Simple detection of outlying short time series

Sets of relatively short time series arise in many situations. One aspect of their analysis may be the detection of outlying series. We examine the performance of standard normal outlier tests applied to the means, or to simple functions of the means, of AR(1) series, not necessarily of equal lengths. Although unequal lengths of series implies that the means have unequal variances, that are only known approximately, it is shown that nominal significance levels hold good under most circumstances. Thus a standard outlier test can usefully be applied, avoiding the complication of estimating the time series' parameters. The test's power is affected by unequal lengths, being higher when the slippage occurs in one of the longer series     

Some weighted mixed portmanteau tests for diagnostic checking in linear time series models

The popular diagnostic checking methods in linear time series models are portmanteau tests based on either residual autocorrelation functions (acf) or partial autocorrelation functions (pacf). In this paper, we device some new weighted mixed portmanteau tests by appropriately combining individual tests based on both acf and pacf. We derive the asymptotic distribution of such weighted mixed portmanteau statistics and study their size and power. It is found that the weighted mixed tests outperform when higher order ARMA models are fitted and diagnostic checks are performed via testing lack of residual autocorrelations. Simulation results suggest to use the proposed tests as complementary to those classical tests found in literature. An illustrative application is given to demonstrate the usefulness of the mixed test.     

Polynomial trends in time series

The J^* test which was previously proposed by the present author for the detection of a trend in a time series does not depend on any quantitative assumptions, but in the case of a polynomial trend it depends on its degree; if this degree is too high, the test cannot be applied. The author finds a bound of the significance level at which the test can be applied when the sample size, as well as a bound of the degree of the trend, are given. Asymptotic results are used only when we trust the asymptotic distribution of J^* under the null hypothesis.     

Modeling nonlinear time series with local mixtures of generalized linear models

The authors consider a novel class of nonlinear time series models based on local mixtures of regressions of exponential family models, where the covariates include functions of lags of the dependent variable. They give conditions to guarantee consistency of the maximum likelihood estimator for correctly specified models, with stationary and nonstationary predictors. They show that consistency of the maximum likelihood estimator still holds under model misspecification. They also provide probabilistic results for the proposed model when the vector of predictors contains only lags of transformations of the modeled time series. They illustrate the consistency of the maximum likelihood estimator and the probabilistic properties via Monte Carlo simulations. Finally, they present an application using real data.     

Structural changes in autoregressive models for binary time series

We study autoregressive models for binary time series with possible changes in their parameters. A procedure for detection and testing of a single change is suggested. The limiting behavior of the test statistic is derived. The performance of the test is analyzed under the null hypothesis as well as under different alternatives via a simulation study. Application of the method to a real data set on US recession is provided as an illustration.     

Change detection in linear regression with time series errors

A new test for detecting a change in linear regression parameters assuming a general weakly dependent error structure is given. It extends earlier methods based on cumulative sums assuming independent errors. The novelty is in the new standardization method and in smoothing when the time series is dominated by high frequencies. Simulations show the excellent performance of the test. Examples are taken from environmental applications. The algorithm is easy to implement. Testing for multiple changes can be done by segmentation. The Canadian Journal of Statistics 38:65–79; 2010 © 2009 Statistical Society of Canada     

Detection of patterns in noisy time series

In the context of a research project in ergonomy, myoelectric signals monitored over two to three hour periods gave rise to long noisy time series, which were smoothed using running medians. Tests developed by the authors show that the patterns displayed by the smoothed time series are not artifacts of smoothed white noise. Indeed, the smoothed series show amplitude fluctuations and short‐term correlations which are larger than those obtained by applying running medians to independent, identically distributed data. The key idea is that of reduction of data to binary signals.     

Self-exciting threshold models for time series of counts with a finite range

In this article, an integer-valued self-exciting threshold model with a finite range based on the binomial INARCH(1) model is proposed. Important stochastic properties are derived, and approaches for parameter estimation are discussed. A real-data example about the regional spread of public drunkenness in Pittsburgh demonstrates the applicability of the new model in comparison to existing models. Feasible modifications of the model are presented, which are designed to handle special features such as zero-inflation.     

Testing time series for interpolability and whiteness

We propose a test to decide if a time series is represented by its linear interpolator better than by its mean value. The same test can be employed to decide if a time series has to be considered white noise. The test is based on a new estimate of the index of linear determinism (Battaglia, 1983, Inverse autocovariances and a measure of linear determinism for a stationary process, J. Time Series Anal. 4, 79-87) and its asymptotic distribution is derived. Comparison with the popular Ljung-Box portmanteau test has been performed based on both asymptotic power and a simulation experiment. The new test     

On consistent testing for serial correlation in seasonal time series models

The author considers serial correlation testing in seasonal time series models. He proposes a test statistic based on a spectral approach. Many tests of this type rely on kernel-based spectral density estimators that assign larger weights to low order lags than to high ones. Under seasonality, however, large autocorrelations may occur at seasonal lags that classical kernel estimators cannot take into account. The author thus proposes a test statistic that relies on the spectral density estimator of Shin (2004), whose weighting scheme is more adapted to this context. The distribution of his test statistic is derived under the null hypothesis and he studies its behaviour under fixed and local alternatives. He establishes the consistency of the test under a general fixed alternative. He also makes recommendations for the choice of the smoothing parameters. His simulation results suggest that his test is more powerful against seasonality than alternative procedures based on classical weighting schemes. He illustrates his procedure with monthly statistics on employment among young Americans.     

