Reg Kulperger

High moment partial sum processes of residuals in GARCH models and their applications

In this paper we construct high moment partial sum processes based on residuals of a GARCH model when the mean is known to be 0. We consider partial sums of kth powers of residuals, CUSUM processes and self-normalized partial sum processes. The kth power partial sum process converges to a Brownian process plus a correction term, where the correction term depends on the kth moment μ k of the innovation sequence. If μ k = 0, then the correction term is 0 and, thus, the kth power partial sum process converges weakly to the same Gaussian process as does the kth power partial sum of the i.i.d. innovations sequence. In particular, since μ 1 = 0, this holds for the first moment partial sum process, but fails for the second moment partial sum process. We also consider the CUSUM and the self-normalized processes, that is, standardized by the residual sample variance. These behave as if the residuals were asymptotically i.i.d. We also study the joint distribution of the kth and (k + 1)st self-normalized partial sum processes. Applications to change-point problems and goodness-of-fit are considered, in particular. CUSUM statistics for testing GARCH model structure change and the Jarque-Bera omnibus statistic for testing normality of the unobservable innovation distribution of a GARCH model. The use of residuals for constructing a kernel density function estimation of the innovation distribution is discussed.

A stochastic forest fire growth model

We consider a stochastic fire growth model, with the aim of predicting the behaviour of large forest fires. Such a model can describe not only average growth, but also the variability of the growth. Implementing such a model in a computing environment allows one to obtain probability contour plots, burn size distributions, and distributions of time to specified events. Such a model also allows the incorporation of a stochastic spotting mechanism.

Option Valuation with Normal Mixture GARCH Models

The class of mixture GARCH models introduced by Haas, Mittnik and Paollela (2004) and Alexander and Lazar (2006) provides a better alternative for fitting financial data than various other GARCH models driven by the normal or skewed t-distribution. In this paper we propose different option pricing methodologies when the underlying stock dynamic is modeled by an asymmetric normal mixture GARCH model with K volatility components. Since under GARCH models the market is incomplete there are an infinite number of martingale measures one can use for pricing. For our mixture setting we analyze the impact of three risk-neutral candidates: a generalized local risk neutral valuation relationship, an Esscher transform and an extended Girsanov principle. We investigate the out-of-sample performance of an asymmetric GARCH model with a mixture density of two normals for Call options written on the S&P 500 Index. The performance under all three transformations is quite impressive when compared to the benchmark GARCH model with normal driving noise. The overall improvement is explained not only by the skewness and leptokurtosis exhibited by the innovation mixture distribution, but also by the richer parametrization used in modeling the dynamics of the multi-component conditional volatility.

On the residuals of autoregressive processes and polynomial regression

The residual processes of a stationary AR(p) process and of polynomial regression are considered. The residuals are obtained from ordinary least squares fitting. In the AR case, the partial sums converge to Brownian motion. In the polynomial case, they converge to generalized Brownian bridges. Other uses of the residuals are considered. Parameter estimation based on approximate log likelihood function of the residuals is considered.

A COMPARISON OF PRICING KERNELS FOR GARCH OPTION PRICING WITH GENERALIZED HYPERBOLIC DISTRIBUTIONS

Under discrete-time GARCH models markets are incomplete so there is more than one price kernel for valuing contingent claims. This motivates the quest for selecting an appropriate price kernel. Different methods have been proposed for the choice of a price kernel. Some of them can be justified by economic equilibrium arguments. This paper studies risk-neutral dynamics of various classes of Generalized Hyperbolic GARCH models arising from different price kernels. We discuss the properties of these dynamics and show that for some special cases, some pricing kernels considered here lead to similar risk neutral GARCH dynamics. Real data examples for pricing European options on the S&P 500 index emphasize the importance of the choice of a price kernel.

Tests of Independence in Time Series

Tests of the hypothesis that a seriesX1...,Xn is a sequence of independent and identically distributed observations are investigated. The tests are based on a smoothing of the plotXi+1 againstXi. Asymptotic distribution theory on the null hypothesis and under contiguous alternatives is derived. The resulting limit distributions are available in published tables. Omnibus quadratic statistics of the Anderson–Darling type are compared with some optimal statistics and are shown to have good asymptotic properties. A Monte Carlo study is provided to show that the tests have good small sample properties

Parametric estimation for simple branching diffusion processes,II

Consider a simple branching diffusion process, which is a branching process in which the individuals move and live and die in space. The offspring distribution has finite moments of all orders. A parametric estimation theory is presented, using time slice data. This involves the use of third order cumulant spectra to identify and estimate the parameters.

Re-colouring the intensity-based bootstrap for point processes

Abstract A method for obtaining bootstrapping replicates for one-dimensional point processes is presented. The method involves estimating the conditional intensity of the process and computing residuals. The residuals are bootstrapped using a block bootstrap and used, together with the conditional intensity, to define the bootstrap realizations. The method is applied to the estimation of the cross-intensity function for data arising from a reaction time experiment.

Bootstrapping empirical distribution functions of residuals from autoregressive model fitting

is a stationary autoregressive model of order p. Data is collected, and ri,nare the residuals after model fitting. : is the empirical distribution function of the residuals. Suppose the innovations sequence has distribution bounded density and four finite moments. Under these conditions the process converges weakly to a Gaussian process, wich depends on the density f. The limit process is not distribution free. Bootstrapping an AR procehss from the raw EDF does not work. A method of estimating quantiles, based on a smoothed EDF bootstrap, is considered. A numerical study of the method is made.

A Remark on a Fourier Bounding Method of Proof for Convergence of Sums of Periodograms

This paper studies sums of periodograms in a random field setting. In a one dimensional or time series setting these can be studied using a method of cumulants, as done by Brillinger. This method does not carry over well to the random field case. Instead one should apply an argument as used by Rosenblatt. In order to have asymptotically correct confidence intervals, one needs to center these sums properly in the random field case.