Copula Functions and Bivariate Distributions:
Applications to Political Interdependence
Alejandro Quiroz Flores, Wilf Department of Politics, NYU
Motivation
Bivariate Distribution Functions for Survival Analysis
Joint Political Survival
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.
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 pi=1. The parameter “theta” is an
association coefficient.
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.
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.
Copula Functions
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’: S1 x S2, where S1 and S2 are subsets of
I containing 0 and 1.
2. C’ is grounded and 2-increasing.
3. For every u in S1 and v in S2, 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 I2.
Gumbel Distribution 1:
Plackett Distribution:
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?
Gumbel Distribution 2:
Ali-Mikhail-Haq Distribution:
Bivariate Exponential Family
Bivariate Weibull Family
Monte Carlo Simulations
Full Maximum Likelihood Estimates
1000 replications
Scale(1)=2, Scale(2)=3
N from 100 to 1000 in increments of 100
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)
Independent Weibulls
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.
Provisional Regression Results
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
Associated Weibulls (a=.9) and (a=.5)
Coefficient Weibull(1) Weibull(2) B.Weibull(1) B.Weibull(2)
Shape 1
Scale 1
Shape 2
Scale 2
Copula Functions
and Random Variables
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.
Assoc
.7452
(.0128)
3.173
(.1012)
.8819
(.0144)
1.882
(.0509)
.7452
(.0124)
3.173
(NA)
.8818
(.0142)
1.882
(NA)
0
(NA)
.7433
(.0126)
3.073
(.0984)
.8776
(.0141)
1.824
(.0489)
1
(.0405)
Future Research
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.