The determinants of entry for black-owned commercial banks.
Market entry plays an important role in determining the nature of long-run market equilibrium. When unimpeded, entry can render the long-run market equilibrium competitive. From a positive and normative viewpoint, such a competitive outcome is ideal, as it can assure an allocation of resources that is both privately and socially desirable. Where there is no case for natural monopoly, the absence of entry has both implications and consequences. One implication is that the absence of entry can result from the strategic behavior of incumbent firms that deters entry and/or the existence of structural entry barriers. The major consequence is that impeded entry can affect a divergence between the private and social optimum with respect to the number of firms.
For commercial banking markets, entry or the lack of it has similar implications and consequences. However, unlike product markets in general, banking markets are more likely to be subject to policy induced entry barriers. This study reports an analysis of Black-owned Commercial Bank (BCB) entry based on a cross section of eighty-nine banking markets. Observed BCB entry is treated as a discrete variable that is a function of the height of entry barriers and disequilibrium conditions. Econometrically, a Poisson regression model is utilized to appropriately accommodate markets without any observed BCB entry.
This study is of interest for several reasons. First, no empirical account exists that explains the entry of BCB.(1) Second, if impeded entry is a result of entry barriers, one implication is that incumbent banks have market power. Hence, it would be useful to evaluate whether the degree of competition, measured indirectly via the level of plausible entry barriers, has explanatory power for the observed entry of BCBs. To the extent that entry barriers impede the formation of BCBs and make it possible for non-Black bank incumbents to engage in discriminatory lending practices, policy interventions that lower entry barriers for BCBs may enhance the availability of credit to Black households and business enterprise. For example, Grown and Bates ( 1991 ) find that commercial banks treat Black-owned construction companies different from non-Black-owned construction companies. The result of this treatment is that Black-owned construction companies are less capitalized, and more likely to fail than non-Black-owned construction companies. Thus, in a broad socioeconomic context, empirical knowledge of the entry barriers faced by BCBs is desirable. Any insights on the empirical significance of entry barriers faced by BCBs can potentially expand the set of policy options designed to combat discrimination in lending to Black households and business enterprise. Finally, the post-1960 period featured a policy designed presumably to increase the participation rate of minority groups in commercial banking. The creation of the Minority Bank Deposit Program (MBDP) was motivated in part by the apparent inability of minority banks to maintain a stable low cost deposit base.(2) Under the MBDP, minority-owned commercial banks were granted special preference as depositories for U.S. Government Deposits. The results reported here will lend some insight into the extent to which the MBDP influenced BCB entry.
I. POISSON PROBABILITY MODEL OF ENTRY
A. Modelling Discrete Entry
The conventional approach to estimating empirical models of entry has been to estimate a log regression on cross-sectional data.(3) This study also relies on the cross-sectional approach under the assumption that industry/market entry barriers are structural characteristics, and unlikely to change significantly over time (Orr, 1974a). However, a Poisson regression is more appropriate for several reasons (Chappel, Kimenyi and Mayer, 1990). First, since observations on entry take on only nonnegative integer values, they are count data. Consequently, an econometric framework based on a discrete probability distribution is more appropriate. Second, a logarithmic specification requires either excluding zero entry observations, or incorporating ad hoc procedures such as the use of dummy variables to account for zero entry observations. In contrast, specifying entry as a discrete random variable permits zero entry observations to be a natural outcome of the econometric specification. Finally, unlike a logarithmic specification, when entry is specified as a discrete random variable, maximum likelihood estimation is permissible, which may afford more efficient estimators of the parameters.
In a Poisson model, discrete observations on BCB entry are viewed as being generated by the probability distribution:
[Mathematical Expression Omitted]
where: [E.sub.i] = Observed entry in industry/market i,
n = 0,1,2,....N,
e = 2.71828, and
[[lambda].sub.i] = Expected value of [E.sub.i] = Variance of [E.sub.i].
