An example on modelling conditional higher moments using maximum entropy density with high frequency data.
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Since the introduction of the Autoregressive Conditional Heteroscedasticity (ARCH) model of Engle (1982), the literature of modelling the conditional second moment has become increasingly popular in the last two decades. This popularity is reflected by the numerous volatility models being proposed in the literature and their multivariate counterparts (see McAleer (2005)) for an excellent survey on the various volatility models and related issues on estimation and specification). Interestingly, the Quasi Maximum Likelihood Estimator (QMLE) with normal density is typically used to estimate the parameters in these models. As such, the higher moments of the underlying distribution are assumed to be the same as the normal distribution. However, various studies reveal that the higher moments, such as skewness and kurtosis of the distribution of financial returns are not likely to be the same as the normal distribution, and in some cases, they are not even constant over time. This has significant implications in risk management, especially in the calculation of Value-at-Risk (VaR), which focuses on the negative quantile of the return distribution. Failed to accurately capture the shape of the negative quantile, which is determined by the skewness and the kurtosis of the distribution, would produce inaccurate measure of risk, and subsequently lead to misleading decision in risk management.This paper proposes a general framework to model the distribution of financial returns using Maximum Entropy Density (MED). The main advantage of MED is that it provides a general framework to estimate the distribution function directly based on a given set of data, and it provides a convenient framework to model higher order moments up to any arbitrary finite order k. However this flexibility comes with a high cost in computation time ask increases, therefore this paper proposes an alternative model that would reduce computation time substantially. Moreover, the sensitivity of the parameters in the MED with respect to the dynamic changes of moments is derived analytically. This result is important as it relates the dynamic structure of the moments to the parameters in the MED. The usefulness of this approach will be demonstrated using 5 minutes intra-daily returns of the Euro/USD exchange rate. 2034
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