Estimation Elizabeth Daykin May 8 2005 Exact Maximum

Estimation Elizabeth Daykin May 8, 2005

• Exact Maximum Likelihood Method (MLE) • Inference for the Margins (IFM) • Canonical Maximum Likelihood (CML)

Maximum Likelihood

• Assume that the usual regularity conditions for asymptotic maximum likelihood theory hold for copula and all of its margins. • Under these conditions, the MLE exists and is consistent and asymptotically efficient. • The MLE verifies the property of asymptotically normal and we have: With the usual Fisher’s information matrix and the usual true value.

Therefore, where, Using the sample data matrix, the log-likelihood function is: where is the set of all parameters: R and The MLE of R is given by (Magnus & Neudecker, 1980)

Multivariate Dispersion Copula with Weibull margins Let be a position parameter, a dispersion parameter, and R a correlation matrix where and for every set of c. d. f.

Multivariate Dispersion Copula Assuming Weibull margins, we obtain the MDC density with, Therefore, the log-likelihood function is with

In the previous example, the log-likelihood function has to be maximized with respect to all parameters using a numerical optimization method. Joint estimation of parameters of the marginal distributions and parameters of the dependence structure represented by the copula can be costly.

Inference for the Margins (IFM) • Log-likelihood function comprises two positive terms: one involving the copula and its parameters, the other involving the margins and all parameters of the copula density.

IFM Steps (Joe and Xu, 1996) • Estimate the margins’ parameters by performing the estimation of the univariate marginal distributions: • Given , perform the estimation of the copula parameter :

The IFM estimator is defined as the vector: We call the entire log-likelihood function, the loglikelihood of the jth marginal, and the log-likelihood for the copula itself. The IFM estimator is the solution of: While the MLE comes from solving:

• Computationally easier to obtain IFM compared to MLE. • Equivalence of MLE and IFM, in general does not hold. • For the MGC with correlation matrix and univariate normal margins, the two coincide.

Asymptotic Normality • It can be shown that IFM under regular conditions verifies the property of asymptotic normality and we have With the Godambe information matrix. Define the score function as Then With Estimation of the covariance matrix requires computing many derivatives. Joe and Xu (1996) suggest using the Jacknife method to estimate it.

Example Consider the joint distribution of the asset returns Alluminum Alloy (AL) and Copper (CU). Assume margins are Gaussian and that the dependence structure is the Frank Copula

The log-likelihood is

IFM We have for n=1, 2 where and are the maximum likelihood estimates. Define The parameter with is estimated by maximizing

The Godambe covariance matrix is computed with the score function:

Canonical Maximum Likelihood (CML) Semi-parametric method Transform sample data into uniform variates using empirical distributions, i. e. , then estimate copula parameters:

Modeling five assets: DAX 30 index S&P 500 index 10 -year total return index for the US bond market 10 -year total return index for the German bond market, and DEM/USD exchange rate Use weekly, average data from January 1992 to June 2001 -- 248 observations Dax 30 S&P 500 Ger 10 y USA 10 y DEM/$ Mean 0. 53 0. 43 0. 30 0. 27 0. 16 Std 3. 56 2. 68 0. 90 1. 04 2. 04 Skew -0. 40 -0. 45 -0. 55 -0. 18 0. 13 Kurtosis 3. 22 4. 00 3. 44 2. 75 3. 38 JB Statistics 0. 0288 0. 0000 0. 0007 0. 3879 0. 3204

Model the joint dependence using a Frank Copula -allows for both positive and negative dependence Use a Student t to capture a high kurtosis -use a non-central Student t to allow for negative skewness Where is the non-centrality parameter and is the parameter for degrees of freedom Marginal estimation Dax 30 S&P 500 Ger 10 y USA 10 y DEM/$ NC Student t δ 78. 10 20. 10 44. 79 300. 00 41. 01 0. 00 -0. 01 300. 00 41. 06 Student t 78. 79 20. 23 45. 40

Estimated α for the Frank Copula with the NC-t marginals Dax 30 S&P 500 Ger 10 y USA 10 y DAX 30 - S&P 500 5. 12 - GER 10 y 1. 17 1. 03 - USA 10 y 0. 23 1. 32 3. 72 - DEM/USD 2. 05 1. 28 0. 44 -1. 10 DEM/$ -

Estimating correlation matrix of the MGC with the CML method. Transform the original data into Gaussian data: (i) estimate the empirical distribution functions (uniform transformation) using order statistics; (ii) generate Gaussian values by applying the inverse of the normal distribution to the empirical distribution functions. Compute the correlation matrix of the transformed data. Dax 30 S&P 500 Ger 10 y USA 10 y DAX 30 1 S&P 500 0. 3035 1 GER 10 y 0. 1998 0. 0551 1 USA 10 y 0. 6494 0. 1501 0. 1941 1 DEM/USD 0. 0019 -0. 2144 0. 4850 0. 1585 DEM/$ 1

Summary • MLE is computationally costly. • IFM is far more efficient than MLE, while still using MLE methods. • MLE and IFM involve specifying parametric form of the margins. • CML avoids specification of margins, but leads to inefficient parameter estimates.

References • Durrleman, V; Nikeghbali, A; Roncalli, T. “Which Copula is the right one? ” August 25, 2000. • Cherubini, Luciano, Vecchiatto. Copula Methods in Finance. Indianapolis, IN: John Wiley and Sons, 2004.

Godambe Information Matrix