A regression model is formulated by specifying the Poisson parameter [[lambda].sub.i] as a deterministic function of exogenous variables with an unknown parameter vector p. Following Hausman, Hall and Griliches (1984), and Chappell, Kimenyi and Mayer (1990) we can specify:
[[lambda].sub.i] = [e.sup.[Z.sub.i][beta]]
where [Z.sub.i] is a vector of exogenous variables. The log likelihood of entry is then (Hausman et al., 1984):
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
The first order condition is nonlinear in [beta] and can be estimated with a numerical maximum likelihood procedure, or as in this study by an iterative nonlinear least squares procedure (Nelder and Wedderburn, 1972). Because the Hessian is negative definite, convergence to a unique solution is assured.(4) For BCB entry, a plausible theoretical specification for [[lambda].sub.i] is:
[Mathematical Expression Omitted]
where: [[pi].sub.i] = expected post entry profit in market i, [EB.sub.ij] = level of entry barrier j in market i, [X.sub.ik] = market growth/disequilibrium variable k in market i.
Peltzman (1965), Throop (1978) and Ladenson and Bombara (1984) all report results showing that bank regulatory policy is a significant determinant of bank entry. Because the cross-sectional approach in this study is not amenable to the inclusion of qualitative variables that are conventionally used to proxy changes in bank regulatory policy, its effects are ignored here. To the extent that bank regulatory policy influences entry of BCBs, the results reported here may be biased. Stigler (1968) and Demsetz (1982) suggest that some entry barriers themselves arise as a result of regulatory restrictions on market conduct. If the entry barriers considered in this study reflect this, then the effects of regulatory policy on BCB entry are indirectly accounted for.
II. SAMPLE, VARIABLES, AND DATA
The sample consists of BCBs that entered in cities where the population was at least 100,000 over the period 1960-1978. Three considerations motivated this sampling procedure. First, the BCB entry observed over the 1960-1978 period occurred in cities in which the percentage of the population that was Black exceeded that of the United States as a whole.(5) This suggests that one of the perceived benefits/determinants of entry for a BCB is a large and/or growing Black population, which likely reflects profitable entry opportunities for a BCB. Detailed census data on the size and growth rate of the Black population were only available for cities in which the total population was at least 100,000. Second, the predominance of BCB entry into cities where the Black population is significant suggests that the relevant market for a BCB is a city.(6) Finally, the years 1960-1978 account for the greatest net entry of BCBs (39), for which data are available. Consequently, the period 1960-1978 contains more cross-sectional observations on BCB entry than any other.
Identification of a city as the relevant market for BCB entry also requires an identification of the incumbents a BCB would compete against following entry. One popular approach in the Structure-Conduct-Performance paradigm is to view all incumbents in the defined industry/market as the relevant post-entry competition. Evidence on market segmentation (Ederington and Skogstad, 1977) and the dissimilar post-entry asset/liability profiles of BCBs relative to non-BCBs (Kwast and Black, 1983), suggest a different approach to incumbent identification for BCBs. To the extent that banking services are segmented, and the average BCB has a balance sheet asset/liability composition significantly different from the average non-BCB, potential post-entry competition for a BCB is not likely to be all the banking firms in a given city. In the standard textbook account of a competitive equilibrium, the long-run equilibrium has the feature that all existing firms have the same characteristics (costs, technology, asset/liability composition etc.). Hence, in the long-run, successful entrants face competition from incumbents with identical characteristics. In a market that is segmented, one could easily imagine a scenario in which there are clusters of submarkets, populated by firms with identical characteristics in competition with one another. Where the short-run entry process is not competitive, potential entraps can presumably face post-entry competition from incumbents with dissimilar characteristics. Acs and Audretsch (1989) find evidence suggesting an entry process in which small firms with similar characteristics enter a market, and compete with incumbents with similar characteristics in similar markets. Finally, MacDonald's (1986) analysis of food manufacturing shows the existence of a competitive fringe in otherwise noncompetitive markets in which there is a substantial amount of entry and exit.
For BCBs, these considerations suggest that the relevant incumbents are those banking firms already in the market with characteristics similar to incumbent BCBs. Hence, to construct the relevant independent variables for the entry model, the sample also includes for each market, observations on those commercial banks with Total Asses less than or equal to that of the average BCB in all markets. In this study, the relevant incumbents for a entering BCB are assumed to be those incumbents with size, as measured by Total Assets, no greater than that of the typical incumbent BCB in any market.
