Smoothed estimates for nonlinear time series models with irregular data

Time series data observed at unequal time intervals (irregular data) occur quite often and this usually poses problems in its analysis. A recursive form of the exponentially smoothed estimated is here proposed for a nonlinear model with irregularly observed data and its asymptotic properties are discussed An alternative smoother to that of Wright (1985) is also derived. Numerical comparison is made between the resulting estimates and other smoothed estimates.     

Inference for linear and nonlinear stable error processes via estimating functions

This paper describes an estimating function approach for parameter estimation in linear and nonlinear times series models with infinite variance stable errors. Joint estimates of location and scale parameters are derived for classes of autoregressive (AR) models and random coefficient autoregressive (RCA) models with stable errors, as well as for AR models with stable autoregressive conditionally heteroscedastic (ARCH) errors. Fast, on-line, recursive parametric estimation for the location parameter based on estimating functions is discussed using simulation studies. A real financial time series is also discussed in some detail.     

Recursive online EM estimation of mixture autoregressions

This paper develops a recursive expectation–maximization (REM) algorithm for estimating a mixture autoregression (MAR) with an independent and identically distributed regime transition process. The proposed method, which is useful for long time series as well as for data available in real time, follows a recursive predictor error-type scheme. Based on a slightly modified system to the expectation–maximization (EM) equations for an MAR model, the REM algorithm consists of two steps at each iteration: the expectation step, in which the current unobserved regime transition is estimated from new data using previous recursive estimates, and the minimization step, in which the MAR parameter estimates are recursively updated following a minimization direction. Details of implementation of the REM algorithm are given and its finite-sample performance is shown via simulation experiments. In particular, the EM and REM provide roughly similar estimates, especially for moderate and long time series.     

Inference for response-limited time-series models

This paper Introduces a robust time series estimation procedure based on a new nonlinear time-series model called the response-limited ARMA model. Under this procedure, In contrast to location or regression, the uncorrected pseudo-observations, which produce the parameter estimates when submitted to a standard analysis, also yield satisfactory standard errors for inference purposes.     

Nonlinear recursive estimation of volatility via estimating functions

For certain volatility models, the conditional moments that depend on the parameter are of interest. Following Godambe and Heyde (1987), the combined estimating function method has been used to study inference when the conditional mean and conditional variance are functions of the parameter of interest (See Ghahramani and Thavaneswaran [Combining Estimating Functions for Volatility. Journal of Statistical Planning and Inference, 2009, 139, 1449-1461] for details). However, for application purposes, the resulting estimates are nonlinear functions of the observations and no closed form expressions of the estimates are available. As an alternative, in this paper, a recursive estimation approach based on the combined estimating function is proposed and applied to various classes of time series models, including certain volatility models.     

A recursive approach to parameter estimation in regression and time series models

In this paper we discuss the recursive (or on line) estimation in (i) regression and (ii) autoregressive integrated moving average (ARIMA) time series models. The adopted approach uses Kalman filtering techniques to calculate estimates recursively. This approach is used for the estimation of constant as well as time varying parameters. In the first section of the paper we consider the linear regression model. We discuss recursive estimation both for constant and time varying parameters. For constant parameters, Kalman filtering specializes to recursive least squares. In general, we allow the parameters to vary according to an autoregressive integrated moving average process and update the parameter estimates recursively. Since the stochastic model for the parameter changes will "be rarely known, simplifying assumptions have to be made. In particular we assume a random walk model for the time varying parameters and show how to determine whether the parameters are changing over time. This is illustrated with an example.     

Efficient algorithms for exact inference in 2 × 2 contingency tables

We propose a simple and robust algorithm for exact inference in 2 × 2 contingency tables. It is based on recursive relations allowing efficient computation of odds-ratio estimates, confidence limits and p-values for Fisher's test. A factor of 3–10 is gained in terms of computer time compared with the classical algorithm of Thomas.     

Approximate Methods for State-Space Models

State-space models provide an important body of techniques for analyzing time-series, but their use requires estimating unobserved states. The optimal estimate of the state is its conditional expectation given the observation histories, and computing this expectation is hard when there are nonlinearities. Existing filtering methods, including sequential Monte Carlo, tend to be either inaccurate or slow. In this paper, we study a nonlinear filter for nonlinear/non-Gaussian state-space models, which uses Laplace's method, an asymptotic series expansion, to approximate the state's conditional mean and variance, together with a Gaussian conditional distribution. This Laplace-Gaussian filter (LGF) gives fast, recursive, deterministic state estimates, with an error which is set by the stochastic characteristics of the model and is, we show, stable over time. We illustrate the estimation ability of the LGF by applying it to the problem of neural decoding and compare it to sequential Monte Carlo both in simulations and with real data. We find that the LGF can deliver superior results in a small fraction of the computing time.     

