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 $k$th powers of residuals, CUSUM processes and self-normalized partial sum processes. The $k$th power partial sum process converges to a Brownian process plus a correction term, where the correction term depends on the $k$th moment $\mu_k$ of the innovation sequence. If $\mu_k=0$, then the correction term is 0 and, thus, the $k$th power partial sum process converges weakly to the same Gaussian process as does the $k$th power partial sum of the i.i.d. innovations sequence. In particular, since $\mu_1=0$, 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 $k$th 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.Comment: Published at http://dx.doi.org/10.1214/009053605000000534 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Change Point Testing for the Drift Parameters of a Periodic Mean Reversion Process

In this paper we investigate the problem of detecting a change in the drift parameters of a generalized Ornstein-Uhlenbeck process which is defined as the solution of $dX_t=(L(t)-\alpha X_t) dt + \sigma dB_t$, and which is observed in continuous time. We derive an explicit representation of the generalized likelihood ratio test statistic assuming that the mean reversion function $L(t)$ is a finite linear combination of known basis functions. In the case of a periodic mean reversion function, we determine the asymptotic distribution of the test statistic under the null hypothesis

A Nonparametric Test of Serial Independence for Time Series and Residuals

AbstractThis paper presents nonparametric tests of independence that can be used to test the independence of p random variables, serial independence for time series, or residuals data. These tests are shown to generalize the classical portmanteau statistics. Applications to both time series and regression residuals are discussed

Change point testing for the drift parameters of a periodic mean reversion process

In this paper we investigate the problem of detecting a change in the drift parameters of a generalized Ornstein-Uhlenbeck process which is defined as the solution of dX_t = (L(t) - alpha X_t)dt + delta dB_t and which is observed in continuous time. We derive an explicit representation of the generalized likelihood ratio test statistic assuming that the mean reversion function L(t) is a finite linear combination of known basis functions. In the case of a periodic mean reversion function, we determine the asymptotic distribution of the test statistic under the null hypothesis

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.simple branching diffusion cumulant cumulant spectra estimation consistency asymptotic normality time slice data

Parametric estimation for a simple branching diffusion process

AbstractA simple branching diffusion process is given as an elementary model of spatial evolution. A parametric estimation theory is presented for this model. As side results, a spatial central limit theorem and spatial strong law of large numbers are also obtained

Parametric estimation for a simple branching diffusion process

A simple branching diffusion process is given as an elementary model of spatial evolution. A parametric estimation theory is presented for this model. As side results, a spatial central limit theorem and spatial strong law of large numbers are also obtained.Branching diffusion spectral density asymptotic likelihood estimation consistency asymptotic normality