maintained model,
M0 ,
validity of the alternative model,
· ....
only,
M1 .
and tells nothing about
the
For example, if the t-statistic
32
on
the
estimated
value
of a implies that
it
is
not
statistically
different from zero (for a given significance level, say 5 percent), .the
implication
validity
of
is
that the null model cannot be rejected.
M1 ,
To
test
the
the roles of the two models must be reversed and
the
test carried out again.
The
empirical
results for the models employed in this
presented and discussed in the next chapter.
study
are
33
CHAPTER 4
EMPI~ICAL
RESULTS
The purpose of model selection tests is to determine which, if any,
of a chosen set of models "best" describe the sample data.
of applying such tests to,
say,
two competing models,
may be rejected while the other cannot,
model
neither
Thus,
model can be rejected.
The results
may be that one
or that both models,
testing each model against
information provided by the alternative provides evidence that
both
models,
are
and
results
models
Chapter 3.
was
of the non-nested hypothesis
tests
the
one,
This chapter presents the
misspecified.
or
or
estimated
discussed
in
The experimental data used in estimating the selected models
discussed
contained
57
in
detail in
Chapter
1.
In
short,
the
experiment
applications of nitrogen (N) and phosphorous (P),
pound increments, in two replicates.
in
40
The data are presented in Table 5.
The Empirical Models
The
quadratic,
transcendental,
translog and Spillman
production
functions were estimated in the following forms, respectively:
( 4. 1 )
( 4. 2)
Y;
( 4. 3) . yi
= f3o<Ni
::
+ d1){31(Pi + &2 ) {32 exp [ {3 3 (Ni + & 1 ) + {34(Pi + &2>) +
f3o<Ni + &1 /1 (Pi + &i)
{3 2
exp[f3 3 1og(Ni + & 1 )log(Pi +
+ .5{3 4 1og(N; + &1) + .5{3 5 1og(P; + 5 2>1
+
E
i '
&2)
Ei
'
and
34
where
Yi
is corn yield (bu),
phosphorous
and
( 3. 3) and ( 3. 4) .
€i
N; is applied nitrogen,
Pi
is .a random disturbance distributed
The subscript i (i = 1,
is
applied
according
to
2, ... ,114) refers to the ith
observation.
Notice
translog
terms,
allow
that the empirical specifications of the transcendental and
functions differ from those presented in Chapter
8 1 and &2 ,
were added to the variable factors,
positive yields
~t
biological standpoint.
applied nitrogen,
Constant
Ni and
Pi,
to
zero levels of the applied nutrients since the
This is reasonable from
data indicated that such a result is possible.
a
2.
The factor,
Ni,
represents the
amount
of
while &1 is the amount of nitrogen already present in
A similar argument can be made for Pi, the amount of applied
the so i 1 .
phosphorous,
and
82 .
Had
the co~stant terms not been added in
this
manner, the functions would have had a value of negative infinity at the
origin and would not
ha~e
been estimable at those points.
The procedure used to estimate equations (4.1)- (4.4) could not be
used
to
estimate the von Liebig model given in
(2.12).
arose because of the nature of the response surface.
The
problem
The model is non-
differentiable at the points, referred to as knots, where the .transition
to
a plateau occurs.
model
as a linearly constrained optimization problem.
model was estimated as
( 4. 5)
This problem was solved by reparameterizing
The von
the
Liebig
35
where
dni
=0
if N; > N;*
if
dpi
= 1.
=0
= 1
if
Ni < Ni*
ff pi > Pi*
pi < Pi*·
The maximum crop response is given by Y* with N* and P* representing the
quantities
The
of
nitrogen and phosphorous corresponding to
reformulation
relationship
that
of
~t
the
model given in
the maximum yield,
(2.12)
Y*
=~1
+
this
output.
was
based' on
~ 2 N*
= (3 3
+
Equation (4.5) was obtained by solving the above relationship for
~
the
f3 4 P*.
~
1 and
Liebig model given in .(4.5) was estimated by solving
the
3 and substituting these back into the original model.
The
von
following nonlinear optimization problem:
(4.6)
subject to
y
-
Y*
-
{32(N
N* )dn ·+ sn
-
€
= o,
y
-
Y*
-
{34(P - P*)dp + sP
-
€
= o,
sP
where
Y,
defined
N,
in (4.5).
variables
tion,
P,
the
dn,
>
o,
sn >
dp and
E
o,
Y* ?:
and
o,
are (114 x 1) vectors of the
variables
The variables Sn and Sp represent vectors of
for the nitrogen and phosphorous regimes.
For each observa-
slack variable of the limiting factor will be zero
slack for the nonlimiting factor will be positive.
attached to the product Sn'Sp,
and
the
An explicit penalty
equal to 10,000 in this case,
the case of both Sni and Spi being positive.
slack
prohibits
The minimum sum of squared
36
residuals was found by searching over values of the knots, N* and P* (in
increments of 5 pounds), since these parameters must be specified before
(4.6) can be solved.
The
Liebig
asymptotic
model
were
vari~nc.es
of the parameter estimates of
obtained from the elements of the
the
von
of
the
minus
the
inverse
information
matrix.
The tnformation matrix is formed from
expectation
of the second-order derivatives of the natural logarithm of
tbe likelihood function.
