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Transcript of Copula Regression
BYRAHUL A. PARSADRAKE UNIVERSITY
&STUART A. KLUGMANSOCIETY OF ACTUARIES
Outline of TalkOLS RegressionGeneralized Linear Models (GLM)Copula RegressionContinuous caseDiscrete CaseExamples
NotationNotation:Y Dependent Variable AssumptionY is related to Xs in some functional form
OLS RegressionY is linearly related to XsOLS Model
OLS RegressionEstimated Model
OLS Multivariate Normal DistributionAssume Jointly follow a multivariate normal distributionThen the conditional distribution of Y | X follows normal distribution with mean and variance given by
OLS & MVNY-hat = Estimated Conditional mean
It is the MLE
Estimated Conditional Variance is the error variance
OLS and MLE result in same values
Closed form solution exists
GLMY belongs to an exponential family of distributions
g is called the link functionx's are not random Y|x belongs to the exponential familyConditional variance is no longer constantParameters are estimated by MLE using numerical methods
GLMGeneralization of GLM: Y can be any distribution (See Loss Models)
Computing predicted values is difficult
No convenient expression conditional variance
Copula RegressionY can have any distribution
Each Xi can have any distribution
The joint distribution is described by a Copula
Estimate Y by E(Y|X=x) conditional mean
CopulaIdeal Copulas will have the following properties:ease of simulation closed form for conditional density different degrees of association available for different pairs of variables.
Good Candidates are:Gaussian or MVN Copulat-Copula
MVN CopulaCDF for MVN is Copula is
Where G is the multivariate normal cdf with zero mean, unit variance, and correlation matrix R.Density of MVN Copula is
Where v is a vector with ith element
Conditional Distribution in MVN CopulaThe conditional distribution of xn given x1 .xn-1 is
Copula RegressionContinuous Case
Parameters are estimated by MLE.
If are continuous variables, then we use previous equation to find the conditional mean.one-dimensional numerical integration is needed to compute the mean.
Copula RegressionDiscrete CaseWhen one of the covariates is discreteProblem:determining discrete probabilities from the Gaussian copula requires computing many multivariate normal distribution function values and thus computing the likelihood function is difficultSolution:Replace discrete distribution by a continuous distribution using a uniform kernel.
Copula Regression Standard ErrorsHow to compute standard errors of the estimates?As n -> , MLE , converges to a normal distribution with mean q and variance I(q)-1, where
I(q) Information Matrix.
How to compute Standard ErrorsLoss Models: To obtain information matrix, it is necessary to take both derivatives and expected values, which is not always easy. A way to avoid this problem is to simply not take the expected value.
It is called Observed Information.
ExamplesAll examples have three variables
R Matrix :
Error measured by
Also compared to OLS
Example 1Dependent X3 - GammaThough X2 is simulated from Pareto, parameter estimates do not converge, gamma model fit
VariablesX1-ParetoX2-ParetoX3-GammaParameters3, 1004, 3003, 100MLE3.44, 161.111.04, 112.0033.77, 85.93
Ex 1 - Standard ErrorsDiagonal terms are standard deviations and off-diagonal terms are correlations
Example 1 - ContMaximum likelihood Estimate of Correlation MatrixR-hat =
Example 2Dependent X3 - GammaX1 & X2 estimated Empirically
VariablesX1-ParetoX2-ParetoX3-GammaParameters3, 1004, 3003, 100MLEF(x) = x/n 1/2n f(x) = 1/nF(x) = x/n 1/2n f(x) = 1/n4.03, 81.04
Example 3Dependent X3 - GammaPareto for X2 estimated by Exponential
VariablesX1-PoissonX2-ParetoX3-GammaParameters54, 3003, 100MLE5.65119.393.67, 88.98
Example 4Dependent X3 - GammaX1 & X2 estimated EmpiricallyC = # of obs x and a = (# of obs = x)
VariablesX1-PoissonX2-ParetoX3-GammaParameters54, 3003, 100MLEF(x) = c/n + a/2nf(x) = a/nF(x) = x/n 1/2n f(x) = 1/n3.96, 82.48
Example 5Dependent X1 - PoissonX2, estimated by Exponential
VariablesX1-PoissonX2-ParetoX3-GammaParameters54, 3003, 100MLE5.65119.393.66, 88.98
Example 6Dependent X1 - PoissonX2 & X3 estimated by Empirically
VariablesX1-PoissonX2-ParetoX3-GammaParameters54, 3003, 100MLE5.67F(x) = x/n 1/2n f(x) = 1/nF(x) = x/n 1/2n f(x) = 1/n