Additive Outlier Detection Via Extreme‐Value Theory

Abstract. This article is concerned with detecting additive outliers using extreme value methods. The test recently proposed for use with possibly non‐stationary time series by Perron and Rodriguez [Journal of Time Series Analysis (2003) vol. 24, pp. 193–220], is, as they point out, extremely sensitive to departures from their assumption of Gaussianity, even asymptotically. As an alternative, we investigate the robustness to distributional form of a test based on weighted spacings of the sample order statistics. Difficulties arising from uncertainty about the number of potential outliers are discussed, and a simple algorithm requiring minimal distributional assumptions is proposed and its performance evaluated. The new algorithm has dramatically lower level‐inflation in face of departures from Gaussianity than the Perron–Rodriguez test, yet retains good power in the presence of outliers.     

Outlier detection in ARMA models

Abstract. We consider an autoregressive moving‐average (ARMA) time series where the observations are perturbed by two kinds of outliers: an additive outlier (AO) or an innovation outlier (IO). Abraham and Yatawara [Journal of Time Series Analysis (1988) Vol. 9, pp. 109–19] investigate a sequential test which successively detects and identifies the outlier type. In this article, we propose an extension of this test, called ‘modified sequential test’, which performs the two procedures simultaneously and coherently. The asymptotic distribution of the test statistic is calculated under the null hypothesis that no outlier is present. Comparison of the two test procedures using simulation experiments shows that the proposed test gives a better power especially in the case of an IO.     

Bootstrap prediction intervals in state–space models

Abstract.  Prediction intervals in state–space models can be obtained by assuming Gaussian innovations and using the prediction equations of the Kalman filter, with the true parameters substituted by consistent estimates. This approach has two limitations. First, it does not incorporate the uncertainty caused by parameter estimation. Second, the Gaussianity of future innovations assumption may be inaccurate. To overcome these drawbacks, Wall and Stoffer [ Journal of Time Series Analysis (2002) Vol. 23, pp. 733–751] propose a bootstrap procedure for evaluating conditional forecast errors that requires the backward representation of the model. Obtaining this representation increases the complexity of the procedure and limits its implementation to models for which it exists. In this article, we propose a bootstrap procedure for constructing prediction intervals directly for the observations, which does not need the backward representation of the model. Consequently, its application is much simpler, without losing the good behaviour of bootstrap prediction intervals. We study its finite-sample properties and compare them with those of the standard and the Wall and Stoffer procedures for the local level model. Finally, we illustrate the results by implementing the new procedure to obtain prediction intervals for future values of a real time series.     

Temporal Aggregation and Bandwidth selection in estimating long memory

This article aims at showing that a temporal aggregation and a specific bandwidth reduction lead to the same asymptotic properties in estimating long memory by Geweke and Porter‐Hudak's [Journal of Time Series Analysis (1983 ) vol. 4, pp. 221–237] and Robinson's [Annals of Statistics (1995b ) vol. 23, pp. 1630–1661] estimators. In other words, irrespective of the level of temporal aggregation, the asymptotic properties of the estimator are uniquely determined by the number of periodogram ordinates used in the estimation, provided some mild additional assumptions are imposed. Monte Carlo simulations show that this result is a good approximation in finite samples. A real example with the daily US Dollar/French Franc exchange rate series is also provided.     

On the Evaluation of the Information Matrix for Multiplicative Seasonal Time‐Series Models

Abstract. This paper gives a procedure for evaluating the Fisher information matrix for a general multiplicative seasonal autoregressive moving average time‐series model. The method is based on the well‐known integral specification of Whittle [Ark. Mat. Fys. Astr. (1953) vol. 2. pp. 423–434] and leads to a system of linear equations, which is independent of the seasonal period and has a closed solution. It is shown to be much simpler, in general, than the method of Klein and Mélard [Journal of Time Series Analysis (1990) vol. 11, pp. 231–237], which depends on the seasonal period. It is also shown that the nonseasonal method of McLeod [Biometrika (1984) vol. 71, pp. 207–211] has the same basic features as that of Klein and Mélard. Explicit solutions are obtained for the simpler nonseasonal and seasonal models in common use, a feature which has not been attempted with the Klein–Mélard or the McLeod approaches. Several illustrations of these results are discussed in detail.     

