Copula Regression

1. Copula Regression By Rahul A. Parsa Drake University & Stuart A. Klugman Society of Actuaries

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 Xi 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 xn given x1 ….xn-1 is Where

14. 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.

15. Copula RegressionDiscrete 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 q and variance I(q)-1, where • I(q) – 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:

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:

23. Example 3 • Dependent – X3 - Gamma • Pareto for X2 estimated by Exponential • Error:

24. Example 4 • Dependent – X3 - Gamma • X1 & X2 estimated Empirically • C = # of obs ≤ x and a = (# of obs = x) Error:

25. Example 5 • Dependent – X1 - Poisson • X2, estimated by Exponential Error:

26. Example 6 • Dependent – X1 - Poisson • X2 & X3 estimated by Empirically Error: