Goodness-of-Fit Tests for Multiplicative Models with Dependent Data

Abstract.  Several classical time series models can be written as a regression model between the components of a strictly stationary bivariate process. Some of those models, such as the ARCH models, share the property of proportionality of the regression function and the scale function, which is an interesting feature in econometric and financial models. In this article, we present a procedure to test for this feature in a non-parametric context. The test is based on the difference between two non-parametric estimators of the distribution of the regression error. Asymptotic results are proved and some simulations are shown in the paper in order to illustrate the finite sample properties of the procedure.     

Bispectral-Based Goodness-of-Fit Tests of Gaussianity and Linearity of Stationary Time Series

Spectral domain tests for time series linearity typically suffer from a lack of power compared to time domain tests. We present two tests for Gaussianity and linearity of a stationary time series. The tests are two-stage procedures applying goodness-of-fit techniques to the estimated normalized bispectrum. We illustrate the performances of the tests are competitive with time domain tests. The new tests typically outperform Hinich's (1982 Hinich , M. J. ( 1982 ). Testing for Gaussianity and linearity of a stationary time series . J. Time Ser. Anal. 3 : 169 – 176 .[Crossref] , [Google Scholar]) bispectral based test, especially when the length of the time series is not large.     

Spectral Density Based Goodness-of-Fit Tests for Time Series Models

A new goodness-of-fit test for time series models is proposed. The test statistic is based on the distance between a kernel estimator of the ratio between the true and the hypothesized spectral density and the expected value of the estimator under the null. It provides a quantification of how well a parametric spectral density model fits the sample spectral density (periodogram). The asymptotic distribution of the statistic proposed is derived and its power properties are discussed. To improve upon the large sample (Gaussian) approximation of the distribution of the test statistic under the null, a bootstrap procedure is presented and justified theoretically. The finite sample performance of the test is investigated through a simulation experiment and applications to real data sets are given.     

Chi-Squared Goodness-of-Fit Tests for Parametric Accelerated Failure Time Models

We give chi-squared goodness-of fit tests for parametric regression models such as accelerated failure time, proportional hazards, generalized proportional hazards, frailty models, transformation models, and models with cross-effects of survival functions. Random right censored data are used. Choice of random grouping intervals as data functions is considered.     

SiZer analysis for the comparison of time series

SiZer (SIgnificant ZERo crossing of the derivatives) is a scale-space visualization tool for statistical inferences. In this paper we introduce a graphical device, which is based on SiZer, for the test of the equality of the mean of two time series. The estimation of the quantile in a confidence interval is theoretically justified by advanced distribution theory. The extension of the proposed method to the comparison of more than two time series is also done using residual analysis. A broad numerical study is conducted to demonstrate the sample performance of the proposed tool. In addition, asymptotic properties of SiZer for the comparison of two time series are investigated.     

基于Copula函数的金融时间序列模型述评

结合当前Copula函数及其应用的热点问题,着重评述了基于Copula函数的金融时间序列模型的应用。鉴于利用Copula可以将边际分布和变量间的相依结构分开来研究这一优良性质,在设定和估计模型时便显得极为方便和灵活。从模型的构造、Copula函数的选择、模型的估计以及拟合优度检验等几方面展开阐述和评价,介绍了Copula模型在金融领域中的几类应用,并对Copula理论和应用的新视角进行了展望。     

Parametric Inference in Stationary Time Series Models with Dependent Errors

This article is concerned with inference for the parameter vector in stationary time series models based on the frequency domain maximum likelihood estimator. The traditional method consistently estimates the asymptotic covariance matrix of the parameter estimator and usually assumes the independence of the innovation process. For dependent innovations, the asymptotic covariance matrix of the estimator depends on the fourth‐order cumulants of the unobserved innovation process, a consistent estimation of which is a difficult task. In this article, we propose a novel self‐normalization‐based approach to constructing a confidence region for the parameter vector in such models. The proposed procedure involves no smoothing parameter, and is widely applicable to a large class of long/short memory time series models with weakly dependent innovations. In simulation studies, we demonstrate favourable finite sample performance of our method in comparison with the traditional method and a residual block bootstrap approach.     

A Note on a Specification Test for Time Series Models Based on Spectral Density Estimation

In a recent paper, Paparoditis [Scand. J. Statist. 27 (2000) 143] proposed a new goodness‐of‐fit test for time series models based on spectral density estimation. The test statistic is based on the distance between a kernel estimator of the ratio of the true and the hypothesized spectral density and the expected value of the estimator under the null and provides a quantification of how well the parametric density fits the sample spectral density. In this paper, we give a detailed asymptotic analysis of the corresponding procedure under fixed alternatives.     

Inference for Time Series Regression Models With Weakly Dependent and Heteroscedastic Errors

Motivated by the need to assess the significance of the trend in some macroeconomic series, this article considers inference of a parameter in parametric trend functions when the errors exhibit certain degrees of nonstationarity with changing unconditional variances. We adopt the recently developed self-normalized approach to avoid the difficulty involved in the estimation of the asymptotic variance of the ordinary least-square estimator. The limiting distribution of the self-normalized quantity is nonpivotal but can be consistently approximated by using the wild bootstrap, which is not consistent in general without studentization. Numerical simulation demonstrates favorable coverage properties of the proposed method in comparison with alternative ones. The U.S. nominal wages series is analyzed to illustrate the finite sample performance. Some technical details are included in the online supplemental material.     

