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Copula Regression. By Rahul A. Parsa Drake University & Stuart A. Klugman Society of Actuaries. Outline of Talk. OLS Regression Generalized Linear Models (GLM) Copula Regression Continuous case Discrete Case Examples. Notation. Notation: Y – Dependent Variable Assumption - PowerPoint PPT Presentation

### Transcript of Copula Regression

&STUART A. KLUGMANSOCIETY OF ACTUARIES

Copula Regression

• 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

Where

• 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

10.70.70.710.70.70.71

• Example 1Dependent X3 - GammaThough X2 is simulated from Pareto, parameter estimates do not converge, gamma model fit

Error:

VariablesX1-ParetoX2-ParetoX3-GammaParameters3, 1004, 3003, 100MLE3.44, 161.111.04, 112.0033.77, 85.93

Copula59000.5OLS637172.8

• Ex 1 - Standard ErrorsDiagonal terms are standard deviations and off-diagonal terms are correlations

Sheet1

R(2,1)R(3,1)R(3,2)

0.2666060.9660670.359065-0.337250.349482-0.33268-0.42141-0.33863-0.29216

0.96606715.509740.390428-0.252360.346448-0.26734-0.37496-0.29323-0.25393

0.3590650.3904280.025217-0.787660.438662-0.35533-0.45221-0.30294-0.42493

-0.33725-0.25236-0.787663.558369-0.384890.4645130.4968530.356080.470009

0.3494820.3464480.438662-0.384890.100156-0.93602-0.34454-0.46358-0.46292

-0.33268-0.26734-0.355330.464513-0.936022.4853050.3656290.4821870.481122

R(2,1)-0.42141-0.37496-0.452210.496853-0.344540.3656290.0100850.4574520.465885

R(3,1)-0.33863-0.29323-0.302940.35608-0.463580.4821870.4574520.010080.481447

R(3,2)-0.29216-0.25393-0.424930.470009-0.462920.4811220.4658850.4814470.009706

Sheet2

Sheet3

• Example 1 - ContMaximum likelihood Estimate of Correlation MatrixR-hat =

10.7110.6990.71110.7130.6990.7131

• Example 2Dependent X3 - GammaX1 & X2 estimated Empirically

Error:

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

Copula595,947.5OLS637,172.8GLM814,264.754

• Example 3Dependent X3 - GammaPareto for X2 estimated by Exponential

Error:

VariablesX1-PoissonX2-ParetoX3-GammaParameters54, 3003, 100MLE5.65119.393.67, 88.98

Copula574,968OLS582,459.5

• Example 4Dependent X3 - GammaX1 & X2 estimated EmpiricallyC = # of obs x and a = (# of obs = x)

Error:

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

CopulaOLSGLM559,888.8582,459.5652,708.98

• Example 5Dependent X1 - PoissonX2, estimated by Exponential

Error:

VariablesX1-PoissonX2-ParetoX3-GammaParameters54, 3003, 100MLE5.65119.393.66, 88.98

Copula108.97OLS114.66

• Example 6Dependent X1 - PoissonX2 & X3 estimated by Empirically

Error:

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

Copula110.04OLS114.66