Statistical Models for Political Science Event Counts: Bias in Conventional Procedures and Evidence for the Exponential Poisson Regression Model * Gary King, Harvard University This paper presents analytical, Monte Carlo, and empirical evidence on models for event count data. Event counts are dependent variables that measure the number of times some event occurs. Counts of international events are probably the most common, but numerous examples exist in every empirical field of the discipline. The results of the analysis below strongly suggest that the way event counts have been analyzed in hundreds of important political science studies have produced statisti- cally and substantively unreliable results. Misspecification, inefficiency, bias, inconsistency, insuffi- ciency, and other problems result from the unknowing application of two common methods that are without theoretical justification or empirical utility in this type of data. I show that the exponential Poisson regression (EPR) model provides analytically, in large samples, and empirically, in small, finite samples, a far superior model and optimal estimator. I also demonstrate the advantage of this methodology in an application to nineteenth-century party switching in the U.S. Congress. Its use by political scientists is strongly encouraged. 1. Introduction This study is concerned with statistical models for event count data. Event counts are variables that have for observation i (i = 1, . . . , N) the number of occurrences of an event in a fixed domain. The domain for each observation may be time-as in a month, year, hour, or some appropriate interval-or space-as in a geographic unit, an individual, or others. Dependent variables of this type exist in every major journal and empirical field in the discipline, often representing central concepts or concerns. The larg- est number of event counts is probably in international relations, where massive databases record the number of actions each nation or political group takes with respect to another (e.g., Azar and Sloan, 1975). But examples from other fields abound: the number of presidential vetoes per year (Rohde and Simon, 1985); the number of congressional staff members engaged in casework services and the number of trips to the home district for each member of the House of Represen- tatives (McAdams and Johannes, 1985); the number of seats the president's party lost in each midterm congressional election (Campbell, 1985); the size of Medic- aid caseloads (Hanson, 1984); the number of persons recorded in LBJ's daily diary as having been present at a particular White House meeting (Sigelman and McNeil, 1980); the number of months that a parliamentary cabinet endures *An earlier version of this paper was presented at the annual meeting of the Political Science Methodology Group, Harvard University, 7- 10 August 1986. I appreciate the helpful comments from participants at that meeting, particularly those of Christopher Achen and Nathaniel Beck. Thanks also for useful conversations with Paul Allison, William H. Green, Herbert Kritzer, Zeev Maoz, Lyn Ragsdale, Elizabeth Rosenthal, and Paul Zarowin. Nasrin Abdolali helped with Figures 1 and 2. The author gratefully acknowledges support from the National Science Foundation (NSFSES-8722715).POLITICAL SCIENCE EVENT COUNTS 839 (Robertson, 1984); the number of members of the House and Senate who switch political parties each year (King and Benjamin, 1985); the number of citizen- initiated and support-related political activities engaged in and reported by Soviet CmigrCs (Di Franceisco and Gitelman, 1984); the number of coups d'etat per year for black African states (Johnson, Slater, and McGowan, 1984). There are many other examples. In the sections that follow, I show the data generation process of event counts to be Poisson (sec. 2). I then introduce the exponential Poisson regression (EPR) model as the appropriate method, directly deducible from the data genera- tion process (sec. 3). The next section shows that the two models used most fre- quently in political science for analyzing this type of data are either misspecified (the OLS model in sec. 4) or biased and inconsistent (the logged OLS model in sec. 5). Since statistical theory is known only for unrealistically large sample sizes, I use Monte Carlo experiments to demonstrate the empirical unbiasedness of the EPR model in finite samples and the bias and inefficiency of the logged OLS model of event counts even in very large samples (sec. 6). A brief analysis of an empirical example (sec. 7) and concluding remarks (sec. 8) are also pro- vided. Appendix A provides details of the proof used in section 3. Appendix B reviews readily available computer programs that can be used to estimate the EPR model. 2. The Data Generation Process of One Event Count Observation The data generation process usually assumed to produce event counts is Pois- son.' This process arises naturally in many situations commonly analyzed by po- litical scientists. This section presents the substance of a proof that all event count data that meet a few modest assumptions arise from a Poisson process (mathematical details appear in Appendix A). Consider a model for what will be one observation in the Poisson regression model to be discussed in later sections. Let y, be the number of recorded events (a nonnegative integer) at an instant in time, t; t does not refer to the observation number, but refers to the time that has passed in recording events within this one observation; y, never decreases, and at certain instances in time, the number of events increases by one. This process is not observed until the end of the observa- tion period, when the total number of events occurring within the period are recorded. 'Next to the normal, and certainly among discrete distributions, the Poisson probability distri- bution is often considered the most important. Interestingly, although he was a mathematician, Si- meon Denis Poisson's (1837) invention was in the context of a "political science" work on the applica- tion of mathematical probability to judicial administration (see Haight, 1967, p. 113). Furthermore, the methodological problems and opportunities presented by event count data are focused in political science. Economics and psychology have occasional applications, and while sociology has a reason- able number, it