Improved multivariate portmanteau test

A new portmanteau diagnostic test for vector autoregressive moving average (VARMA) models that is based on the determinant of the standardized multivariate residual autocorrelations is derived. The new test statistic may be considered an extension of the univariate portmanteau test statistic suggested by Peňa and Rodríguez (2002) . The asymptotic distribution of the test statistic is derived as well as a chi‐square approximation. However, the Monte–Carlo test is recommended unless the series is very long. Extensive simulation experiments demonstrate the usefulness of this test as well as its improved power performance compared to widely used previous multivariate portmanteau diagnostic check. Two illustrative applications are given.     

Two Simple Procedures for Testing for a Unit Root When There are Additive Outliers

This paper presents some results on testing for a unit root in the presence of additive outliers. Two procedures are proposed. The first procedure is to ignore the possibility of additive outliers and use modified Phillips–Perron statistics. The second procedure uses a new and very simple outlier detection statistic to identify outliers and then properly adjust standard Dickey–Fuller unit root tests. Simulations show that these procedures are robust to additive outliers in terms of size and power.     

ROBUST BAYESIAN ESTIMATION OF AUTOREGRESSIVE--MOVING-AVERAGE MODELS

A Bayesian approach is presented for modeling a time series by an autoregressive--moving-average model. The treatment is robust to innovation and additive outliers and identifies such outliers. It enforces stationarity on the autoregressive parameters and invertibility on the moving-average parameters, and takes account of uncertainty about the correct model by averaging the parameter estimates and forecasts of future observations over the set of permissible models. Posterior moments and densities of unknown parameters and observations are obtained by Markov chain Monte Carlo in O( n ) operations, where n is the sample size. The methodology is illustrated by applying it to a data set previously analyzed by Martin, Samarov and Vandaele (Robust methods for ARIMA models. Applied Time Series Analysis of Economic Data , ASA--Census--NBER Proceedings of the Conference on Applied Time Series Analysis of Economic Data (ed. A. Zellner), 1983, pp. 153--69) and to a simulated example.     

Recursive adjustment,unit root tests and structural breaks

In this paper, we introduce unit root tests for time series with a potential structural break computed from test regressions in which the deterministic components have been recursively adjusted. We present finite sample critical values as well as Monte Carlo results on the size and power performance of the new procedures, and compare these with other available tests in the literature, such as OLS and quasi‐differenced based tests (see, for instance, Perron, (1997) Perron and Rodriguez, (2003) and Carrion‐i‐Silvestre et al. (2009) ). The small sample behaviour of the tests is evaluated in a known and an unknown break date context allowing for negligible and non‐negligible initial conditions. In the unknown break date case, two break date estimation procedures are considered, one based on the minimum unit root t‐statistic and the other based on the minimum sum of squared residuals obtained from a regression on a set of deterministic variables. The size and power performance of the recursive adjustment based procedure in the unknown break date case is encouraging. A further result of this paper relates to the aditional finite sample evidence on the performance of quasi‐differenced unit root tests, complementing the results in Perron and Rodriguez (2003) .     

Range Unit‐Root (RUR) Tests: Robust against Nonlinearities,Error Distributions,Structural Breaks and Outliers

Abstract. Since the seminal paper by Dickey and Fuller in 1979, unit‐root tests have conditioned the standard approaches to analysing time series with strong serial dependence in mean behaviour, the focus being placed on the detection of eventual unit roots in an autoregressive model fitted to the series. In this paper, we propose a completely different method to test for the type of long‐wave patterns observed not only in unit‐root time series but also in series following more complex data‐generating mechanisms. To this end, our testing device analyses the unit‐root persistence exhibited by the data while imposing very few constraints on the generating mechanism. We call our device the range unit‐root (RUR) test since it is constructed from the running ranges of the series from which we derive its limit distribution. These nonparametric statistics endow the test with a number of desirable properties, the invariance to monotonic transformations of the series and the robustness to the presence of important parameter shifts. Moreover, the RUR test outperforms the power of standard unit‐root tests on near‐unit‐root stationary time series; it is invariant with respect to the innovations distribution and asymptotically immune to noise. An extension of the RUR test, called the forward–backward range unit‐root (FB‐RUR) improves the check in the presence of additive outliers. Finally, we illustrate the performances of both range tests and their discrepancies with the Dickey–Fuller unit‐root test on exchange rate series.     

