engleski

Maximization of the Likelihood Function in Financial Time Series Models

Many financial time series such as stock returns or foreign exchange rates, observed on daily basis, have showed stylized facts. These facts include serially uncorrelated returns with zero mean, time-varying variance (heteroscedasticity), leptokurtic distribution of returns and volatility clustering. In empirical research we find that these characteristics can be parametrically described using GARCH(p, q) models (Generalized AutoRegrresive Conditional Heteroscedasticity models). In practice these models are used in forecasting market risk. However, parameter estimation in symmetric GARCH(1, 1) model, assuming Gaussian distribution of returns is not that simple. Maximum likelihood estimation (MLE) is usually concerned in evaluating the parameters. Analytical solution of maximization of the likelihood function using first and second derivatives is too complex when the variance of innovations is not constant. Therefore, we present usefulness of quasi-Newton iteration procedure in parameter estimation of the conditional variance equation within BHHH algorithm. Namely, the advantage of BHHH algorithm in comparison to the other numerical optimization algorithms will be presented. To simplify optimization procedure algorithm uses the approximation of the matrix of second derivatives (Hessian). Within BHHH algorithm Hessian matrix is approximated according to information identity. When assumption of normality is unrealistic the estimates are still consistent, but robust standard errors should be used. Solutions of the numerical optimization algorithms are sensitive to the initial values and convergence criteria. Optimization procedure will be illustrated by modeling daily returns of the most liquid stock in first quotation on Zagreb Stock Exchange. In final step, from the evaluated model, prognostic values of expected return and expected standard deviation are estimated. These prognostic values can be used to estimate alternative risk measures, such as Value at Risk (VaR) or Conditional Value at Risk (CVaR). Even so, from estimated GARCH(1, 1) model we can reveal the intensity of volatility reaction on past information, and volatility persistence (time for shocks in volatility to die out).

Log-likelihood ; GARCH model ; BHHH algorithm ; information identity

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