Over the period 1960-1978, BCB entry occured in 29 cities in which the population was at least 100,000. Estimation of the entry model with just these 29 observations poses an often overlooked difficulty. Chappell et al. (1990) note that to include just markets where there is positive entry is wholly inadequate. It is those markets in which no entry occurs that reveal the configuration of profits and other market characteristics that make entry exclusion possible. In general, the exclusion of zero entry markets could bias parameter estimates. To overcome this difficulty, the sample was augmented by including observations on commercial banks in all cities where no BCB entry occurred, and where the population was at least 100,000. For these markets, the independent variables were constructed by considering only those commercial banks with Total Asses less than or equal to that of the average BCB in all markets. The augmentation of the sample in this manner also complements the Poisson regression approach of this study since zero is a natural outcome of the Poisson specification.
For the sample considered in this study, the cross sectional variables are constructed as follows:
A. Entry: observations were measured as the total number of BCB entries in market i over the period 1960-1978.
B. The Independent Variable: were measured as average levels in market i over the period 1960-1978.
A consideration of BCB entry in a time-series would result in an insufficient number of observations for reliable time-series estimation. The data begin in 1960, and are reported annually for 4 years (1960-1964), semi-annually for 11 years (1965-1975) and quarterly thereafter (1976-present). Fifty-six observations could be obtained with the quarterly data. However, a consideration of this time period would perhaps be relatively unimportant in assessing the determinants of entry for BCBs. The post 1976 period is characterized by negative net entry, and the entry that did occur was less dramatic than the entry that occurred over the 1960-1978 period.
Utilizing end of year annual data over the period 1960-1978, the sample selection criteria resulted in 89 city-market observations on entry. Twenty-nine of the observations account for markets in which there was actual BCB entry. The remaining observations represent markets where no BCB entry occurred, presumably reflecting the configuration of profits and market characteristics that make BCB entry exclusion possible.
B. Definition of Variables
The empirical literature on entry has used a set of exogenous variables that has become quite standard in estimating entry models. Entry is generally specified to be a function of variables that reflect (1) market structure characteristics that inhibit entry, or entry barriers; (2) growth and profitability measures that induce entry; (3) the effect of technology on the conditions of entry; and (4) the risk preferences of shareholders and managers making decisions about bank capital. The role that technology has on commercial bank entry has not been considered empirically. Available data on the commercial bank industry do not report outlays on items such as research and development, the usual proxy for technology in empirical entry analysis.
As an empirical model of BCB entry, the following Poisson specification is considered:
(1) [[lambda].sub.i] = exp([[beta].sub.0] + [[beta].sub.1][APROF.sub.i] + [C.sub.1][AHH1.sub.i]
+ [C.sub.2][ADD1.sub.i] + [C.sub.3][ASUPE.sub.i] + [C.sub.4][CAPREQ.sub.i]
+ [C.sub.5][APROF.sub.i] + [D.sub.1][AVGPOP.sub.i] + [D.sub.2][AGBPOP.sub.i] + [D.sub.3][ADGRO.sub.i] + [D.sub.4][AUSDGRO.sub.i])
where: [E.sub.i] = gross entry in market i
[APROF.sub.i] = average profitability in market i
[AHH1.sub.i] = average Herfindahl-Hirschman index of Loan concentration in market
[ADD1.sub.i] = average Herfindahl-Hirschman index of Deposit concentration in
[ASUPE.sub.i] = average supply effect in market i
[CAPREQ.sub.i] = average capital requirements in market i
[AVGPOP.sub.i] = average Black population in market i [AGBPOP.sub.j] = growth rate of Black population in market i [ADGRO.sub.i] = average growth rate of Total Deposits in market i [AUSDGRO.sub.i] = average growth rate of U.S. Government Deposits in market i [AVPROF.sub.i]= average risk in market i
Orr (1974a) found that entry into Canadian manufacturing decreased with respect to the standard deviation of profits. Thus [AVPROF.sub.i] is a proxy for risk in market i. In Equation 1, APROFi measures (expected) profitability, [AHH1.sub.i], [ADD1.sub.i], [ASUPE.sub.i], [CAPREQ.sub.i], measure entry barriers, [AVGPOP.sub.i], [AGBPOP.sub.i], [ADGRO.sub.i], [AUSDGRO.sub.i], measure market growth and disequilibrium, and [AVPROF.sub.i] measures capital risk.