Improved Prediction Limits For AR(p) and ARCH(p) Processes

Abstract. A new simulation‐based prediction limit that improves on any given estimative d‐step‐ahead prediction limit for a Markov process is described. This improved prediction limit can be found with almost no algebraic manipulations. Nonetheless, it has the same asymptotic coverage properties as the Barndorff‐Nielsen and Cox [Inference and Asymptotics (1994) Chapman and Hall, London] and Vidoni [Journal of Time Series Analysis Vol. 25, pp. 137–154.] (2004) improved prediction limits. The new simulation‐based prediction limit is ideally suited to those Markov process models for which the algebraic manipulations required for the latter improved prediction limits are very complicated. We illustrate the new method by applying it in the context of one‐step‐ahead prediction for a zero‐mean Gaussian AR(2) process and an ARCH(2) process.     

Robust discrimination between long‐range dependence and a change in mean

In this article we introduce a robust to outliers Wilcoxon change‐point testing procedure, for distinguishing between short‐range dependent time series with a change in mean at unknown time and stationary long‐range dependent time series. We establish the asymptotic distribution of the test statistic under the null hypothesis for L₁ near epoch dependent processes and show its consistency under the alternative. The Wilcoxon‐type testing procedure similarly as the CUSUM‐type testing procedure (of Berkes I., Horváth L., Kokoszka P. and Shao Q. 2006. Ann.Statist. 34:1140–1165), requires estimation of the location of a possible change‐point, and then using pre‐ and post‐break subsamples to discriminate between short and long‐range dependence. A simulation study examines the empirical size and power of the Wilcoxon‐type testing procedure in standard cases and with disturbances by outliers. It shows that in standard cases the Wilcoxon‐type testing procedure behaves equally well as the CUSUM‐type testing procedure but outperforms it in presence of outliers. We also apply both testing procedure to hydrologic data.     

An improved methodology for outlier detection in dynamic datasets

A time series Kalman filter (TSKF) is proposed that successfully handles outlier detection in dynamic systems, where normal process changes often mask the existence of outliers. The TSKF method combines a time series model fitting procedure with a modified Kalman filter to deal with additive outlier and innovational outlier detection problems in dynamic process dataset. Compared with current outlier detection methods, the new method enjoys the following advantages: (a) no prior knowledge of the process model is needed; (b) it is easy to tune; (c) it can be applied to both univariate and multivariate outlier detection; (d) it is applicable to both on‐line and off‐line operation; (e) it cleans outliers while maintains the integrity of the original dataset. © 2014 American Institute of Chemical Engineers AIChE J, 61: 419–433, 2015     

Interventions in INGARCH processes

We study the problem of intervention effects generating various types of outliers in a linear count time‐series model. This model belongs to the class of observation‐driven models and extends the class of Gaussian linear time‐series models within the exponential family framework. Studies about effects of covariates and interventions for count time‐series models have largely fallen behind, because the underlying process, whose behaviour determines the dynamics of the observed process, is not observed. We suggest a computationally feasible approach to these problems, focusing especially on the detection and estimation of sudden shifts and outliers. We consider three different scenarios, namely the detection of an intervention effect of a known type at a known time, the detection of an intervention effect when the type and the time are both unknown and the detection of multiple intervention effects. We develop score tests for the first scenario and a parametric bootstrap procedure based on the maximum of the different score test statistics for the second scenario. The third scenario is treated by a stepwise procedure, where we detect and correct intervention effects iteratively. The usefulness of the proposed methods is illustrated using simulated and real data examples.     

Maximum likelihood estimation of higher‐order integer‐valued autoregressive processes

Abstract. In this article, we extend the earlier work of Freeland and McCabe [Journal of time Series Analysis (2004) Vol. 25, pp. 701–722] and develop a general framework for maximum likelihood (ML) analysis of higher‐order integer‐valued autoregressive processes. Our exposition includes the case where the innovation sequence has a Poisson distribution and the thinning is binomial. A recursive representation of the transition probability of the model is proposed. Based on this transition probability, we derive expressions for the score function and the Fisher information matrix, which form the basis for ML estimation and inference. Similar to the results in Freeland and McCabe (2004) , we show that the score function and the Fisher information matrix can be neatly represented as conditional expectations. Using the INAR(2) specification with binomial thinning and Poisson innovations, we examine both the asymptotic efficiency and finite sample properties of the ML estimator in relation to the widely used conditional least squares (CLS) and Yule–Walker (YW) estimators. We conclude that, if the Poisson assumption can be justified, there are substantial gains to be had from using ML especially when the thinning parameters are large.     

