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 varianceOLS and MLE result in same valuesClosed form solution exists
8 GLM Y belongs to an exponential family of distributions g is called the link functionx's are not randomY|x belongs to the exponential familyConditional variance is no longer constantParameters are estimated by MLE using numerical methods
9 GLM Generalization of GLM: Y can be any distribution (See Loss Models) Computing predicted values is difficultNo convenient expression conditional variance
10 Copula Regression Y can have any distribution Each Xi can have any distributionThe joint distribution is described by a CopulaEstimate Y by E(Y|X=x) – conditional mean
11 Copula Ideal Copulas will have the following properties: ease of simulationclosed form for conditional densitydifferent degrees of association available for different pairs of variables. Good Candidates are:Gaussian or MVN Copulat-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 isWhere v is a vector with ith element
13 Conditional Distribution in MVN Copula The conditional distribution of xn given x1 ….xn-1 isWhere
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 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.
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, whereI(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 byAlso compared to OLS10.7
19 Example 1 Dependent – X3 - Gamma Though X2 is simulated from Pareto, parameter estimates do not converge, gamma model fitError:VariablesX1-ParetoX2-ParetoX3-GammaParameters3, 1004, 300MLE3.44,1.04,3.77, 85.93CopulaOLS
20 Ex 1 - Standard ErrorsDiagonal terms are standard deviations and off-diagonal terms are correlations
21 Example 1 - Cont Maximum likelihood Estimate of Correlation Matrix 1 0.7110.6990.713R-hat =