The log-likelihood function for the von Liebig
mode 1 is
- 1/2a2 [I<Yi - Y*) 2 + I<Yi - Y* - ~ 2 CNi - N*)) 2
Nl
N2
=k
L
( 4 •. 7)
+ I<Yi - Y*- ~4(P~ - P*)) 2 + I<Yi - Y*- p2(Ni - N*)) 2
N3
N4
+ I<Yi - Y* - ~4(Pi - P*))2],
N5
where
k
=
(-NT/2J[log(2u) ~ log(a 2 >] and N1 represents the observations
for which N;
and Pi
<
P*,
~
N* and Pi > P*, N2 are the observations
N3 are the observations for which N;
~
fo~
N* and Pi
are the observations for which Ni < N*, Pi < P* and (Y* +
and
N3 ,
N4
N5
which N; < N*
~
P*,
N4
2 CNi - N*)) <
are the observations for which
and N5 represent the total number of sample
<
Ni < N*,
observations,
Nr·
The asymptotic standard errors of the parameter estimates were
obtained
by
the
inverted
It is important to recognize that these
standard
taking
information
the
square root of the diagonal elements
matrix.
of
errors are conditional on the estimated values of the knots,
N* and P*.
n
The
sums in (4.7) depend directly on N* and P•.
differentiable
with
respect
to
the likelihood
function
is
Further,
the derivatives for the remaining parameters are not
However,
if
these
not
Thus,
these
parameters.
defined.
we fix the values of N* and P* at their estimated
derivatives are defined.
of
understatement
asymptotically
the
This assumption probably results in an
estimated
standard
errors.
this difference is expected to be zero.
function estimates and related
values,
~tatistics
However,
The
production
are presented and discussed in
the following section.
Empirical Results
The empirical results are presented in Table 1. Estimation of model
parameters
for the quadratic,
transcendental,
translog
and
Spillman
functions was carried out using a nonlinear least squares procedure (SAS
Institute
Inc.,
estimated
1984).
using
a
The
.'para~eters
Fortran-based
of the von Liebig
software
for
mddel
solving
were
linearly
constrained optimization problems (Murtagh and Saunders).
The
estimated regression coefficients for the
quadratic
function
are all statistically significant at the 5 percent level, except for the
intercept.
Forth~
statistically
terms,
B1
function
except p0 ,
significant
significant at the 5 percent level,
and a2 ,
are
transcendental function, all parameter estimates are
all
and p0 .
a2 ,·
constant
The estimated parameters for the translog
statistically significant at the
the intercept
except the
and p2 . The fact that
5
a1
per~~nt
level,
is statistically
for the estimated translog function implies that the sample
data indicate a positive amount of nitrogen already present in the soil.
Table 1.
Production Function Estimates and Related Statisticsa
Quadratic:
Y1
=-7.509
+
(6.637)
.584N 1
.664P 1
+
(.0635)
(. 0635)
(.000353)
(.000353)
R2 = .832
d.f. = 108
(.000155)
SSE
= 40728.9
Transcendental:
Y.1 =· 1. 791CH 1 + 9.0369)
(1.235)
.655
CP; + .142)
(5. 0914 )( .144)
.307
[
]
exp -.00288CH; + 9.0369) - .0012HP 1 + .142)
(.232)(.0776) (.000690)
(5.0914)
(.000488)
(.0776)
SSE·= 17570.8
Trans log:
.0412CH 1 + 24.380)
(.105)
d.f.
2.554
CP 1 + 2.0877)
(9. 340 )(. 949)
= 106
(1. 729)(
R2
.304
[
exp .1231og(H; + 24.380llog(P; +
.355)
= .935
( .0366)
(9.340)
2.0877)
(1.729)
SSE
15847.3
SSE
18551.8
Spillman:
yl .= 127 -~28(1
(2.423)
d.f.
= 109
- . 775( .981 lN, HI - .857(.973)
( .0265)(.00216)
P,
)
(.0275)(;00313)
R2
= .924
(.5).583Clog(H 1 + 24.380)) 2 ·- (.5).169(1og(P 1 + 2.0877!12]
(.183)
(9.340)
( .0712)
(1. 729)
Table 1 (continued)
von Liebig:
Y1 = Hin(124.579 I
29.1200 + .95459N 1 ; 19.9814 + !.23056P 1 l
(1.7025) (19.1035) (.0370)
d.f.
109
(17.3305)
(.0423)
SSE
= 21260.5
w·
10
aNumbers In parentheses are standard errors •.
40
All the estimated parameters of -the Spillman function are
significant at the 5 percent level.
Liebig
model
are
except the terms
With
it
is
~
The estimated parameters of the von
statistically significant at the
1 and
~
statistically
percent
5
2•
regard to the parameter estimates of the quadratic
interesting
level,
to note that the standard errors of
equal, as are the standard errors of
~
3 and
~
~
function,
1 and
~
are
2
This is a result of the
4•
The observations on nitrogen and phosphorous
experimental design.
are
such that they occur in the same number of levels (40 pound increments),
the same number of times, Thus, lNi
restrictions
parameter
The
= IP 1
implied
monotonicity and concavity conditions,
for
by
= IP1.
the
satisfaction
as outlined in Chapter
Therefore,
each of the selected models.
well-behaved
and IN1
each of the
and consistent with an interior solution for
of
2,
hold
function~
is
a . perfectly
competitive.profit-maximizing or cost-minimizing equilibrium for a firm.
The
parameter restrictions which are in the form of a set of inequality
constraints,
in generali are difficult to test.
Although no measure of
the statistical precision of the estimates is presented,
constraints
If
~4
can be examined at specific points to check if
the inequality can be shown to hold at a point,
general.
<
they
hold.
it is satisfied
in
In particular, the set of .constraints [~1 > 0, ~2 > O, ~3 < O,
0 ,~ 3 ~ 4 > ~g] holds for the estimated quadratic
[~ 4 < 0,
the inequality
~ 5 < 0,
0 < [~ 1
+ ~ 4 1ogCX 1 >
function.
+ ~ 3 logCX 2 >] < 1,
0
Also,
< [~
2
+
~slogCX 2 ) + ~ 3 1ogCX 1 >] < 1] hdlds fa~ the estimated translog function.
The parameter estimates of
are
·.· ..·.
given in Table 1.
