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Copula Functions and Bivariate Distributions: Applications to Political Interdependence Alejandro Quiroz Flores, Wilf Department of Politics, NYU Motivation Copula Functions and Random Variables Bivariate Exponential FamilyBivariate Weibull Family Joint Political Survival Provisional Regression Results Future Research Functions that join or couple multivariate distribution functions to their one dimensional marginal distribution functions. A two-dimensional subcopula is a function C’ with the following properties: 1. Dom C’: S 1 x S 2, where S 1 and S 2 are subsets of I containing 0 and 1. 2. C’ is grounded and 2-increasing. 3. For every u in S 1 and v in S 2, then C’(u,1)=u and C’(1,v)=v. Note that for every (u,v) in Dom C’, 0<=C’(u,v)<=1; so Ran C’ is also a subset of I. A two dimensional copula is a subcopula C whose domain is I 2. Consider a political system with n-components. These components are subject to shocks that increase their conditional probability of failure. We want to estimate the impact of these shocks both directly to the subjects and indirectly through the failure of one of these subjects. In other words, there might be interdependence between the survival time of both components. This interdependence can be modeled as a full system of equations with non-normal disturbances. In spite of the importance of these events, political science has made little progress in the development or application of the methods to estimate this interdependence. What needs to be done is to derive multivariate distributions in order to estimate the parameters that govern this interdependence. There are several ways of deriving multivariate distributions. However, copula functions seem to be the best way to do so. Define a pair of random variables X and Y with cumulative distribution functions F(x) and G(y), respectively, and a joint cumulative distribution function H(x,y). Sklar’s Theorem: There exists a copula C such that for all x,y in R, H(x,y)=C[F(x),G(y)] Frechet-Hoeffding Bounds: Max[F(x)+G(y)-1,0]<=H(x,y)<=Min[F(x),G(y)] Scale Invariant Measures of Dependence Focus not on correlation coefficients as linear dependence between random variables, but on scale invariant “measures of association”. Measures of Association Kendall’s Tau and Spearman’s Rho focus on estimating probability of concordance (large values of one variable associated with large values of another variable) and discordance between random variables. Bivariate Distribution Functions for Survival Analysis How to produce bivariate distributions? The key resides in having nice marginal distributions and a copula function. How do we find the copula? There are several methods: inversion, geometric, and algebraic methods. The following are Weibull distributions derived from different copula functions. Exponentials are given by assuming p i =1. The parameter “theta” is an association coefficient. Gumbel Distribution 1: Plackett Distribution: Gumbel Distribution 2: Ali-Mikhail-Haq Distribution: Monte Carlo Simulations Full Maximum Likelihood Estimates 1000 replications Scale(1)=2, Scale(2)=3 N from 100 to 1000 in increments of 100 Independent Exponentials Associated Exponentials (a=.15) Monte Carlo Simulations Full Maximum Likelihood Estimates 1000 replications Scale(1)=2, Scale(2)=3 N from 100 to 1000 in increments of 100 Independent Weibulls Associated Weibulls (a=.9) and (a=.5) It is not uncommon for governments to receive external shocks. Indeed, international crises, as well financial shocks, political scandals, labor strikes, and transportation breakdowns, among other problems, threaten the political survival of leaders. Often times, some leaders manage to reduce the impact of these shocks by “firing” the minister “responsible” for the policy failure. Yet, some leaders shamelessly keep those ministers in spite of their obvious failure in the area they are responsible for. Why do some leaders fire some of their agents in some crises but not in others? Can leaders effectively respond to shocks by manipulating the composition of their cabinet? Does this depend on the type of political system? In order to answer this question I collected data on the tenure in office of 7,428 foreign ministers in 181 countries, spanning the years 1696-2004. When combined with data on leaders, we can answer some of the questions presented above. Although I am working on the theory, in this poster I have summarized the necessary methods to empirically test hypotheses on the joint political survival of leaders and foreign ministers. The dependent variable is a vector with two elements: the total survival time of a leader and the median survival time of ministers that held office with that particular leader. The mean survival time for leaders is 3.83 years with a standard deviation of.1316; whereas the mean survival time for median ministers is 2.02 years with a standard deviation of.0668. Sample size is 1966. The correlation for the survival times is.1802 Coefficient Weibull(1) Weibull(2) B.Weibull(1) B.Weibull(2) Shape 1.7452.7452.7433 (.0128) (.0124) (.0126) Scale 1 3.173 3.173 3.073 (.1012) (NA) (.0984) Shape 2.8819.8818.8776 (.0144) (.0142) (.0141) Scale 2 1.882 1.882 1.824 (.0509) (NA) (.0489) Assoc 0 1 (NA) (.0405) 1.Develop more PDFs to compare distributions through simulation. 2.Check for better algorithms for maximization procedure. So far R does a good job, but we can do better. 3.Incorporate time varying covariates in the likelihood. 4.Bivariate censoring is not a problem. But how to deal with univariate censoring? 5.The PDFs are unstable for certain values of the parameters. How to solve this problem? 6.Compare Copula Functions with other procedures for the generation of multivariate distributions, such as the conditional distributions, transformation, and rejection approaches. 7.Further develop theory for joint political survival. So far I am working on a nested principal-agent model.

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Lecture II-2: Probability Review

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