Nonlinear filters based on taylor series expansions ∗

The nonlinear filters based on Taylor series approximation are broadly used for computational simplicity, even though their filtering estimates are clearly biased. In this paper, first, we analyze what is approximated when we apply the expanded nonlinear functions to the standard linear recursive Kalman filter algorithm. Next, since the state variable α_t and α_t-t are approximated as a conditional normal distribution given information up to time t - 1 (i.e., I_t-1) in approximation of the Taylor series expansion, it might be appropriate to evaluate each expectation by generating normal random numbers of α_t and α_t-1 given I_t-1 and those of the error terms θ and η_t. Thus, we propose the Monte-Carlo simulation filter using normal random draws. Finally we perform two Monte-Carlo experiments, where we obtain the result that the Monte-Carlo simulation filter has a superior performance over the nonlinear filters such as the extended Kalman filter and the second-order nonlinear filter.     

Assessing the bias of maximum likelihood estimates of contaminated garch models

It is well known that Gaussian maximum likelihood estimates of time series models are not robust. In this paper we prove this is also the case for the Generalized Autoregressive Conditional Heteroscedastic (GARCH) models. By expressing the Gaussian maximum likelihood estimates as Ψ estimates and by assuming the existence of a contaminated process, we prove they possess zero breakdown point and unbounded influence curves. By simulating GARCH processes under several proportions of contaminations we assess how much biased the maximum likelihood estimates may become and compare these results to a robust alternative. The t-student maximum likelihood estimates of GARCH models are also considered.     

Robust estimation of a multilevel model with structural change

We postulate a spatiotemporal multilevel model and estimate using forward search algorithm and MLE imbedded into the backfitting algorithm. Forward search algorithm ensures robustness of the estimates by filtering the effect of temporary structural changes in the estimation of the group-level covariates, the individual-level covariates and spatial parameters. Backfitting algorithm provides computational efficiency of estimation procedure assuming an additive model. Simulation studies show that estimates are robust even in the presence of structural changes induced for example by epidemic outbreak. The model also produced robust estimates even for small sample and short time series common in epidemiological settings.     

RECURSIVE ESTIMATION OF THE GENERAL LINEAR MODEL WITH DEPENDENT ERRORS AND MULTIPLE ADDITIONAL OBSERVATIONS1

The techniques for recursive estimation of the general linear model with dependent errors and known second order properties, is generalised to allow for simultaneous addition of an arbitrary number of additional observations. Computational formulae for recursive updating of parameter estimates are derived, together with a sequence of univariate recursive residuals for testing the constancy of the regression relation over time.     

Estimation of Periodic Bilinear Time Series Models

In this article, a new class of models is proposed for modeling nonlinear and nonstationary time series. This new class of models, referred to as the periodic bilinear models, has a state space representation and can be characterized by a set of recursive equations. Condition for the stationarity is presented. Procedures for parameter estimation using the cumulants of order less than four are described and the accuracy of the proposed method is demonstrated in the Monte Carlo simulations.     

Conditional inference in linear versus nonlinear models for binary time series

The modelling of discrete such as binary time series, unlike the continuous time series, is not easy. This is due to the fact that there is no unique way to model the correlation structure of the repeated binary data. Some models may also provide a complicated correlation structure with narrow ranges for the correlations. In this paper, we consider a nonlinear dynamic binary time series model that provides a correlation structure which is easy to interpret and the correlations under this model satisfy the full?1 to 1 range. For the estimation of the parameters of this nonlinear model, we use a conditional generalized quasilikelihood (CGQL) approach which provides the same estimates as those of the well-known maximum likelihood approach. Furthermore, we consider a competitive linear dynamic binary time series model and examine the performance of the CGQL approach through a simulation study in estimating the parameters of this linear model. The model mis-specification effects on estimation as well as forecasting are also examined through simulations.     