~
3,
~
4 and
~
It can be seen that
5 for the quadratic function
~
3~4
= .0000113
is
always
41
greater than ~~
= .000000658.
Further,
cx 1 = 160, x2 = 160),
tions for nitrogen and phosphorous
that
using the means of the observa~t
can be shown
the inequality constraints for the translog function hold at
that
point. Using the estimated values of the parameters given in Table 1 and
the
means
of
p4 log(160)
the sample observations,
p31og(160)
+
become 0 < 0.219 < 1 and
< 1
and 0 <
~
the constraints
2 +
~
5 1og(160) +
0 < 0.071 < 1, respectively.
~
~1
+
3 log(160) <
1
0
<
Thus, it appears
that the translog function in concave at that point.
Notice that the results for the .von Liebig function in Table 1
reported
model
in
gfven
problem
the more general form
in (4.5).
4
= 1.2305)
= 124.579)
reported.
= 100
by
resulted
variances
solving the
in
and P*
= 85
minimization
in addition
(~
= .9545
2
rel~tionships
(conditional
calculated from Var<P 1 >
and
of p1
on
Y*
= 29.1200
the
= Var(Y*-
The estimated intercepts were
= p1
and
+ ~ 2 N*
p3
=~3
=
+
19.9814.
estimated knots) for p1
p2 N*) and
Varc~
31
·p 4P*.
Further,
p3 were
and
= Var(Y*-
commonly used to describe how well an
This
p
4 P*)~
estimated
model
fits the observed data is the multiple correlation coefficient, R2 .
value
to
Trese values were used to derive estimates for
estimates
A measure
reformulated
constrained
and slope coefficients
the intercepts p1 and p3 given in (2.12).
found
rather than the
The solution to the
(4.6) implied estimates of N*
the maximum yield (Y*
~
(2~12)
are
of R2 is presented for each of the models in Table 1.
The
While the
quadratic function appears to fit the data poorly, the values of the R2statistic
for the transcendental,
functions
are
appears
that
translog,
Spillman and
all high and comparable in magnitude.
In
these four models are equally suitable
for
von
Liebig
general,
it
representing
42
corn
yield response to fertilizer inputs,
given
in terms of
However,
regularity conditions and goodness of fit.
criterion
for
satisfying
judging the specification of the models is
a
the
sharper
provided
by
non-nested hypothesis tests.
The results of non-nested hypothesis tests,
or more of the models to be rejected.
of competing models.
in general,
allow one
This allows us to narrow the
set
If none of the specifications can be rejected, the
tests indicate that the models are equally suitable for representing· the
production
process under consideration.
The results of the non-nested
tests are presented and. discussed in the following section.
Model Selection Test Results
As outlined earlier,
out on a pairwise basis.
the
specification
the non-nested hypothesis tests were
carried
In this manner, each model was tested against
of each alternativa model.
The selection· of
a
nested models depends on a test of the hypothesis H0 :
=0
non-
in (3.6) or
(3.7), depending on whether the currently maintained model was linear or
nonlinear. If Ho is rejected, the currently held null model is rejected.
However,
nothing
can
be
inferred about the truth 6r falsity
currently held alternative model.
of
the
The test is based on an asymptotic t-
statistic with a significance level of S percent.
The
model selection test results are presented in
Table
columns represent the models when they serve as the maintained
2.
The
hypothe-
sis, while the rows represent the models when they serve as the alternative.
The
reported
( 3. 7).
The
numbers
statistics
in
are the estimates of a in
parentheses are standard
errors.
(3.6)
or
43
Table
2.
Model Selection Test Resultsa
Maintained
Hypothesis
Alternative
Hypothesis
Quadratic
Quadratic
Transcendental
.0568
(.115)
Transcendental
Trans log
Spillman
von Liebig
-.00791
.133
.1875
( .127)
( .120)
( .0807)
;964
. 7250
( .406)
( .1578)
-.224
( .0753)
(,637)
.983
.900
( .0753)
( .289)
.954
.8371
( .08039)
4
Spillman
I .00639
( .0779)
von Liebig
Trans log
1.00062
(.235)
-.552
-.662
(.683)
( .556)
.2621
( .1634)
.01388
( .1919)
rne reported sta~lstlcs are estimates of a In (3.6) or (3.7).
standard errors.
·... ·.
.8875·
( .2202)
.7250
( .1801)
.2588
( .1755)
Numbers In
parentheses
are
«
Takin~
compa~ison·
the
between the quadratic and
the
alternative
models first (column 1 in Table 2), we see that the null hypothesis, a=
0 in (3.6),
which
is rejected in every case. In other words, the t-statistic,
defined by the estimated.value of a divided by
is
error, is greater than 2.0 for all cases.
function
cannot be rejected,
the quadratic.
is clearly rejected ( at-value
maintained
percent
However,
rejects
Williams
of
null
level
(ro~
This outcome
1 and column 5).
of
in particular the
a
seems
Liebig
clearly
quadratic.
that the von Liebig specification cannot reject the quadratic
depend
on
the
calculated.
method in which the
asymptotic
As mentioned earlier,
5
Ackello-Ogutu,
which finds that the von
polynomial specifications,
=
when the von Liebig serves as
in light of the recent work
(1985),
the
0.8371/0.08039
the null hypothesis cannot be rejected at
particularly
and
fact
model,
significance
unlikely,
the
excepi in the case of the von Liebig vs.
10.413 in row 5 and column 1).
Paris
Similarly, when the quadratic
When the quadratic serves as the maintained model,
hypothesis
the
standard
serves as the alternative model (row 1 in Table 2),
hypothe~is
null
its
standard
errors
The
may
were
the standard errors are conditional
on the estimated values of N* and P*, .which results in an understat~ment
of
their. value.
