Statistical inference for time-varying ARCH processes

In this paper the class of ARCH$(\infty)$ models is generalized to the nonstationary class of ARCH$(\infty)$ models with time-varying coefficients. For fixed time points, a stationary approximation is given leading to the notation ``locally stationary ARCH$(\infty)$ process.'' The asymptotic properties of weighted quasi-likelihood estimators of time-varying ARCH$(p)$ processes ($p<\infty$) are studied, including asymptotic normality. In particular, the extra bias due to nonstationarity of the process is investigated. Moreover, a Taylor expansion of the nonstationary ARCH process in terms of stationary processes is given and it is proved that the time-varying ARCH process can be written as a time-varying Volterra series.Comment: Published at http://dx.doi.org/10.1214/009053606000000227 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A recursive online algorithm for the estimation of time-varying ARCH parameters

In this paper we propose a recursive online algorithm for estimating the parameters of a time-varying ARCH process. The estimation is done by updating the estimator at time point $t-1$ with observations about the time point $t$ to yield an estimator of the parameter at time point $t$. The sampling properties of this estimator are studied in a non-stationary context -- in particular, asymptotic normality and an expression for the bias due to non-stationarity are established. By running two recursive online algorithms in parallel with different step sizes and taking a linear combination of the estimators, the rate of convergence can be improved for parameter curves from H\"{o}lder classes of order between 1 and 2.Comment: Published at http://dx.doi.org/10.3150/07-BEJ5009 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

The quantile spectral density and comparison based tests for nonlinear time series

In this paper we consider tests for nonlinear time series, which are motivated by the notion of serial dependence. The proposed tests are based on comparisons with the quantile spectral density, which can be considered as a quantile version of the usual spectral density function. The quantile spectral density 'measures' sequential dependence structure of a time series, and is well defined under relatively weak mixing conditions. We propose an estimator for the quantile spectral density and derive its asympototic sampling properties. We use the quantile spectral density to construct a goodness of fit test for time series and explain how this test can also be used for comparing the sequential dependence structure of two time series. The method is illustrated with simulations and some real data examples

Statistical inference for stochastic coefficient regression models (Technical Report)

Abstract The classical multiple regression model plays a very important role in statistical analysis. The typical assumption is that changes in the response variable, due to a small change in a given regressor, is constant over time. In other words, the rate of change is not influenced by any unforeseen external variables and remains the same over the entire time period of observation. This strong assumption may, sometimes, be unrealistic, for example, in areas like social sciences, environmental sciences etc. In view of this, we propose stochastic coefficient regression (SCR) models with stationary, correlated random errors and consider their statistical inference. We assume that the coefficients are stationary processes, where each admits a linear process representation. We propose a frequency domain method of estimation, the advantage of this method is that no assumptions on the distribution of the coefficients are necessary. These models are fitted to two real data sets and their predictive performance are also examined

On multiple regression models with nonstationary correlated errors

S We consider the estimation of parameters of a multiple regression model with nonstationary errors. We assume the nonstationary errors satisfy a time-dependent autoregressive process and describe a method for estimating the parameters of the regressors and the time-dependent autoregressive parameters. The parameters are rescaled as in nonparametric regression to obtain the asymptotic sampling properties of the estimators. The method is illustrated with an example taken from global temperature anomalies. Some key words: Asymptotic normality; Consistency; Heteroscedastic errors; Local least squares; Local stationarity; Multiple regression; Nonstationary; Temperature anomaly; Time series. I In many fields of research a time series {X t } is observed together with certain regressors which are believed to have a linear effect on the time series. The time series is then fitted with the multiple regression model where the regressors { f t,i : t=1, . . . , N, i=1, . . . , q} are observed. Often it is assumed that the errors {e t } are independent and identically distributed, and ordinary least squares is used to estimate the parameters {a i }. However, it is quite plausible that the errors are dependent, in which case treating the errors as if they were independent and proceeding with the estimation could result in a severe loss of efficiency in the estimator. In this direction, several authors, see for example Durbin (1960) and Pierce (1971), have considered the case of dependent stationary errors, where the errors are assumed to satisfy a linear model. A cause for concern is if the time series {X t } were observed over a long period of time, in which case it would seem quite likely that exogeneous factors could affect the stationarity of the errors. For example, it is believed that gas emissions over the past century may have caused &apos;global warming&apos;. Such an effect would almost certainly lead to a change in the structure of the temperature time series. In this case it would seem quite reasonable to change our working hypothesis and include the class of nonstationary models. Nonstationary time-varying autoregressive models have previously been studied by Subb

Time series analysis: time series analysis methods and applications

The field of statistics not only affects all areas of scientific activity, but also many other matters such as public policy. It is branching rapidly into so many different subjects that a series of handbooks is the only way of comprehensively presenting the various aspects of statistical methodology, applications, and recent developments. The Handbook of Statistics is a series of self-contained reference books. Each volume is devoted to a particular topic in statistics, with Volume 30 dealing with time series. The series is addressed to the entire community of statisticians and scientists in various disciplines who use statistical methodology in their work. At the same time, special emphasis is placed on applications-oriented techniques, with the applied statistician in mind as the primary audience. Comprehensively presents the various aspects of statistical methodology Discusses a wide variety of diverse applications and recent developments Contributors are internationally renowened experts in their respective areas