基于秩方法的TAM估计

一、问题的提出作为统计学一个重要分支学科 ,现代时间序列分析的发展十分惊人 ,尤其是近二十年来 ,人们已不再满足于平稳、线性的时间序列分析 ,如AR、MA、ARMA、ARIMA等 ,越来越多的人将视野投向非平稳时间序列、谱分析、时间序列的线性系统、非线性时间序列及非线性系统、空间序列、不等间隔抽样等问题的研究。TAM模型属于非线性时间序列分析的范围 ,是我国香港地区的学者汤家豪 (参见 [12 ,13])先生于 1978年提出来的 ,由于该模型具有一些重要的性质特征 ,如比 :设置“门坎”(门限 ) ,然后通过门限的控制作用 ,保障模型自身的稳…     

Local Fourier tests for structural change based on residuals

We propose a structural change test based on the recursive residuals with the local Fourier series estimators. The statistical properties of the proposed test are derived and the empirical properties are shown via simulation. We also consider other structural change tests based on CUSUM, MOSUM, moving estimates (ME), and empirical distribution functions with the recursive residuals and the ordinary residuals. Empirical powers are calculated in various structural change models for the comparison of those tests. These structural change tests are applied to South Korea's gross domestic product (GDP), South Korean Won to US Dollar currency exchange rates, and South Korea's Okun's law.     

Boosting techniques for nonlinear time series models

Many of the popular nonlinear time series models require a priori the choice of parametric functions which are assumed to be appropriate in specific applications. This approach is mainly used in financial applications, when sufficient knowledge is available about the nonlinear structure between the covariates and the response. One principal strategy to investigate a broader class on nonlinear time series is the Nonlinear Additive AutoRegressive (NAAR) model. The NAAR model estimates the lags of a time series as flexible functions in order to detect non-monotone relationships between current and past observations. We consider linear and additive models for identifying nonlinear relationships. A componentwise boosting algorithm is applied for simultaneous model fitting, variable selection, and model choice. Thus, with the application of boosting for fitting potentially nonlinear models we address the major issues in time series modelling: lag selection and nonlinearity. By means of simulation we compare boosting to alternative nonparametric methods. Boosting shows a strong overall performance in terms of precise estimations of highly nonlinear lag functions. The forecasting potential of boosting is examined on the German industrial production (IP); to improve the model’s forecasting quality we include additional exogenous variables. Thus we address the second major aspect in this paper which concerns the issue of high dimensionality in models. Allowing additional inputs in the model extends the NAAR model to a broader class of models, namely the NAARX model. We show that boosting can cope with large models which have many covariates compared to the number of observations.     

Estimating and Testing Nonlinear Local Dependence Between Two Time Series

ABSTRACT

The most common measure of dependence between two time series is the cross-correlation function. This measure gives a complete characterization of dependence for two linear and jointly Gaussian time series, but it often fails for nonlinear and non-Gaussian time series models, such as the ARCH-type models used in finance. The cross-correlation function is a global measure of dependence. In this article, we apply to bivariate time series the nonlinear local measure of dependence called local Gaussian correlation. It generally works well also for nonlinear models, and it can distinguish between positive and negative local dependence. We construct confidence intervals for the local Gaussian correlation and develop a test based on this measure of dependence. Asymptotic properties are derived for the parameter estimates, for the test functional and for a block bootstrap procedure. For both simulated and financial index data, we construct confidence intervals and we compare the proposed test with one based on the ordinary correlation and with one based on the Brownian distance correlation. Financial indexes are examined over a long time period and their local joint behavior, including tail behavior, is analyzed prior to, during and after the financial crisis. Supplementary material for this article is available online.     

The Performance of a Robust Multistage Estimator in Nonlinear Regression with Heteroscedastic Errors

In this article, a robust multistage parameter estimator is proposed for nonlinear regression with heteroscedastic variance, where the residual variances are considered as a general parametric function of predictors. The motivation is based on considering the chi-square distribution for the calculated sample variance of the data. It is shown that outliers that are influential in nonlinear regression parameter estimates are not necessarily influential in calculating the sample variance. This matter persuades us, not only to robustify the estimate of the parameters of the models for both the regression function and the variance, but also to replace the sample variance of the data by a robust scale estimate.     

Simultaneous estimation of number of signals and signal parameters of superimposed sinusoidal model: A robust sequential bivariate M-periodogram approach

Accurate estimation of the parameters of superimposed sinusoidal signals is an important problem in digital signal processing and time series analysis. In this article, we propose a simultaneous estimation procedure for estimation of the number of signals and signal parameters. The proposed sequential method is based on a robust bivariate M-periodogram and uses the orthogonal structure of the superimposed sinusoidal model for sequential estimation. Extensive simulations and data analysis show that the proposed method has a high degree of frequency resolution capability and can provide robust and efficient estimates of the number of signals and signal parameters.