However,
likelihood estimates,
the
due
to
the consistency
of
this bias should be very small.
the
In
maximum
particular,
likelihood ratio statistic corresponding to the restriction a=
O,
which is not conditional on N* and P*, implies an asymptotic t-statistic
of
2.24.
This
restriction.
"conditional"
·.· ..·.
result
Thus,
is
there
t-statistic.
valid since there
is
very
little
is
only
one
difference
implicit
with
the
It follows that the asymptotic t-ratio may
45
be
larger
variables.
compared
that
than
it
b~
would
However;
the
if N* and
P*
were
treated
as
random
relatively small value of 0.1875 for a
to the other values in the fifth column of Tabla 2)
(when
indicates
quadratic is only "marginally" preferred to the von Liebig.
~he
a significance of 0.01 were chosen,
general,
we would reject the quadratic.
If
In
all of the chosen models provide evidence against the truth of
the quadratic specification.
Similar results were obtained for the von
Liebig.
The
von Liebig specification is clearly rejected by the
transcendental and translog functions.
Spillman;
When it serves as the maintained
model,
the null hypothesis is rejected for all cases.
Liebig
appears to be a better specification than the quadratic,
inferior
to the more
flexible
specifications,
Although the von
e.g.,
it
is
transcendental,
translog and Spillman. The Spillman model rejects both the quadratic and
von Liebig specifications. However, when tested against the other flexible forms, it is rejected.
The
When
models.
hypothesis
is
I
comparison is between the transcendental
final
the
and
transcendental
the translog as the
rejected (t-value
different
models
from
zero
are reversed,
rejected.
= 3.114).
function serves as
alternativ~,
That is,
the
translog
the
maintained
null
hypothesis
a in (3.7) is
at the 5 percent level.
and
statistically
When the roles
of
the result is that the null hypothesis cannot
the
be
At. the 5 percent level, a·.is not statistically different from
zero.
The
translog
·~
..
results
function
of the non-nested hypothesis test indicate
is
a superior specification in that
it
that
the
cannot
be
46
rejected
in
it
Thus,
pairwise comparisons with any of the
appears
production
alternative
to provide the most satisfactory explanation of
the
A comparison of R2-statistics indicated that
the
process.
performance of the transcendental,
translog, Spillman and von Liebig in
terms of fitting the data, were very similar.
The non-nested hypothesis
tests provided an additional criterion with which to judge the
cation
model.s.
of the competing models and,
in fact,
allowed one
specifi-
model,
the
translog, to be chosen above the rest.
The
translog specification has frequently been used in
production
studies
technology.
In
a
because. of
its flexibility
in
comparison with several commonly
industrial
representing
used
any
agricultural
production functions, it appears that it can also be used, successfully,
to represent crop response to fertilizer inp~ts. A graphical representation of the response surface of the estimated translog function is given
in
Figure 4.
It can be seen that the function exhibits a plateau with
respect to phosphorous,
creases
at
P.
In terms of nitrogen,
N, the function in-
at a decreasing rate for low levels of the input and
an increasing rate at higher levels.
From a technical
decreases
standpoint,
this implies that excessive applications of P will not decrease
yields,
while the opposite is true for N t within the range of the sample data).
The
rejection of the von Liebig by the translog specification indicates
that the hypothesis of input non-substitution is also rejected.
This chapter has presented the results of the non-nested hypo~~esis
tests for the five models discussed in this study.
that
the translog model is a
supert~r
The results indicate
specification for the
pr6duction
process under consideration. With regard to the von Liebig function, the
47
results
of
superior
quadratic,
following
the
non-nested
specification
but
inferior
chapter
to
t~sts
lower
indicate that it
order
appears
polynomials,
such
to the more flexible functional
presents a summary and conclusions
together with recdmmendations for further study.
to
of
be
~
as
the
forms.
The
this
study,
y
132.59
92.08
A
00
51.57
360
11.05
360
120
N
Figure 4.
Estimated Response Surface for the Translog Function
0
49
CHAPTER 5
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
The purpose of this study was to address the problem of
function
specification
competing
models
functional
forms
choice when faced with several
arising
from different
parametric
alternative
families.
representing observed and maintained
quadratic,
transcendental,
translog,
Spillman
or
Five
differences
in
They included
plant yield response to nutrient inputs were considered.
the
production
and
von
Liebig
production functions.
There were three specific objectives for this research.
was
to
establish the parameter·restrictions necessary to
The first
sati.sfy
the
regularity conditions consistent with optimizing behavior of individuals
or firms.
The second was to estimate the parameters of each model using
experimental
data
fertilization.
parameter
on
This
estimates
~orn
yield response to nitrogen
included
with
evaluating
accepted
the
production
and
phosphorous
consistency
theory.
of
The
the
final
objective was to statistically test the restrictions implied by
produc-
tion
between
theory
on
the
parameters of each functional
form
and
functional forms using non-nested hypothesis testing procedures.
Estimation was carried out using a nonlinear least squares procedure
and a procedure for solving linearly constrained optimization
·...•.
The
parameter restrictions implied by the satisfaction of
and
concavity conditions held for each of the choseri models.
problems.
monotonicity
That is,
50
each
functional . form was consistent with
perfectly
for
The hypothesis tests were conducted using
procedure
against
Cox.
The
involved nesting two models in one comprehensive model.