The above variables attempt to capture the perceived benefits and costs of entry for BCBs. APROF, AVGPOP AGBPOP ADGRO, and AUSDGRO represent the benefits. BCB entry is expected to be higher in banking markets where incumbents earn high profit(s) (APROF), and where the markets are large and/or growing (AVGPOP, AGBPOP, ADGRO, AUSDGRO). The costs of entry are captured by variables reflecting entry barriers and risk; AHH1, ADD1, ASUPE, CAPREQ, and AVPROF. BCB entry is expected to be lower in markets where: concentration is relatively high (AHH1, ADD1), scale economies are relatively large (ASUPE), capital requirements are high (CAPREO), and capital risk is high (AVPROF). The expectations for the coefficients are therefore: [[beta].sub.0] [greater than or equal to] 0, [[beta].sub.1] > 0, [C.sub.1] < 0, [C.sub.2] < 0, [C.sub.3] < 0, [C.sub.4] < 0, [D.sub.1] > 0, [D.sub.2] > 0, [D.sub.3] > 0, [D.sub.4] > 0, and [C.sub.5] < 0.
For the empirical specification in Equation 1 entry ([E.sub.i]) is a measure of gross new BCB entry over a long period of time. Orr (1974b) utilized a net entry measure, which due to measurement error, can bias parameter estimates (Duetsch, 1984). Because of this, Khemani and Shapiro (1986) note that gross entry is a better measure to use. ASUPE is the fraction of market output (Loans) accounted for by a bank of Minimum Efficient Scale (MES). If scale economies are large, an entrant's output can depress market price below the entrants average cost of production. ASUPE is therefore a rough approximation to scale economies in a given market. Orr (1974b) considered an MES bank to be the smallest profitable bank in a normal year of economic activity. In this study, business cycle effects will be ignored, and it is assumed that an MES bank in market i at time t is the smallest profitable bank.
Some of the variables considered above are questionable from a theoretical perspective. Stigler (1968), Schmalensee (1981), and Baumol and Willig (1981) provide theoretical reasons why scale economies and/or capital intensity are not necessarily entry barriers. Nevertheless, the approach in this study will follow that of Orr (1974b), where the reported results show that commercial bank entry is inversely related to measures of scale economies, and capital intensity.
All variables except [AVGPOP.sub.i] and [ACBPOP.sub.i] are constructed from data reported in the Reports of Income and Condition (ROC), and Income and Dividend Statements (IDS) compiled by the Federal Reserve Board of Governors. Both the ROC and IDS data were obtained from the RCRI series tapes catalogued at the Federal Reserve Bank of Chicago. [AVGPOP.sub.i] and [AGBPOP.sub.i] are constructed from census data.
IV. RESULTS AND CONCLUSIONS
A. Fit of the Model
Table 1 reports the results on Equation 1. CAPREQ is measured in natural logarithms and all reported standard errors are asymptotic.(8) The results indicate that only one variable (AVGPOP) is significant, and three (APROF, AHH1, CAPREQ) have signs that are contrary to a priori expectations, suggesting that Equation 1 is a poor candidate for an empirical model of BCB entry. In particular, Equation 1 may suffer from overdispersion, or contain independent variables that are not sufficient for discriminating between entry and no entry across the 89 markets in the sample.
Table 1. POISSON REGRESSION ESTIMATES: EQUATION 1 Variable Parameter Estimate Standard Error T-statistics Intercept -4.09 3.18 1.29 APROF -25.03 46.78 .53 AHH1 .28 2.08 .13 ADD1 -1.94 1.78 1.09 ASUPE -1.07 .87 1.23 CAPREQ .52 .49 1.06 AVGPOP .03 .01 3.0(a,b,c) AGBPOP .003 .003 1.0 ADGRO .0002 .0003 .67 AUSDGRO .002 .002 1.0 AVPROF -14.82 24.97 .59 Notes: a. Significant at the .10 level b. Significant at the .05 level c. Significant at the .01 Ievel
Table 2 reports on overdispersion tests for Equation 1.(9) Cameron and Trivedi (1986, 1990) suggest that in cases where there is overdispersion, a good candidate for a probability distribution that features overdispersion is the negative binomial.(1u) The results in Table 2 are for testing the Poisson mean-variance equality against the negative binomial mean-variance relationship under the condition: [phi]([[lambda].sub.i])[alpha] = [[lambda].sub.i].(11) In Table 2, the results indicate that the null hypothesis cannot be rejected for the assumed form of overdispersion. Failure to reject the null hypothesis suggests that as an econometric model of BCB entry, Equation 1 is compatible with specifications such as the Poisson, that have mean-variance equality.
