Unit root testing with slowly varying trends

A unit root test is proposed for time series with a general nonlinear deterministic trend component. It is shown that asymptotically the pooled OLS estimator of overlapping blocks filters out any trend component that satisfies some Lipschitz condition. Under both fixed‐b and small‐b block asymptotics, the limiting distribution of the t‐statistic for the unit root hypothesis is derived. Nuisance parameter corrections provide heteroskedasticity‐robust tests, and serial correlation is accounted for by pre‐whitening. A Monte Carlo study that considers slowly varying trends yields both good size and improved power results for the proposed tests when compared to conventional unit root tests.     

The impact of the initial condition on robust tests for a linear trend

This article examines the behaviour of some recently proposed ‘robust’ (to the order of integration of the data) tests for the presence of a deterministic linear trend in a univariate times series in situations where the magnitude of the initial condition of the series is non‐negligible. We demonstrate that the asymptotic size and/or local power properties of these tests are extremely sensitive to the initial condition. Straightforward modifications to the trend tests are suggested, based on the use of trimmed data, which are shown to help reduce this sensitivity.     

The power of unit root tests against nonlinear local alternatives

This article extends the analysis of local power of unit root tests in a nonlinear direction by considering local nonlinear alternatives and tests built specifically against stationary nonlinear models. In particular, we focus on the popular test proposed by Kapetanios et al. (2003, Journal of Econometrics 112, 359–379) in comparison to the linear Dickey–Fuller test. To this end, we consider different adjustment schemes for deterministic terms. We provide asymptotic results which imply that the error variance has a severe impact on the behaviour of the tests in the nonlinear case; the reason for such behaviour is the interplay of non‐stationarity and nonlinearity. In particular, we show that nonlinearity of the data generating process can be asymptotically negligible when the error variance is moderate or large (compared to the ‘amount of nonlinearity’), rendering the linear test more powerful than the nonlinear one. Should however the error variance be small, the nonlinear test has better power against local alternatives. We illustrate this in an asymptotic framework of what we call persistent nonlinearity. The theoretical findings of this article explain previous results in the literature obtained by simulation. Furthermore, our own simulation results suggest that the user‐specified adjustment scheme for deterministic components (e.g. OLS, GLS, or recursive adjustment) has a much higher impact on the power of unit root tests than accounting for nonlinearity, at least under local (linear or nonlinear) alternatives.     

Detecting misspecifications in autoregressive conditional duration models and non‐negative time‐series processes

We develop a general theory to test correct specification of multiplicative error models of non‐negative time‐series processes, which include the popular autoregressive conditional duration (ACD) models. Both linear and nonlinear conditional expectation models are covered, and standardized innovations can have time‐varying conditional dispersion and higher‐order conditional moments of unknown form. No specific estimation method is required, and the tests have a convenient null asymptotic N(0,1) distribution. To reduce the impact of parameter estimation uncertainty in finite samples, we adopt Wooldridge's (1990a) device to our context and justify its validity. Simulation studies show that in the context of testing ACD models, finite sample correction gives better sizes in finite samples and are robust to parameter estimation uncertainty. And, it is important to take into account time‐varying conditional dispersion and higher‐order conditional moments in standardized innovations; failure to do so can cause strong overrejection of a correctly specified ACD model. The proposed tests have reasonable power against a variety of popular linear and nonlinear ACD alternatives.     

Fractional cointegration in the presence of linear trends

Abstract. We consider bivariate regressions of nonstationary fractionally integrated variables dominated by linear time trends. The asymptotic behaviour of the ordinary least square (OLS) estimators in this case allows limiting normality to arise at a faster rate of convergence than if the individual series were detrended, increasing in this way the power of the tests for fractional cointegration. We also show that the limiting distribution of the t‐ratio of the slope coefficient depends upon the presence or not of a deterministic trend in the conditional regressor. We introduce the concept of local fractional trend to explain the apparently diverging asymptotic theories that apply when a trend is either present or absent in our set‐up.     

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.     