The
made
on a pairwise basis so that
results
alte~native
each
model
any
models.
of the tests indicated that the
of
the alternative
translog
production
industrial
translog
models,
including
function has been used
almost
studies,
that
it
appears
successfully to represent crop response to fertilizer
of its more general,
of
its superior
an
a wide range
including non-substitution.
the results
than
Ackello-Ogutu,
Liebig.
exclusively
it
can
be
inputs.
substitution
Paris
the
and
confirm~d
quadratic.
used
Because
possibilities
was
given the
sample
that the von Liebig is a
better
This supports the recent
Williams (1985),
the quadratic and square root functions.
these
in
This may be one reason for
which found that
performed better than more traditional polynomial
exhibit
von
results of this study indicated that the von Liebig model
However,
Liebig
the
rejected
performance~
specification
that
of
inferior specification of the production process,
data.
production
flexible form, the translog production function is
representing
between inputs,
The
was. tested
was a superior specification in that it could not be
Although ·the
capable
equilibrium
of
technique based on the initial works
against the specifications of the
function
a
artificial·
were
The
· for
an
regression
tests
solu~ion
competitive profit-maximizing or cost-minimizing
a firm.
linear
interior
~n
work
of
the
von
forms,
e·.g.,
However, the authors concluded
polynomials "should be abandoned" in favor of models
the agronomic principles of input nonsubstitution
and
which
plateau
51
The
response.
results of this study suggest that other specifications
may be superior to the von Liebig.
In particular, both the translog and
Spillman functions fit the data better than the von Liebig and allow for
input subs.titution and plateau response.
nonsubstitution
reached
Thus,
by Ackello-Ogutu,
the conclusion of input
Paris and Williams
is
not
supported by the results of this study.
The
the
objective of model specification is not only to best
production
resource use.
process,
but also to provide
information
describe
on
optimum
If the optimization criterion is to maximize profits, it
would be of interest to know the actual cost (in terms of lost profits),
of
model
bushel,
nitrogen,
each
misspecification .
1.61,
and
the
the current
price
current prices of fertilizer in
of
corn
the
per
form
of
0.21, and phosphorous (P 2 0 5 ), 0.233, the profit equations for
of the five models,
rived.
Using
The
given in equations (2.8) - (2.12),
profit-maximizing
levels of inputs,
output
were
and
de-
maximum
profit were obtained and are presented in table 3.
Table 3.
Profit Maximizing Values for the Various Functions
output
(bu.)
Function
Quadratic
Transcendental
Trans log
Spillman
von Liebig
Referring
137.01
119.75
116.41
115.23
124.57
to
,.··.
191.80
156.00
134. 10
137.20
100.00
Table 3,
large application rates,
·.
N
( 1bs. )
p
Profit
( 1bs. )
188.00
126.80
106.40
108.60
85.00
the quadratic function implies
( $)
136.51
130.50
134.46
1 31. 40
159.75
relatively
while the von Liebig implies relatively
small
52
application rates.
The
transc~ndental,
translog and Spillman functions
are similar with respect to the profit maximizing input mix.
Of course,
the choice of the "true" application rate depends on which specification
is "true".
The
results of the non-nested hypothesis tests indicated that
the
translog function was the superior specification in that it could not be
rejected
Thus,
it
in
pairwise comparisons with any of the
alternative
appears to provide the most satisfactory explanation
production process.
models.
of
the
It would be valuable to know the cost, in terms of
profits,
of choosing one of the alternative models.
translog
is the true specification,
Assuming that the
but using the input mix implied by
the alternative functions, yields the results given in Table'4.
Table 4.
Values Implied by Translog Functiona
Function
Quadratic
Transcendental
Spillman
von Liebig
p
( 1bs. )
( 1bs.)
( $)
128.3'3
121.34
117.11
106.68
191.80
156.00
137.00
100.00
188.00
126.80
108.60
85.00
122.52
133.06
134.43
130.95
N
aAssumes that the translog ·is the true specification.
the optimal values. implied by profit maximization of
models.
The cost of model misspecification,
be
obtained
by
·.· ..·.
implied
Input values are
the alternat.ive
in terms of lost profits,· can
comparing the results presented in Table
maximum profit implied by the translog model.
mix
Prof it
Output
(bu.)
4
with
the
For example, if the input
by profit maximization of the quadratic function
is
used
53
rather than that implied by the· translog, the loss in profits, per
is
$11.94
implied
($134.43 minus $122.52).
acr~,
The result of using the input
by the von Liebig function is smaller,
$3.51
per
mix
acre.
The
smallest loss occurs when the input mix implied by the Spillman function
is
used,
only
trans log,
used,
$0.03
the
but
Thus,
per acre.
if the true
model
were
resource mix implied by the Spillman function
the
were
the cost of misspecification would be very small.
Recommendations for Further Study
Application
models
of non-nested hypothesis tests to production
However,
is relatively new.
this area of research
insights into the problem of model specification.
this
study,
function
can
offer
During the course of
a number of possibilities for further research have become
evident.
First,
forms,
future
studies need to examine a wider range of functional
including more general flexible forms.
functions·
alternative
relationship.
of
Mitscherlich-Baule
cubic,
which
Bray,
which
allow
forms
and
the
represents
the
theory
Higher-order
repre~ents
the
developed
by
model
of
relative
polynomials,
which
yield
such as
more generality and flexibility in the
the
parameters
In addition, there exists a class of flexible
are
capable
of
representing
production possibilities and technical properties.
· . ·..··:.
fertilizer-yield
the
substitution
Russell).
should also be included.
functional
analyzing
such as the Leontief function which
nutrient
(Balmukand,
for
Some of these express specific agronomic principles held
by soil scientists,
absence
exist
For example, a number of
a
vari~ty
of
54
The set of flexible forms includes the generalized power production
function
(DeJanvry),
the
generalized Leontief and generalized
production functions (Diewert),
a
number
Laurent,
of
additional
linear
the Fourier flexible form (Gallant) and
second-Qrder
flexible
forms
including
These models have appeared
translog and Box-Cox.
the
relatively
recently in the literature ahd are receiving attention because they have
enough parameters to permit a wide variety of technical properties.
the
example,
DeJanvry
generalized
(1972),
power
includes
as
production
special
function, introduced
cases
the
Cobb-Douglas
transcendental
production functions and allows variability in a
of properties,
including returns to scale,
elasticities
of
substitution.
