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

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Presentation on theme: "BY RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES Copula Regression."— Presentation transcript:

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

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

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

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

5 OLS Regression Estimated Model

6 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

7 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

8 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

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

10 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

11 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

12 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

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

14 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.

15 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.

16 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.

17 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.”

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

19 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, , , Copula OLS

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

21 Example 1 - Cont Maximum likelihood Estimate of Correlation Matrix R-hat =

22 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, Copula595,947.5 OLS637,172.8 GLM814,

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

24 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, CopulaOLSGLM 559, , ,708.98

25 Example 5 Dependent – X1 - Poisson X2, estimated by Exponential Error: VariablesX1-PoissonX2-ParetoX3-Gamma Parameters54, 3003, 100 MLE , Copula OLS114.66

26 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 Copula OLS114.66


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