Fitting Time Series Models by Minimizing Multistep-ahead Errors: a Frequency Domain Approach

This paper brings together two topics in the estimation of time series forecasting models: the use of the multistep-ahead error sum of squares as a criterion to be minimized and frequency domain methods for carrying out this minimization. The methods are developed for the wide class of time series models having a spectrum which is linear in unknown coefficients. This includes the IMA(1, 1) model for which the common exponentially weigh-ted moving average predictor is optimal, besides more general structural models for series exhibiting trends and seasonality. The method is extended to include the Box–Jenkins `air line' model. The value of the multistep criterion is that it provides protection against using an incorrectly specified model. The value of frequency domain estimation is that the iteratively reweighted least squares scheme for fitting generalized linear models is readily extended to construct the parameter estimates and their standard errors. It also yields insight into the loss of efficiency when the model is correct and the robustness of the criterion against an incorrect model. A simple example is used to illustrate the method, and a real example demonstrates the extension to seasonal models. The discussion considers a diagnostic test statistic for indicating an incorrect model.     

The LASSO Method for Bilinear Time Series Models

In this article we propose a method called GLLS for the fitting of bilinear time series models. The GLLS procedure is the combination of the LASSO method, the generalized cross-validation method, the least angle regression method, and the stepwise regression method. Compared with the traditional methods such as the repeated residual method and the genetic algorithm, GLLS has the advantage of shrinking the coefficients of the models and saving the computational time. The Monte Carlo simulation studies and a real data example are reported to assess the performance of the proposed GLLS method.     

Tests for Structural Changes in Time Series of Counts

We propose methods for detecting structural changes in time series with discrete‐valued observations. The detector statistics come in familiar L2‐type formulations incorporating the empirical probability generating function. Special emphasis is given to the popular models of integer autoregression and Poisson autoregression. For both models, we study mainly structural changes due to a change in distribution, but we also comment for the classical problem of parameter change. The asymptotic properties of the proposed test statistics are studied under the null hypothesis as well as under alternatives. A Monte Carlo power study on bootstrap versions of the new methods is also included along with a real data example.     

Multiresolution anomaly detection method for fractional Gaussian noise

Driven by network intrusion detection, we propose a MultiResolution Anomaly Detection (MRAD) method, which effectively utilizes the multiscale properties of Internet features and network anomalies. In this paper, several theoretical properties of the MRAD method are explored. A major new result is the mathematical formulation of the notion that a two-scaled MRAD method has larger power than the average power of the detection method based on the given two scales. Test threshold is also developed. Comparisons between MRAD method and other classical outlier detectors in time series are reported as well.     

Estimation Procedure for a Multiple Time Series Model

Given a multiple time series sharing common autoregressive patterns, we estimate an additive model. The autoregressive component and the individual random effects are estimated by integrating maximum likelihood estimation and best linear unbiased predictions in a backfitting algorithm. The simulation study illustrated that the estimation procedure provides an alternative to the Arellano–Bond generalized method of moments (GMM) estimator of the panel model when T > N and the Arellano–Bond generally diverges. The estimator has high predictive ability. In cases where T ≤ N, the backfitting estimator is at least comparable to Arellano–Bond estimator.     

Structural Change Monitoring for Random Coefficient Autoregressive Time Series

A monitoring scheme is proposed to sequentially detect a structural change in random coefficient autoregressive time series of order p (RCA(p)) after a training period of size T. It extends structural change monitoring to RCA(p) time series. The asymptotic properties of our monitoring statistic are established under both the null of no change in parameters and the alternative of a change in coefficient. The finite sample properties are investigated by a simulation study.     

Nonparametric Trend Estimation for Periodic Autoregressive Time Series

A periodically stationary time series has seasonal variances. A local linear trend estimation is proposed to accommodate unequal variances. A comparison of this proposed estimator with the estimator commonly used for a stationary time series is provided. The optimal bandwidth selection for this new trend estimator is discussed.     

Forecasting Economic Time Series With Structural and Box-Jenkins Models: A Case Study

The basic structural model is a univariate time series model consisting of a slowly changing trend component, a slowly changing seasonal component, and a random irregular component. It is part of a class of models that have a number of advantages over the seasonal ARIMA models adopted by Box and Jenkins (1976). This article reports the results of an exercise in which the basic structural model was estimated for six U.K. macroeconomic time series and the forecasting performance compared with that of ARIMA models previously fitted by Prothero and Wallis (1976).     

Issues in Fitting Inverse Gaussian First Hitting Time Regression Models for Lifetime Data

Inverse Gaussian first hitting time regression models sometimes provide an attractive representation of lifetime data. Various authors comment that dependence of both parameters on the same covariate may imply multicollinearity. The frequent appearance of conflicting signs for the two coefficients of the same covariate may be related to this. We carry out simulation studies to examine the reality of this possible multicollinearity. Although there is some dependence between estimates, multicollinearity does not seem to be a major problem. Fitting this model to data generated by a Weibull regression suggests that conflicting signs of estimates may be due to model misspecification.     

Tests for time series of counts based on the probability-generating function

We propose testing procedures for the hypothesis that a given set of discrete observations may be formulated as a particular time series of counts with a specific conditional law. The new test statistics incorporate the empirical probability-generating function computed from the observations. Special emphasis is given to the popular models of integer autoregression and Poisson autoregression. The asymptotic properties of the proposed test statistics are studied under the null hypothesis as well as under alternatives. A Monte Carlo power study on bootstrap versions of the new methods is included as well as real-data examples.     

Dynamic Copula-Based Markov Time Series

This article examines a test procedure for checking the constancy of serial dependence via copulas for Markov time series data. It also provides a copula-based modeling approach for the dynamic serial dependence. Various parametric families of copulas offering different dependent structures are investigated. A score test is proposed for checking the constancy of a copula parameter. The score test is constructed and its asymptotic null distribution established under a two-stage estimation procedure. The test does not require specification of the probability distribution for the copula parameter. To capture the dynamics of dependence structure over time, autoregressive moving average and exponential type models are proposed. Illustrations are given based on simulated data and historic coffee prices data.