The
partial
elasticities
of
by
and
number
marginal productivities and
generalized Leontief
production functions proposed by Diewert (1971),
of
For
and
linear
can attain any
number
yet
substitution,
remain
relatively
parsimonious in parameters.
In general, the increased flexibility of this class of functions is
achieved
This
through
may
cause
problems
testing,
to
Conceptually,
conceptually.
and
in
agronomic principles,
shown
introduces
•,.
·.
in
hypothesis
Because
of the
parameters.
large
number
there may be a greater problem
calculating
F
in
both
equation
of
with
( 3. 7).
it may be difficult to explain some interactions in terms
if the problem ls one such as the fertiltzer-
yield problem in this study.
been
arise
and interaction terms,
multicollinearity
of
of
and
computationally
parameters
a sighificant increase in the number
to
favor more
In addition, some testing procedures have
parsimonious
model
specifications.
a second area where further research would be
of
This
interest.
55
That
is,
the
varidus
procedures which exist for
testing
non~nested
hypotheses and their ability to reject false hypotheses.
Several
developed
Most
procedures
that
work
for
non-nested hypothesis testing
might be used for the problem of
has· focused
on
the
likelihood
model
ratio
have
been
specification.
and
comprehensive
was
Cox's
intention to develop a procedure which provided high power of one
model
approaches
against
initially
proposed
of
Deaton (1978),
a~d
Fisher
(1961,
Cox's test,
for example Pesaran
Davidson and MacKinnon (1981,
McAleer
(1981),
for
(1974),
the
than 60 observations.
Their 'Wald-type'
explore
the
In
test
(W-test)
for instance samples
Such a test may offer some improvement for
sample size used in this study.
Godfrey and Pesaran showed in
Carlo simulation that the J-test lacked power for
samples.
test
to
Godfrey and Pesaran (1983) developed small sample adjustments
tests of non-nested hypotheses.
Monte
and
in terms of robustness.
appears to be superior for moderate sized samples,
less
Pesaran
1983), McAleer (1981) and
it would be of interest
advantages provided by the different tests,
addition,
It
1962).
Because of the existence of many large-sample
an alternative.
equivalents
by Cox
these
a
smaller
However, the difference between the power of the J-test and W-
decreased
as
sample
size
increased
to
60.
Therefore,
this
difference may be small for sample sizes greater than 110.
It
has
(1983),as
been
well
as
suggested
others,
by
Mizon and
that there is room
inferences to be drawn from these tests.
test
of
Davidson
and
Richard
MacKinnon
(1981)
for
(1982)
and
conflict
H~ll
in· the
McAleer (1981) shows that the
favors
models
with
parameters than the test developed by Fisher and McAleer (1981).
fewer
Quandt
56
(1974)
shows
scheme
may have the greatest ability to reject
addition,
that a comprehensive model formed by a
there
alternative
MacKinnon
exist
hypotheses
against
several
alternative
useful
for
However,
false
simultaneously,
for
hypotheses.
example,
models simultaneously.
Davidson
Therefore,
it
higher power may come at the cost of
and
know
testing
if
against
waul d
empirical applications to use more than
achieving
In
several
It is of interest to
a single model is more powerful than
future
weighting
procedures for testing a model against
(1981) and MacKinnon (1983).
testing
linear
one
be
test.
computational
ease.
In
addition
examining
to
range
of
functional
forms,
the power of various tests and applying more than one test in
empirical applications,
testing
including .a wider
there is another area of non-nested
which has received little attention,
proposed
three
procedure
has
yet is of interest.
Cox
The
testing
non-nested
hypothes~s.
received the least
attention,
is
procedures
which
hypothesis
for
the
Bayesian
approach.
The
forms
Bayesian approach to choosing between
is
models
alternative
based an a comparison of the posterior probabilities
under
The
consideration.
probability is chosen.
model with
the
highest
approaches,
that
the researcher must arbitrarily choose the maintained
outcome
considered
of
the
of
the
posterior
One of the qisadvantages of the more frequently
used
The
functional
the log-likelihood ratio and artificial
these tests is directly dependent on
maintained
hypothesis and a result
accepts both models is possible.
nesting,
which
which
is
hypothesis.
mod~l
is
rejects
or
57
With the Bayesian approach,
is
not
model,
the
to
outcomes
non-nested
are
hypothesis.
pares
That
two
One application of the
cost
data,
and Fourier..
form.
together
comparisons
of
The
using two
,.·.
and
the
the
be
decision
maintained
Bayesian
approach
He
com-
calculated for aggregate
U.S.
flexible
functional
forms,
Bayesian approach
to
non-nested
the
hypothesis
with a wider variety of functional forms
existing
for
He found that the posterior odds ratios favored
procedures provide a wide range
topics deserving further study.
•,
This ratio can
on which model is considered
cost-share equation systems,
Fourier
testing,
particular
between non-nested models is given by Rossi (1984).
manufacturing
the
a
which
model which is favored by the posterior odds ratio is
model.
-:
translog
models without modification
inde~endent
preferred
choosing
models,
prior probability and sample evidence for
are summarized by the posterior odds ratio.
applied
the
The posterior probabilities of the
possible.
represent
acceptance or rejection of both. models
and
of
power
possible
58
··~
REFERENCES
·
.....
59
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65
APPENDICES
·.·,.·.
66
APPENDIX A
DATA SET
67
Table
5.
Experimental Yields of Corn For Varying Levels of
Inputs a
Fertilizer
Nttrogen
(I bs.)
