iter1 =

u2/n 0 0 2rj4

(A.6.6)

A.6.4 MLEs for Non-normal Density Functions

Maximum likelihood estimation is a very flexible method for estimating the parameters of any distribution. Distributions of returns in financial markets are often fat-tailed, that is, they have positive excess kurtosis (§10.1). This is particularly apparent in high-frequency data in foreign exchange markets, but in other markets, particularly equity markets, market crashes or other extreme events produce outliers that give far more weight in the tails that can be modelled by a single normal distribution. In §10.2 a number of different distributions were introduced to model the fat-tailed nature of financial returns, and the parameters of these distributions are normally estimated by maximum likelihood.

Many financial distributions are skewed as well as fat-tailed, particularly those that relate to credit risk. With a typical portfolio of many credits, each having a very small probability of loss, the credit loss distribution will be heavily skewed. When a credit loss is modelled by exposure, recovery rate and default probability distributions, each of these distributions can be regarded as skewed and fat-tailed. Because normal probability measures are inappropriate, some models characterize these distributions by beta or gamma distributions, and maximum likelihood estimation of their parameters is standard.

When non-normal distributions are assumed, the computation of second derivatives for the covariance matrix of MLEs can be a very tricky problem. For normal variates substituting in (A.6.6) the MLE of rj2 gives the estimated covariance matrix, and the information matrix was particularly easy to compute in this case. But for non-normal densities the information matrix may be non-diagonal and the expected values of the second derivatives of the log-likelihood very complex. Often they have to be approximated by taking their actual values, rather than expected values, at the maximum likelihood estimates. Analytic second derivatives may be impossible to compute, in which case an alternative form for the information matrix

can prove most useful computationally. An estimated covariance matrix is obtained taking the inverse of the estimated information matrix above, e\aluated at the MLEs. Only the first derivatives of the log-likelihood are required, and these have to be computed anyway for the first-order conditions that define the MLEs.

1(0) = [ \ )/ ][ \ 0)/<90]

Maximum likelihood estimation is a standard procedure in statistical packages. It is not as straightforward as least squares, since the likelihood optimization is

an iterative procedure (often a gradient method) and it may not converge. For example, if there are many parameters in the model then the likelihood is a multi-dimensional surface that can be very flat indeed. In such circumstances an iterative search algorithm can yield a local rather than a global optimum, or it may hit a boundary if there are constraints on parameters. Because differences in starting values and optimization methods, as well as sometimes small differences in sample data can all lead to different results, a maximum likelihood procedure should always be monitored more closely than its simple least squares relative.

References

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Alexander, CO. (1994) Cofeatures in international bond and equity markets. University of Sussex Discussion Papers in Economics no. 94/1, available from ww w. ismacen tre.rdg.ac.uk

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Alexander, CO. (1996) Evaluating the use of RiskMetrics™ as a risk measurement tool for your operation: What are its advantages and limitations. Derivatives Use, Trading and Regulation 2(3), 277-285.

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