Density-dependent dynamics in regulated industries: founding rates of banks and life insurance companies

Administrative Science Quarterly,  March, 1991  by James Ranger-Moore,  Jane Banaszak-Holl,  Michael T. Hannan

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[Mathematical Expression Omitted] (11) and assumes that [epsilon.sub.t] follows a gamma distribution (see McCullagh and Nelder, 1983). The presence of [epsilon.sub.t] produces overdispersion.

We have contrasted the fits of two parameterizations of overdispersion. In one, the ratio of the variance of the count of foundings to the mean is constant; in the second, this ratio increases linearly with the mean of the process. For both populations we found that the latter fit better. Thus we use a specification of overdispersion of this form

[Mathematical Expression Omitted] (12)

This parameterization is implemented in LIMDEP (Greene, 1988) and has been programmed in GAUSS by Barron (1990).

In each analysis reported below, we contrasted the fit of negative binomial and Poisson models. The negative binomial improved significantly over the Poisson model for both bank and life insurance company populations, indicating a substantial degree of overdispersion. Significant overdispersion, however, does not characterize the founding processes of savings banks and mutual life insurance companies, presumably because many fewer foundings took place in these subpopulations and there cannot be many years with exceptionally high counts of foundings. Therefore, we report estimates of Poisson regression models for these subpopulations. Our estimates of Poisson regressions use the method of maximum likelihood and a correction to produce heteroscedasticity-consistent standard errors, as implemented in King's (1990) program COUNT.

In estimating models from yearly counts of foundings, we do not include the observation for a population's (or subpopulation's) first year, because the first observation has a peculiar status. By definition, it is the first positive value of the count of yearly foundings; no subsequent year's value is so constrained. Each subsequent year can have zero foundings or some positive number. Thus the first year's count cannot be assumed to be a realization of the same probability mechanism as counts for subsequent years. For this reason, we condition on the appearance of a population (or subpopulation) in a particular year by beginning the record in that year and analyzing the flow of foundings in all subsequent years.

A related issue concerns treatment of years before the first founding in the more recent subpopulation in each pair. As we noted above, the first savings bank in Manhattan was founded 28 years after the first commercial bank; the lag between the two life insurance subpopulations was also 28 years. One possible approach, used by Hannan and Freeman (1987, 1989), is to begin both subpopulations at the same time and treat the 28 consecutive counts of zero foundings per year as valid outcomes. Instead, we begin each subpopulation in the year after its first founding so as to treat subpopulations symmetrically with complete populations. Our reasoning is that the subpopulations might be so distinct as to be bounded populations with little interaction over the boundary. If this were the case, it would be appropriate to treat the date of initiation of each as the starting time of the process. Since we do not want to prejudice the issue of independence of subpopulations by design, we start the record on each with the year following the first observed event.