P205
(1 bs. )
0
40
ao
120
160
200
240
2aO
320
0
24.5
6.2
23.9
11 .a
2a.7
6.4
25.1
24.5
17.3
4.2
7.3
10.0
16.2
6.a
26.a
7. 7
25.1
19.0
40
26.7
29.6
60.2
a2.5
96.0
107.0
95.4
95.4
80
22.1
30.6
120
44.2
21.9
160
12.0
34.0
96.2
ao.7
200
37.7
34.2
at.1
51.0
240
38,0
35 .o·
280
32.4
27.4
320
5.3
17.9
99.5
115.4
102.2
IOa.5
115.9
72.6
113.6
102 .I
133.3
124.4
129.7
116.3
105.7
115.5
128.7
109.3
140.3
142.2
'127 .6
125.a
97.2
79.5
39.7
116.9
a3.6
112.4
125.6
119.4
97.3
101 .a
129.5
125.2
134.4
127.6
135.7
121.5
122.9
122.7
81.9
76.4
130.5
124.3
129.0
a2.0
114.9
129.2
124.6
a3.0
123.6
142.5
135.6
122.7
136.0
11 a.2
121 .I
114.2
13a.7
126.1
127.3
139.5
aTwo numbers are shown In each cell si.nce treatme.nts were replicated.
130.9
144.9
130.0
141.9
124.a
114 .I
131.a
Ill. 9
127.9
tla.a
68
APPENDIX 8
CALCULATION OF THE STANDARD ERRORS
FOR THE VON LIEBIG MODEL
·. ,.·.
69
The
asymptotic
standard
errors
of the
von
Liebig
model
were
calculated using the well known result
(1)
where
e is a vector of parameters,
matrix
and
Cov(B) is the asymptotic covariance
I is the Fisher information
matrix.
The
matrix,
I,
is
estimate
of
defined as
( 2)
wher~
e1 and ej are t~e ;th and jth elements of a.
An
Cov(B) can ·be obtained by setting all parameters equal to their
likelihood
estimates.
The log-likelihood function .for the von
maximum
Liebig
model is
L = k - c1/2 cr2 >[ I cv, - Y*)2 + l<Y; - Y* - p2 CN; - N*) )2
N2
1 N1
( 3)
+ l<Y; - Y* - {34(P;
N3
P*))2 + L(Yi - Y* - (3 2 CNi
N4
-
N*) )2
L(Y; - Y* - (3 4 CP; - P*))2],
+
Ns
where
k
= (-NT/2)[log(2n)
for which N;
and P; < P*,
~
+ log(cr 2 >] and N1 represents the observations
N* and Pi > P*, .N 2 are the observations for which N; < N*
N3 are the observations for which N;
~
N* and P; < P*,
N4
are the observations for which N; < N*, P; < P* and (Y* + (3 2 CN; - N*)) <
(Y* + p 4 cPi - P*)),
and
N5
are the observations for which
Ni < N*,
70
N3 , N4 and N5 represent the total number of sample observations, NT.
The
required
derivatives of the log- likelihood
function
(after
simplification) are:
(4)
aLtav*
= 1t~2 (I<Yi
Nl
Y*-
(5)
8L/a~ 2
- Y*) + I<Yi - Y*- ~ 2 <Ni - N*)) + I<Yi N2
N3
~4CP;
= 11~2 (I<Y;
- P*)) +ICY; - Y*N4
.- Y* -
N2
~2(Ni
P2 <N;
I<Yi - Y*- p2(Ni - N*))(Ni - N*>)
N4
(6)
aL/a~ 4
= 11~
2
[I<Y; - Y* -
p2 <P;
N3
J(Y; - Y*- p2 (Pi - P*))(Pi - P*)J
Ns
( 8)
a 2 uav*a~ 2
(9)
a 2 uav*a~ 4 = - 1to-2[I<P; - P*) + I<P; - P*>]
=-
11 0"2 [ I ( Ni - N*) + I<N;
N2
N4
Ns
N3
(10)
a 2 ua~~
( 11 ) a 2 Lta~~
=-
11 0"2 [ I ( Ni
N2
<12 > a2 Ltap 2ap 4 = l2l
.·:
N*)2 + I<N;
N4
N*l 2)
= 11 o-2 [I cP' - P*l2 + ICP; - P*> 2 ],
N3
·..
N*l)
Ns
and
- N*)) +ICY; Ns
71
Expressions
(7)-(12)
fully deflne (2) since the Information matr·lx
is
symmetric.
Notice
that the derivatives with respect to ~ 2 are not
presented.
These derivatives are not needed to calculate the standard errors of Y*,
~
2
the
and
~
4 since the information matrix is also block
matrix of derivatives defined by (7)-(12) can be
accounting
SSE/NT,
for ~ 2 .
where
However,
inverted
Thus,
without
the maximum likelihood estimator of ~ 2 is
SSE represents the error sums of squares and NT
total number of observations.
the
diagonal.
is
the
This latter estimate was used (along with
maximum likelihood estimates of Y*,
~
2 and
estimates of the needed asymptotic variances.
~
4 1 to determine
point
72
APPENDIX C
CALCULATIOM OF THE NON-NESTED TEST
FOR THE VON LIEBIG MODEL
. . .. ..
73
The non-nested test for a linear null model is
where gi is the predicted Yi under the alternative model.
For the von Liebig model this equation is
( 2)
Y;
= (1
- o:)Min(Y*, ~1 + ll2Ni' {33 + /l4P;) + ag; +
€
iJ
which can be rewritten as
( 3)
= (1
Y;
where dni
=0
=
'
- o:)Min(Y*
if Ni
+ {l2(Ni
-
N*ldn,
~4(P;
Y* +
-
P*)dp) + agi +
Ei
> Ni*
if
Ni < Ni*
if
pi
if
pi < Pi*·
and
dpi
=0
=1
> Pi*
In order to test a= 0, the standard error
can
This
be
corresponding
(assuming
accomplished
to
the
the
error
by
setting
log-likelihood
terms to
be
up
of a must be estimated.
the
function
normally
information
for
the
matrix
above
distributed).