Local Gaussian Autocorrelation and Tests for Serial Independence

The traditional and most used measure for serial dependence in a time series is the autocorrelation function. This measure gives a complete characterization of dependence for a Gaussian time series, but it often fails for nonlinear time series models as, for instance, the generalized autoregressive conditional heteroskedasticity model (GARCH), where it is zero for all lags. The autocorrelation function is an example of a global measure of dependence. The purpose of this article is to apply to time series a well‐defined local measure of serial dependence called the local Gaussian autocorrelation. It generally works well also for nonlinear models, and it can distinguish between positive and negative dependence. We use this measure to construct a test of independence based on the bootstrap technique. This procedure requires the choice of a bandwidth parameter that is calculated using a cross validation algorithm. To ensure the validity of the test, asymptotic properties are derived for the test functional and for the bootstrap procedure, together with a study of its power for different models. We compare the proposed test with one based on the ordinary autocorrelation and with one based on the Brownian distance correlation. The new test performs well. Finally, there are also two empirical examples.     

Synthesis of poly [(methylsi1oxane)-co-(dimethylsilazane)] copolymers as precursors of ceramic materials

Summary A series of new crosslinked poly[(methylsiloxane)-co-(dimethylsilazane)] copolymers were synthesized by cationic ring opening polymerization of the cyclic monomers 1,3,5,7 – tetramethylcyclotetrasiloxane and 2,2,4,4,6,6 – hexamethylcyclotrisilazane. These copolymers contain different concentrations of methylsiloxane and silazane co-monomer units. The thermal stability is strongly related to the concentration of methylsiloxane co-units in the copolymers. The weight loss curve of the copolymers lies in a temperature range between its two homopolymers. FT-IR studies are in good agreement with the proposed structure, either for the copolymers or for the residual powder ceramic obtained by pyrolysis. Two cured processes (thermal and UV) were used. The pyrolysis of the copolymers were carried out under argon atmosphere at 1200°C and in air at 1000 °C. Kinetic decomposition parameters of the copolymers such as a pre-exponential factor, the decomposition reaction orders and the activation energies were determined. The molecular weights were determined by osmometry. Received: 21 August 2002/Revised version: 20 February 2003/ Accepted: 21 February 2003 Correspondence to Mario Rodríguez-Baeza     

A HYBRID BOOTSTRAP APPROACH TO UNIT ROOT TESTS

This article proposes a hybrid bootstrap approach to approximate the augmented Dickey–Fuller test by perturbing both the residual sequence and the minimand of the objective function. Since innovations can be dependent, this allows the inclusion of conditional heteroscedasticity models. The new bootstrap method is also applied to least absolute deviation‐based unit root test statistics, which are efficient in handling heavy‐tailed time‐series data. The asymptotic distributions of resulting bootstrap tests are presented, and Monte Carlo studies demonstrate the usefulness of the proposed tests.     

A sequential procedure to determine the number of breaks in trend with an integrated or stationary noise component

Perron and Yabu (2009a) consider the problem of testing for a break occurring at an unknown date in the trend function of a univariate time series when the noise component can be either stationary or integrated. This article extends their work by proposing a sequential test that allows one to test the null hypothesis of, say, l breaks versus the alternative hypothesis of (l + 1) breaks. The test enables consistent estimation of the number of breaks. In both stationary and integrated cases, it is shown that asymptotic critical values can be obtained from the relevant quantiles of the limit distribution of the test for a single break. Monte Carlo simulations suggest that the procedure works well in finite samples.     

Oracle Properties,Bias Correction,and Bootstrap Inference for Adaptive Lasso for Time Series M‐Estimators

We derive new theoretical results on the properties of the adaptive least absolute shrinkage and selection operator (adaptive lasso) for possibly nonlinear time series models. In particular, we investigate the question of how to conduct inference on the parameters given an adaptive lasso model. Central to this study is the test of the hypothesis that a given adaptive lasso parameter equals zero, which therefore tests for a false positive. To this end, we introduce a recentered bootstrap procedure and show, theoretically and empirically through extensive Monte Carlo simulations, that the adaptive lasso can combine efficient parameter estimation, variable selection, and inference in one step. Moreover, we analytically derive a bias correction factor that is able to significantly improve the empirical coverage of the test on the active variables. Finally, we apply the adaptive lasso and the recentered bootstrap procedure to investigate the relation between the short rate dynamics and the economy, thereby providing a statistical foundation (from a model choice perspective) for the classic Taylor rule monetary policy model.