Table 2. OVERDISPERSION TEST: EQUATION 1 Variable Parameter Estimate Standard Error T-statistics [[lambda].sub.i] -.11 .13 .85
The poor fit of the model may result from inclusion of exogenous variables not related to BCB entry. To investigate this possibility, the entry specification in Equation 1 was subjected to a stepwise discriminant analysis.(12) The results indicated that the variables ADD1, AUSDGRO,AVGPOP, and CAPREQ provide the best discrimination amongst the observations on BCB entry in the data. APROFs lack of discriminatory power suggests that as a measure of expected profitability, it has no discriminatory power in the data.
The above considerations suggest the following empirical model of entry for BCBs:
(2) [[lambda].sub.i] = exp([[beta].sub.0] + [C.sub.2][ADD1.sub.i] + [C.sub.4][CAPREQ.sub.i]
+ [C.sub.1][AVGPOP.sub.i] + [C.sub.2][AUSDGRO.sub.i])
The results of Equation 2, reported in Table 3 indicate that all but one variable are statistically significant and have signs that conform to a priori expectations. Table 4 reports the results of an overdispersion test on Equation 2. Again, the Poisson restriction that the mean equals the variance cannot be rejected. In tandem, the results reported in Tables 3 and 4 suggest that as an empirical model of BCB entry, the specification in Equation 2 is an adequate empirical model of BCB entry.
Table 3. POISSON REGRESSION ESTIMATES: EQUATION 2 Parameter Standard T- Variable Estimate Error statistic Intercept -5.02 3.04 1.65 AVGPOP .03 .01 3.0(a,b,c) ADDI -2.28 .66 3.65(a,b,c) CAPREQ .67 .45 1.49 AUSDGRO .003 .001 3.0(a,b,c)
B. The Determinants of BCB Entry
The results in Table 3 suggest that BCBs are attracted to markets in which the Black population is large, and where U.S. Government Deposis are growing. In markets where deposits are highly concentrated, BCB entry is discouraged. The estimated sample mean of BCB entry ([[lambda].sub.i]) is .4453. Evaluating all variables at their respective means, the elasticities of BCB entry with respect to the average size of the Black population (AVGPOP), deposit market concentration (ADD1), and the growth rate of U.S. Government Deposits (AUSDGRO) are .0003, -2.85, and .023 respectively. Given that the actual average of BCB entry in the sample is .3820, and not .4453; the Poisson entry model overpredicts BCB entry on average. The recent analysis of Chappell et al. (1990) suggests that the extent to which an entry model over or under predicts is not the sole criterion for assessing the adequacy of the model. One implication of limit pricing in a dynamic setting is that incumbents face a continuum of pricing strategies linked to the probability of entry (Kamien and Schwartz, 1971; Gaskins, 1971). For example, if market concentration is high, the necessary entry deterring price decrease is likely to be small enough to render limit pricing feasible for incumbents. This suggests that the probability of entry decreases as market concentration increases.
Table 5 reports average Poisson probabilities of BCB entry across two market structures. Based upon the significance of ADD1 in the model, two market structures were created using the average value of ADD1 in the sample. The probabilities were calculated as:
[Mathematical Expression Omitted]
The results indicate that the probability of BCB entry decreases with respect to concentration in the deposit market for all levels of entry considered. For initial entry ([E.sub.i] > 0), BCB entry is 2.1 times more likely in markets where deposit concentration is below average, relative to markets in which deposit concentration is above average. The relative probabilities increase to 6 when entry of more than 5 BCBs is considered. In general, the estimated probabilities are consistent with the probabilities one expects to find in a limit price model of entry. Incumbents in markets where concentration is high are more likely and/or able to engage in entry deterring behavior that reduces the probability of entry. Within the sample the mean ADD1 is .52 in markets that experienced BCB entry in contrast to .78 where no BCB entry occured. Hence the actual pattern of BCB entry generally conforms to the pattern of estimated entry probabilities.
