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H. Mori et al. /International Food and Agribusiness Management Review 3 (2000) 189-205
from 1980 to 1990 alone, a part of which may result from the difference in birth cohort, since
those who were 30-39 years old in 1990 were born in the 1950s.
There is an exact linear relationship among cohort, age, and period: a subject i years of age
at the jth year of the study belongs to birth cohort j 2 i. This linear relationship presents a
problem in model identification, and results in infinitely many least squares solutions for
parameter estimates (Mason & Fienberg, 1985). Overparameterized models have been
widely studied by statisticians (Graybill, 1978). Nakamura (1986) presents a “Bayesian
approach” to selecting among the class of least squares estimates. Nakamura begins with the
usual cohort model:
mij 5 b 1 biA 1 bjP 1 bkC 1 ei (1)
ij,
where mij represents the consumption of age group i, in survey period j, and eij represents
sampling error in the survey. Nakamura then adds an assumption of “gradual changes”
(zenshin-teki henka) in the parameter estimates:
biA2 biA11 < 0,
bjP2 bjP11 < 0,
bkC2 bkC11 < 0 (2)
along with the usual sum to zero constraints:
SbiA 5 SbjP 5 SbkC 5 0 (3)
With these added constraints the parameters can be uniquely estimated using least squares
techniques. Specifically, we find the parameter estimates that minimize
S@mij2~b1biA1bjP1bkC!#2 (4)
subject to the side constraints that
1/s2AS(biA 2 biA11)2 1 1/s2PS(bjP 2 bjP11)2 1 1/s2CS(bkC 2 bkC11)2 (5)
The side constraints require an iterative solution, and are evaluated using the ABIC (Akaike’s
Bayesian Information Criterion) function. Nakamura uses a grid search over the range of the
hyperparameters s2 from 227 to 27. In this study, the range is restricted to 222 to 22, for
reasons explained in Mori and Gorman (1999, pp. 85- 88).
4. Results for selected commodities
Using the individual consumption derivation method provided by Mori and Inaba (1997)
and Lewis et al. (1997), several commodities were analyzed. The method was refined and
more recent data were included (Tanaka & Mori, 1998; Mori, 1998; etc.). Cohort analysis
was performed on resulting individual consumption estimates by age classes for the various