Some Remarks on the “Generalized Event Count” Distribution

King (1989) presented the "Generalized Event Count" (GEC) model as a means of dealing with event count data when the analyst is un sure whether the data are "underdispersed" or "overdispersed." Here I establish several useful properties of the GEC model and make some practical suggestions for estimation. The GEC Model Often political analysts are interested in settings in which the dependent variable is an "event count" such as the number of veto challenges in a given Congress or the number of coups happening in a given country during a given year. This type of data occupies a methodological limbo between dichotomous dependent variables, which can be analyzed using models such as probit and logit, and the continuous dependent variable setting suitable for regression analysis. The Poisson model has been widely applied to event count data, and it can readily adapt to handle covariates. However, Poisson models impose a particularly restrictive assumption: for each data point the expected value of the dependent variable and its variance are identical. King (1989) proposed a generalization of the Poisson model, the "Gen eralized Event Count" model, which allows the variance to differ from the expectation. Author's address: John Londregan, Department of Political Science, Univer sity of California at Los Angeles, Los Angeles CA 90095-1472, USA. Internet johnlond@sscnet.ucla.edu This content downloaded from 157.55.39.45 on Fri, 02 Sep 2016 05:15:53 UTC All use subject to http://about.jstor.org/terms 214 Political Analysis Equation 26 of King 1989 defines the probability density for the GEC model, fgec(yi | At-, icr2)The variable ytmust take on nonnegative integer values, and it is interpreted as a "count" of the number of ob served occurrences of an event such as a coup d'etat or a challenge to a presidential veto. The parameter A,indexes the expected value of the distribution, with a qualification that will be discussed in detail. The cr2 parameter calibrates the ratio of the variance to the expectation of the distribution. A value of cr2 = 1 corresponds to the Poisson distri bution. The literature refers to values of a2 in excess of 1 as cases of "overdispersion," while "underdispersion" corresponds to values of a2 below 1. If yi e {1,2,3, ...} then fgedVi I A,-, a2) = fgec(0 | A,-, (72)fi(y,| A,, a2) where " ]Xi + (a2-l)(j n( 1 for a2 = 1. for a2 a} that is, [a) -f 1 is the smallest integer greater than or equal to a. While the nonnegative integers constitute the support of the GEC distribution when a2 > 1, in the so-called underdispersed case, with a2 1. While the nonnegative integers constitute the support of the GEC distribution when a2 > 1, in the so-called underdispersed case, with a2 1 [5*0+? f '! 0). (4) This specification uses p(y,-17,0) to denote fgec(yi I ^> A,). For difference equation 4 to be stable, 7 cannot exceed 1. For a given choice of (7,0), a pair ( (T2) >0 This content downloaded from 157.55.39.45 on Fri, 02 Sep 2016 05:15:53 UTC All use subject to http://about.jstor.org/terms 216 Political Analysis and Ylf9ec{yi\X%^2) = 1. The first part of this claim is trivial, simply verify that fgec(0 | A,, a2) > 0 for all possible values of A, and a2. The second part of the claim takes only slightly more work. Claim 1. Ni Proof of Claim 1. If a2 > 1, so that Ni = oo, then for y, > 0 fgec{yi i a,, -lr j=0 j=0 r((^l)+i)(a3-l)i ((*2-i)) However, the summation term is in fact a Taylor's series expansion (Hogg and Tanis 1977, 76): So we have i 1 -A./(<r3-l) E/,ec(i|ai),T2) = (^)Ai/('S-1) -A,/(<r2-l) = 1 This content downloaded from 157.55.39.45 on Fri, 02 Sep 2016 05:15:53 UTC All use subject to http://about.jstor.org/terms Remarks on the "Generalized Event Count" Distribution 217 If a2 = 1, N is again equal to oo and we are in the familiar Poisson case, for which However, the Taylor's series expansion for eXi is