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BY RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES Copula Regression

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Outline of Talk OLS Regression Generalized Linear Models (GLM) Copula Regression Continuous case Discrete Case Examples

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Notation Notation: Y – Dependent Variable Assumption Y is related to X’s in some functional form

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OLS Regression Y is linearly related to X’s OLS Model

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OLS Regression Estimated Model

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OLS Multivariate Normal Distribution Assume Jointly follow a multivariate normal distribution Then the conditional distribution of Y | X follows normal distribution with mean and variance given by

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OLS & MVN Y-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

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GLM Y belongs to an exponential family of distributions g is called the link function x's are not random Y|x belongs to the exponential family Conditional variance is no longer constant Parameters are estimated by MLE using numerical methods

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GLM Generalization of GLM: Y can be any distribution (See Loss Models) Computing predicted values is difficult No convenient expression conditional variance

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Copula Regression Y can have any distribution Each X i can have any distribution The joint distribution is described by a Copula Estimate Y by E(Y|X= x ) – conditional mean

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Copula Ideal 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 Copula t-Copula

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MVN Copula CDF 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

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Conditional Distribution in MVN Copula The conditional distribution of x n given x 1 ….x n-1 is Where

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Copula Regression Continuous 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.

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Copula Regression Discrete Case When one of the covariates is discrete Problem: determining discrete probabilities from the Gaussian copula requires computing many multivariate normal distribution function values and thus computing the likelihood function is difficult Solution: Replace discrete distribution by a continuous distribution using a uniform kernel.

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Copula Regression – Standard Errors How to compute standard errors of the estimates? As n -> ∞, MLE, converges to a normal distribution with mean and variance I( ) -1, where I( ) – Information Matrix.

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How to compute Standard Errors Loss 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.”

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Examples All examples have three variables R Matrix : Error measured by Also compared to OLS 10.7 1 1

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Example 1 Dependent – X3 - Gamma Though X2 is simulated from Pareto, parameter estimates do not converge, gamma model fit Error: VariablesX1-ParetoX2-ParetoX3-Gamma Parameters3, 1004, 3003, 100 MLE3.44, 161.111.04, 112.0033.77, 85.93 Copula59000.5 OLS637172.8

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Ex 1 - Standard Errors Diagonal terms are standard deviations and off- diagonal terms are correlations

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Example 1 - Cont Maximum likelihood Estimate of Correlation Matrix 10.7110.699 0.71110.713 0.6990.7131 R-hat =

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Example 2 Dependent – X3 - Gamma X1 & X2 estimated Empirically Error: VariablesX1-ParetoX2-ParetoX3-Gamma Parameters3, 1004, 3003, 100 MLEF(x) = x/n – 1/2n f(x) = 1/n F(x) = x/n – 1/2n f(x) = 1/n 4.03, 81.04 Copula595,947.5 OLS637,172.8 GLM814,264.754

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Example 3 Dependent – X3 - Gamma Pareto for X2 estimated by Exponential Error: VariablesX1-PoissonX2-ParetoX3-Gamma Parameters54, 3003, 100 MLE5.65119.393.67, 88.98 Copula574,968 OLS582,459.5

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Example 4 Dependent – X3 - Gamma X1 & X2 estimated Empirically C = # of obs ≤ x and a = (# of obs = x) Error: VariablesX1-PoissonX2-ParetoX3-Gamma Parameters54, 3003, 100 MLEF(x) = c/n + a/2n f(x) = a/n F(x) = x/n – 1/2n f(x) = 1/n 3.96, 82.48 CopulaOLSGLM 559,888.8582,459.5652,708.98

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Example 5 Dependent – X1 - Poisson X2, estimated by Exponential Error: VariablesX1-PoissonX2-ParetoX3-Gamma Parameters54, 3003, 100 MLE5.65119.393.66, 88.98 Copula108.97 OLS114.66

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Example 6 Dependent – X1 - Poisson X2 & X3 estimated by Empirically Error: VariablesX1-PoissonX2-ParetoX3-Gamma Parameters54, 3003, 100 MLE5.67F(x) = x/n – 1/2n f(x) = 1/n F(x) = x/n – 1/2n f(x) = 1/n Copula110.04 OLS114.66

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