That the coefficient of AUSDGRO is less than the coefficient of AVGPOP, implies that the disequilibrium in the market for deposits created by the Minority Bank Deposit Program (MBDP) is less effective in encouraging BCB entry than are increases in expected profitability. There was a presumption that deposits received by BCBs via MBDP would be cheaper relative to other types of deposits (Scott, Gardner and Mills, 1988). Lower deposit costs, ceteris paribus, should increase profitability and make entry feasible for BCBs eligible to receive them. In the sample, a one percent increase in the growth rate of U.S. Government Deposits increases the expected value of BCB entry by approximately 2.3 x [10.sup.-3]. A one percent increase in the average size of the Black population increases the expected value of BCB entry by approximately 3.0 x [10.sup.-4]. The coefficient on ADD1 suggests that the most important determinant of BCB entry is concentration in the deposit market. In the sample, a one percent increase in deposit market concentration decreases the expected value of BCB entry by 2.85 percent on average. Whether or not a BCB enters a particular market is by and large a function of market structure. The Poisson entry probabilities reported in Table 5 lend some credibility to this notion.
The estimated model of BCB entry indicates that the determinants of entry are expected profitability, the growth rate of U.S. Government Deposits, and concentration in the deposit market. Overall, the expected value of BCB entry appears to be most sensitive to changes in deposit market concentration. This suggests that at the policy level if the objective is to encourage the formation of BCBs, a policy that renders banking markets more competitive may be most effective. From a social welfare perspective, more competitive banking markets could facilitate more BCB entry, and reduce the incentives that incumbent banking firms have to engage in discriminatory lending practices. If deposit market concentration depends upon the charter issuing regulatory authority and not on the behavior of incumbents in banking markets, then a liberalized chartering policy to the extent that it does not conflict with the regulatory objectives of a sound banking system may be beneficial to the formation of BCBs. As for the MBDP, the results reported here show that the effect of deposits received under this program have an effect of less magnitude than the effect associated with decreases in deposit market concentration. This does not imply that the MBDP is a cost ineffective way to stimulate BCB entry, and should therefore be abandoned. The results reported here only suggest that relative to a policy that makes U.S. Government Deposits available to BCBs, a policy that decreases deposit market concentration provides a greater stimulus to BCB entry.
The results reported here are not necessarily robust, and are therefore subject to some caveats. First, the sampling procedure was motivated primarily by a desire to generate an adequate number of observations on BCBs to fit a model, and to capture a time period that accounted for the highest amount of gross BCB entry. Thus, a fitting of the model to a time period alternative to the years 1960-1978 with fewer BCBs could result in coefficient estimates different from those reported above. Finally, the data utilized in this study are based upon year end reporting. As the model is fitted as such, issues of causality may be confounded. For example, it could be that U.S. Government deposits grow in a market because a BCB entered earlier in the year. It is quite possible that this may be the direction of causality. If so, the reported results on the effects of the growth of U.S. Government Deposits on BCB entry should be interpreted with caution.
(*) Direct all correspondence to: Gregory N. Price, North Carolina A&T State University, Department of Economics, Greensboro, NC 27411.
(1.) For a general account of BCB formation after the abolition of slavery, an interesting account is provided by Abram Harris (1936).
(2.) See Price (1993) for an overview of the Minority Bank Deposit Program.
(3.) See the studies by Orr (1974a,1974b) and Khemani & Shapiro (1986).
(4.) Nelder and Wedderburn (1972) have shown that a Poisson regression model is a member of the class of Generalized Linear Models (GLIM). Hence, maximum likelihood estimates of the parameter vector can be obtained by a nonlinear weighted least squares procedure with the weights being the reciprocal of the variances.
(5.) In cities where BCB entry occured, the percentage of the population that is Black averaged 24.9 percent over the 1960-1980 period. This is in contrast to 11.3 percent for the United States as a whole for the same period.