The
model
log-
likelihood function is
(4)
L
=
k -( 1/2a2 >[Icvi- (1- o:)Y*- o:g 1 12 + I<Yi - (1- a)(Y*
Nl
N2
{J 2 CN;
- N*)) - agi >2
+ I<Yi - (1 - o:)(Y* + {3 4 (Pi - P*))
N3
- o:g;J 2 + I<Yi- (1- o:)(Y* + {l2(N;- N*))- o:g;) 2
N4
+
74
--
+ l<Yi - (1 - a)(Y* + /:1 4 (Pi - P*)) - agi )2),
Ns
.
where the sums are defined in Appendix B.
The required derivatives are:
( 5)
aLlaY* =
((1
- a ) I a2
) [ I (Y;
Nl
-
-
b2 (Ni
N*))
-
ag;)
-
}:<Y;
+
-
(1
-
- a)(Y*
(1
- N*))
+ /:12(N;
~
+ l<Y;
-
+ /:14<P; - P*)) - ag;)
( 1 - a) ( Y*
(1
- a) (Y*
+
a) ( Y* + P4 <P; - P*))
N3
- agi) + l<Y;
N4
-
- a)Y* - ag;) + }:CYi
N2
(1
-
ag;)
1
Ns
( 6)
3LI3{3 2
( ( 1 - a) I 0"2 ) [l (Yi
=
-
- a) ( Y* +
(1
b2 CNi
N2
-
- ag i ) ( Ni
N*)
+ l<Y;
-
(1
N4
- N*))
- a)(Y* + {3 2 CN;
- N*))
- agi)(Ni - N*l)
( 7)
aLtap 4
= ((1
- a ) I 0"2
) [ L(Yi
N3
- ag i ) (Pi
-
- P*)
3L/3a =
- <
u
u2 )
[I< v; -
(1
-
- a)(Y*+
+ }:<Y;
-
(1
b4 (P;
- a) ( Y* + {34(P;
ci) Y* -
ag; )( Y* - 9; )
Nl
(Y* + b2(Ni
-
N*))
- P*))
- P*))
Ns
- ag;)(P; - P*lJ
( 8)
(1
-
ag; )( Y*
+ {32(N;
-
+ }:<Y; - ( 1 - a)
N2
N*)
-
gi )
+ l<Y;
N3
(1
- a)(Y* + {34(P;
-1 (Y i
(1
-
N4
-:- g i ) + }:<Y;
Ns
- P*))
-
ctQ;)(Y* + f34(P; - P*)
a)(Y* + p2 (N; - N*))
-
(1
- er:)(Y* +
13 4 <P;
a:g i ) ( Y* + b2 ( N;
- P*)) - ag i ) ( Y*
..
- gi
+
)
N*)
75
(10)
a 2 uav*a13 2
=-
(1
- ex)21cr2[}:<N; - N*) + }:CN; - N*)]
N2
N4
( 11 )
a 2 uav*a13 4
=-
(1
- ex)21cr2[I<P; - P*) + I<P; - P*l]
N3
Ns
( 12)
a 2LiaY*aex
=
- 1/ cr2 [ }:Y i -
2Nr< 1 - ex)Y* +
Nr
2(1
-
ex)f32<I<N; - N*) + I<N;
N2
N4
(1
- ex)I9; Nr
N*)) -
2(1 - ex)f3 4 C}:CP; - P*) + I<P; - P*))
N3
Ns
( 13)
a2 L1 a13~
=-
(1
(14)
a 2 ua13~
=-
(1
(16)
a 2 L1a13 2 aex
=-
- ex ) I cr2 [ I ( Ni - N*)2 + I<N; - N*l 2 ]
N2
N4
-
ex ) I cr2 [l (P i - P*)2 + I<P; - P*l 2 ]
N3
Ns
11cr2 (CIY 1 CN; - N*) + IY;(N; - N*))
N2
N4
- 2(1 - ex)Y*<I<N; - N*) + I<N; - N*))
N2
N4
- 2(1 - ex)f32C}:(N; - N*) 2 + I<N; - N*)2)
N2
N4
+ (1- 2exHt9;CN;- N*) + }:g 1 CN;- N*ll)
N2
(17)
a 2 L1a13 4 aex
=-
N4
llcr2 (ciY;(P; - P*) + }:Y;CP; - P*))
N3
N5
- 2(1 - ex)Y*<I<P; - P*) + I (Pi - P*))
N3
Ns
- 2(1 - ex)f34C}:CP; - P*l2 + I<P; - P* )2)
I
N3
Ns
76
+ (1- 2c£)(}:g;(P;- P*) + }:g;(P;- P*))]
N3
(18)
a2 Ltaa2
=-
lt~ 2 [NrY* 2
+ ~~<ICN;
+
Ns
2Y*~ 2 C}:CN; - N*) + }:CN; - N*))
N2
N4
- N*> 2
I<N; - N*) 2 )
+
N2
N4
-2P 2 C}:g 1 CN; - N*)
N2
+
N4
I (pi
2Y*P 4 <}:<P; - P*)
+
P~ <I< P; - P*) 2
}:CP; - P*)2)
Ns
N3
+
}:g;CN; - N*))
+
Ns
+
N3
-.
-2P4<L9;CP; - P*)
N3
-
- P*))
+
L9;CP; - P*))
Ns
- 2Y*L9; + l9~]
NT
Expressions
the
. NT
(9) - (18) were used to estimate the covariance matrix
estimated
parameters
in
(2) as
outlined
in
Appendix
estimated asymptotic standard error for a was used to test H0 :
( 2) •
·... ·.
B.
for
The
a= 0 in