SOUTHERN JOURNAL OF AGRICULTURAL ECONOMICS
JULY, 1981
SPLINE FUNCTIONS: AN ALTERNATIVE TO ESTIMATING
INCOME-EXPENDITURE RELATIONSHIPS FOR BEEF
Chung-Liang Huang and Robert Raunikar
Income-expenditure relationships are impor-
tant components in many economic models used
to project food expenditure and to understand
food-expenditure behavior. The empirical esti-
mation of income-expenditure relations has con-
centrated on the effects of income in explaining
the variations of the household food expenditure.
However, the problem of structural or paramet-
ric homogeneity for Engel curves in the analysis
of household food expenditure behavior has re-
ceived less attention in the applied demand litera-
ture.
Agarwala and Drinkwater argue that the famil-
iar Engel curve results require modification when
applied in situations in which the structure of the
population and economy is diverse and changing.
When economic and socioeconomic characteris-
tics change, policies predicated on forecasts of
such change cannot be based on parameter esti-
mates from models that implicitly or explicitly
assume that such variations cannot occur. There-
fore, meaningful applications of even the sim-
plest income-expenditure parameters to policy
analysis should be conditioned on evidence of
structural or parametric homogeneity.
The traditional approach to test the assump-
tion of structural homogeneity for Engel curves
is based on sample partitions. Forsyth studied
the income-expenditure relationships by stratify-
ing the sample according to numbers of persons
in the household. Hassan and Johnson examined
the parametric homogeneity for Engel curves in
Canada across sample partitions based on cities,
family income, life cycles, age of family head,
tenure in home, and education of family head.
With few exceptions, their results show a lack of
homogeneity of the Engel curve coefficients
across sample partitions. Stratifying the sample
by socioeconomic characteristics is cumbersome
because it can result in many estimated relation-
ships. Moreover, partitioning the sample into dif-
ferent socioeconomic groups substantially re-
duces the degrees of freedom for the estimated
relationships fitted to the subsamples and, hence,
reduces the estimates reliability.
This study develops an alternative approach to
account for the effect of socioeconomic charac-
teristics upon food expenditures. Specifically,
spline functions were developed to reflect differ-
ences in income-expenditure relationships by al-
lowing different functional forms within the vari-
ous subintervals of income and household size
variables. The authors demonstrate how spline
functions capture various empirical economic re-
lationships and test the hypothesis that consum-
ers react differently at different income levels.
THE STATISTICAL MODEL
Adopted from the engineering discipline,
spline functions have been applied to several
economic problems in recent years (Barth, Kraft,
and Kraft). The development of spline theory
and piecewise regression models are well known
and discussed elsewhere (Poirier; Smith; Wold).
Recently, Buse and Lim have shown that spline
functions can be regarded as a special case of
restricted least squares. They demonstrate how
the continuity restrictions and the validity of the
restrictions can be tested using restricted least
squares; and prove that under a common set of
restrictions, the two procedures are equivalent.
An alternative way of handling the restricted
least squares problem is to incorporate the re-
strictions in the fitting process so that the esti-
mated coefficients satisfy the restrictions ex-
actly. This can be done by working out directly
the special form of the estimating equations, the
approach employed by Suits, Mason, and Chan
which related interest rates to money supply and
inflation. By using appropriately defined com-
posite variables, they demonstrated that the mul-
tivariate spline functions can be treated as a least
squares regression model and fitted by standard
ordinary least squares (OLS) procedures.
The development and formulation of spline
functions for estimating income-expenditure re-
lationships are briefly discussed to show how this
procedure is used for investigating the structural
homogeneity of household expenditure behavior
with respect to household income and size. For
simplicity, household income is employed to in-
troduce the'procedure.
To begin with, one may choose to fit a piece-
wise linear regression; that is, one linear segment
Authors are Assistant Professor and Professor, respectively, in the Department of Agricultural Economics, University of Georgia, College of Agriculture, Georgia
Experiment Station, Experiment, Georgia.
The authors are grateful to Stanley M. Fletcher and to the anonymous reviewers for their helpful comments and suggestions.
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