(6.) The notion of a city being the relevant market for banking is not without controversy. For a consideration of other market definitions and the theoretical issues involved, see Simmons (1988).
(7.) AVGPOP and AGBPOP are constructed from census data reported in Statistical Abstract of the United States. A data appendix is available upon request from the author.
(8.) The studies by Orr (1974a, 1974b), Khemani and Shapiro (1986), and Chappel et al. (1990) all include the logarithm of capital requirements as an explanatory variable for entry. The model is estimated with a SAS macro nonlinear procedure for Generalized Linear Models (GLIM), which includes the Poisson regression model [Nelder and Wedderburn]. For a description of GLIM procedures see: SAS/STAT User's Guide, Vol. 2, Version 6, 4th ea., (1990) pp. 1168-1169.
9. If the Poisson regression is correctly specified, this consistent coefficient estimates are obtained. However if the restriction that A, = Expected value of [E.sub.i] = Variance of E, is not satisfied, the variances for the parameter estimates will be inconsistent, rendering hypothesis tests invalid. To test for this restriction, and hence the adequacy of the Poisson specification, the following auxiliary OLS regression is estimated:
[Mathematical Expression Omitted]
where [[lambda].sub.i] equals the estimate of Ai obtained from the estimated parameter vector, [phi]([[lambda].sub.i]) equals the assumed form of overdispersion such that expected value of [E.sub.i] [not equal] variance of [E.sub.i], and [v.sub.i] is a white noise error term. Cameron and Trivedi (1990) show that testing for the mean-variance equality in the Poisson regression is equivalent to a standard t-test on a. If the hypothesis that [alpha] = 0 cannot be rejected, the Poisson model along with its mean-variance equality cannot be rejected.
10. Cameron and Trivedi (1986) note that the negative binomial regression model is simply an example of a generalized Poisson regression model. A common approach to generalizing the Poisson model is to specify some unexplained randomness in the Poisson parameter A [[lambda].sub.i] as being generated by a gamma distribution. The resulting distribution for [[lambda].sub.i] is the negative binomial where:
E([[lambda].sub.i]) = (i
[Mathematical Expression Omitted]
where [v.sub.i] = (1/[alpha])([[phi].sub.i][).sup.k]. substituting for [v.sub.i]:
E([[lambda].sub.i]) = [[phi].sub.i]
[Mathematical Expression Omitted]
Note that for specific choices of the constant k, when [alpha] = 0, we have the basic Poisson regression model.
11. The results in Tables 2 and 3 are for testing the Poisson mean-variance equality against the negative binomial mean-variance relationship under the condition: [phi]([[lambda].sub.i])[alpha]=[[lambda].sub.i]. This specification allows for heterogeneity in the relationship between the mean and variance under the hypothesis test:
[H.sub.0]: Var([E.sub.i] = ([[lamda].sub.i])
[H.sub.A]: Var([E.sub.i]) = [[lamda].sub.i] + g([[lamda].sub.i])[alpha]
[H.sub.A] implies a test for [alpha] = 0 in the auxiliary OLS regression in note 10 above.
12. Discriminant analysis is valid when the dependent variables are distributed multivariate normal. The observations on BCB entry are assumed to be Poisson distributed. In this context, Maddala (1983, p. 54) notes that a Poisson distribution can be written approximately as the product of normal density functions. Hence the [E.sub.i]'s in a Poisson distribution can be viewed as being distributed approximately multivariate normal. Stepwise Discriminant Analysis selects a subset of quantitative variables to produce a good discrimination model. In the context of BCB entry, the discrimination is between markets with and without entry. Variables are chosen to enter or leave the model on the basis of one of two criteria: (1) The significance level of an F-test from an analysis of covariance, where the variables chosen act as covariates and the variable under consideration is the dependent variable; and (2)The squared partial correlation for predicting the dependent variable under consideration, controlling for the effects of the variables already selected for the model. The Stepwise Discriminant Analysis was done with the SAS procedure STEPDISC. See SAS/STAT User's Guide, Vol. 2, Version 6, 4th ea., (1990) pp.1493-1509.
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|Author:||Price, Gregory N.|
|Publication:||Quarterly Review of Economics and Finance|
|Date:||Sep 22, 1995|
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