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.     

Nonparametric confidence intervals for location in time series data

We extend the confidence interval construction procedure for location for symmetric iid data using the one-sample Wilcoxon signed rank statistic (T⁺) to stationary time series data. We propose a normal approximation procedure when explicit knowledge of the underlying dependence structure/distribution is unknown. By conducting extensive simulations from linear and nonlinear time series models, we show that the extended procedure is a strong contender for use in the construction of confidence intervals in time series analysis. Finally we demonstrate real application implementations in two case studies.     

Missing data and forecasting in multivariate time series: An application of the common components dynamic linear model

Summary The paper deals with missing data and forecasting problems in multivariate time series making use of the Common Components Dynamic Linear Model (DLMCC), presented in Quintana (1985), and West and Harrison (1989). Some results are presented and discussed: exploiting the correlation between series, estimated by the DLMCC, the paper shows as it is possible to update state vector posterior distributions for the unobserved series. This is realized on the base of the updating of the observed series state vectors, for which the usual Kalman filter equations can be applied. An application concerning some Italian private consumption series provides an example of the model capabilities.     

Using difference-based methods for inference in nonparametric regression with time series errors

Summary. We show that difference-based methods can be used to construct simple and explicit estimators of error covariance and autoregressive parameters in nonparametric regression with time series errors. When the error process is Gaussian our estimators are efficient, but they are available well beyond the Gaussian case. As an illustration of their usefulness we show that difference-based estimators can be used to produce a simplified version of time series cross-validation. This new approach produces a bandwidth selector that is equivalent, to both first and second orders, to that given by the full time series cross-validation algorithm. Other applications of difference-based methods are to variance estimation and construction of confidence bands in nonparametric regression.     

On testing for multivariate ARCH effects in vector time series models

Using a spectral approach, the authors propose tests to detect multivariate ARCH effects in the residuals from a multivariate regression model. The tests are based on a comparison, via a quadratic norm, between the uniform density and a kernel‐based spectral density estimator of the squared residuals and cross products of residuals. The proposed tests are consistent under an arbitrary fixed alternative. The authors present a new application of the test due to Hosking (1980) which is seen to be a special case of their approach involving the truncated uniform kernel. However, they typically obtain more powerful procedures when using a different weighting. The authors consider especially the procedure of Robinson (1991) for choosing the smoothing parameter of the spectral density estimator. They also introduce a generalized version of the test for ARCH effects due to Ling & Li (1997). They investigate the finite‐sample performance of their tests and compare them to existing tests including those of Ling & Li (1997) and the residual‐based diagnostics of Tse (2002).Finally, they present a financial application.     

Test of parameter changes in a class of observation-driven models for count time series

Abstract

This paper investigates the parameter-change tests for a class of observation-driven models for count time series. We propose two cumulative sum (CUSUM) test procedures for detection of changes in model parameters. Under regularity conditions, the asymptotic null distributions of the test statistics are established. In addition, the integer-valued generalized autoregressive conditional heteroskedastic (INGARCH) processes with conditional negative binomial distributions are investigated. The developed techniques are examined through simulation studies and also are illustrated using an empirical example.     

On testing for causality in variance between two multivariate time series

Verifying the existence of a relationship between two multivariate time series represents an important consideration. In this article, the procedure developed by Cheung and Ng [A causality-in-variance test and its application to financial market prices, J. Econom. 72 (1996), pp. 33–48] designed to test causality in variance for univariate time series is generalized in several directions. A first approach proposes test statistics based on residual cross-covariance matrices of squared (standardized) residuals and cross products of (standardized) residuals. In a second approach, transformed residuals are defined for each residual vector time series, and test statistics are constructed based on the cross-correlations of these transformed residuals. Test statistics at individual lags and portmanteau-type test statistics are developed. Conditions are given under which the new test statistics converge in distribution towards chi-square distributions. The proposed methodology can be used to determine the directions of causality in variance, and appropriate test statistics are presented. Monte Carlo simulation results show that the new test statistics offer satisfactory empirical properties. An application with two bivariate financial time series illustrates the methods.     

Tests for outliers in the inverse Gaussian distribution,with application to first hitting time models

The inverse Gaussian (IG) distribution is often applied in statistical modelling, especially with lifetime data. We present tests for outlying values of the parameters (μ, λ) of this distribution when data are available from a sample of independent units and possibly with more than one event per unit. Outlier tests are constructed from likelihood ratio tests for equality of parameters. The test for an outlying value of λ is based on an F-distributed statistic that is transformed to an approximate normal statistic when there are unequal numbers of events per unit. Simulation studies are used to confirm that Bonferroni tests have accurate size and to examine the powers of the tests. The application to first hitting time models, where the IG distribution is derived from an underlying Wiener process, is described. The tests are illustrated on data concerning the strength of different lots of insulating material.