equation is represented by additive splines in
household size and income. That is4
3
(5) E = a + ɪ ∕3k(H-H1)k +
k=l
m
∑ ‰-ft+1) (H-H1)3Dh +
j=2
3
Σ γk(Y-Y1)k +
k=l
n
∑ (%+2-yi+1) (Y-Yi)3Si-1 + U,
i = 2
where H is the number of persons in the house-
hold. E and Y represent household food expendi-
ture and income, respectively, as previously de-
fined. H1 and H1, j=2,3,...,m, define the knots
where household size is divided into m segments.
Y1 and Y1, i = 2,3,...,n, define the knots for
household income. Di.1 is a dummy variable with
D1.1=l, if H⅛Hi, and 0, otherwise; Sh repre-
sents another set of dummy variables, with
Sh = 1, if Y≡≡Yi, and 0, otherwise.
Thus, equation (5) represents a multiple re-
gression of E on a set of composite variables.
Estimates of coefficients in equation (5) are ob-
tained directly from the regression analysis. With
this formulation, the analogy of the spline meth-
od to the adaptive regression model suggested by
Cooley and Prescott becomes evident.5 Cooley
and Prescott argue that the parameters in most
economic models cannot, in general, be expected
to be constant over all the observations. In time-
series studies, there can be variation over time in
the parameters. In cross-section studies, there
can be heterogeneity in the parameters across
different cross-section units. Since structural re-
lationships of household food expenditure were
postulated to change as the level of household
income and size change, equation (5) can be re-
garded as an alternative to varying-parameter
models.6 This analogy implies that the use of
spline functions is an appropriate procedure for
application in the present study.
THE DATA AND ESTIMATION PROCEDURE
Household food purchase data from a con-
sumer panel consisting of approximately 120 re-
porting households in Griffin, Georgia, during
the 1975-77 period were used for this analysis.
Four beef expenditure categories were examined
with separate regression equations: (a) fresh beef
(includes all types of beef that were purchased in
fresh form, such as ground beef, beef roasts,
steaks, stew beef, short ribs and other beef);
(b) ground beef (includes all types, e.g., ham-
burger, ground chuck, extra lean); (c) beef roasts
(includes chuck roast, rib roast and other roasts);
and (d) beef steaks (includes round steak, sirloin
steak, T-bone steak and other steaks).
To estimate equation (5) statistically, knot lo-
cations were specified, using an empirical ap-
proach to determine the appropriate position of
the knots. Therefore, the knots are located at
points separating selected intervals within which
the scatter of observations is distributed in simi-
lar patterns. In addition, since each additional
interval used to fit the function involves an addi-
tional variable in the regression equation and loss
of an additional degree of freedom in the re-
sidual, it is also desirable to keep the number of
knots as small as possible. For convenience and
simplicity, the same number and position of
knots were chosen for each beef expenditure cat-
egory, although the number and location of the
knots may vary among different equations.
Based on these considerations, equation (5) was
fitted to the sample data of each beef expenditure
category with household income divided into
three segments, such that $l,285=sY<$10,000,
S10,000≤Y<S25,000, and Y⅛S25,000; and
household size was divided into two intervals of
1≤H≤3 and H>3.
A spline function of equation (5) was specified
and estimated by OLS for each beef expenditure
category. Within the framework of least squares,
the existence of significantly different fit between
two spline models of different degrees in poly-
nomials can be tested. The test procedure in-
volves the F-test, which compares the difference
in error sum of squares between the two models.
The coefficient of partial determination, partial
R2, associated with additive splines in income
and household size, respectively, can also be
calculated and their significance tested by using
the F-Statistic. In addition, the significance of an
individual coefficient can be determined by test-
ing the validity of the occurrence of a structural
change at the endpoints of the polynomial seg-
ment in a particular interval. For example, in
equation (5), the null hypothesis tested is
whether β1+2=∕31+1, or y1+2=¾+ι∙ Because this is
a linear restriction, the standard test using the
t-statistic is appropriate.
RESULTS AND DISCUSSION
Results obtained by applying spline functions
to household beef expenditures are presented in
Table 1. The F-test was used to determine
whether the additive cubic splines in household
income and household size were significantly dif-
4 Although the cubic splines were specified both for household income and size, there is nothing about either the theory or the practice that requires all individual segment's
to be fitted by polynomials of the same degree. Equation (5) can be reduced to quadratic or linear splines simply by adding and deleting the appropriate terms.
5The adaptive regression suggested by Cooley and Prescott allows the constant term to vary in an autoregressive fashion to account for structural change. They argue that
for most economic time-series, their model gives better results for economic forecasting in practice.
6SeveraI models for tackling the problem of variational parameters in addition to the adaptive regression model are discussed in Maddala (Chap. 17).
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