A FIXED‐ b TEST FOR A BREAK IN LEVEL AT AN UNKNOWN TIME UNDER FRACTIONAL INTEGRATION

In this paper, we propose a test for a break in the level of a fractionally integrated process when the timing of the putative break is not known. This testing problem has received considerable attention in the literature in the case where the time series is weakly autocorrelated. Less attention has been given to the case where the underlying time series is allowed to be fractionally integrated. Here, valid testing can only be performed if the limiting null distribution of the level break test statistic is well defined for all values of the fractional integration exponent considered. However, conventional sup‐Wald type tests diverge when the data are strongly autocorrelated. We show that a sup‐Wald statistic, which is standardized using a non‐parametric kernel‐based long‐run variance estimator, does possess a well‐defined limit distribution, depending only on the fractional integration parameter, provided the recently developed fixed‐b asymptotic framework is applied. We give the appropriate asymptotic critical values for this sup‐Wald statistic and show that it has good finite sample size and power properties.     

Bootstrap procedures for detecting multiple persistence shifts in heteroskedastic time series

This article proposes new bootstrap procedures for detecting multiple persistence shifts in a time series driven by non-stationary volatility. The assumed volatility process can accommodate discrete breaks, smooth transition variation as well as trending volatility. We develop wild bootstrap sup-Wald tests of the null hypothesis that the process is either stationary [I(0)] or has a unit root [I(1)] throughout the sample. We also propose a sequential procedure to estimate the number of persistence breaks based on ordering the regime-specific bootstrap p-values. The asymptotic validity of the advocated procedures is established both under the null of stability and a variety of persistence change alternatives. A comparison with existing tests that assume homoskedasticity illustrates the finite sample improvements offered by our methods. An application to OECD inflation rates highlights the empirical relevance of the proposed approach and weakens the case for persistence change relative to existing procedures.     

IV‐BASED COINTEGRATION TESTING IN DEPENDENT PANELS WITH TIME‐VARYING VARIANCE

The distributions of cointegration tests are affected when the innovation variance varies over time. In panels, one must also pay attention to dependence among units. To obtain a panel cointegration test robust to both heteroskedasticity and dependence, we adapt the nonlinear instruments method proposed for the Dickey–Fuller test by Chang (2002, J Econometrics 110, 261–292) to an error‐correction framework. We show that IV‐based testing of the no error‐correction null in individual equations yields standard normal test statistics when computed with heteroskedasticity‐robust standard errors. The result holds under endogenous regressors, irrespective of the number of integrated covariates and for any variance profile. A non‐cointegration test combining single‐equation tests retains these nice properties. In panels of fixed cross‐sectional dimension, such test statistics from individual units are shown to be asymptotically independent even under dependence, leading to panel tests robust to dependence and heteroskedasticity. The tests perform well in finite panels.     

A NOTE ON TESTING FOR SEASONAL COINTEGRATION USING PRINCIPAL COMPONENTS IN THE FREQUENCY DOMAIN

Abstract. This paper deals with testing for cointegration at any frequency with a focus on the bounds tests proposed by Joyeux (Tests for seasonal cointegration using principal components. J. Time Ser. Anal. 13 (1992), 109–18). It is shown that this class of tests has asymptotic size equal to one because the author does not take into account non-contemporaneous cointegration at frequencies other than zero and π. The consequences of this size distortion with finite samples are investigated by a Monte Carlo experiment. Bounds tests for contemporaneous cointegration are also proposed. Finally, an empirical example of testing for seasonal cointegration in monthly time series is presented and discussed.     

Seasonal Unit Root Tests Under Structural Breaks*

Abstract. In this paper, several seasonal unit root tests are analysed in the context of structural breaks at known time and a new break corrected test is suggested. We show that the widely used HEGY test, as well as an LM variant thereof, are asymptotically robust to seasonal mean shifts of finite magnitude. In finite samples, however, experiments reveal that such tests suffer from severe size distortions and power reductions when breaks are present. Hence, a new break corrected LM test is proposed to overcome this problem. Importantly, the correction for seasonal mean shifts bears no consequence on the limiting distributions, thereby maintaining the legitimacy of canonical critical values. Moreover, although this test assumes a breakpoint a priori, it is robust in terms of misspecification of the time of the break. This asymptotic property is well reproduced in finite samples. Based on a Monte‐Carlo study, our new test is compared with other procedures suggested in the literature and shown to hold superior finite sample properties.     

A Sequential and Iterative Testing Procedure to Identify the Nature of a Time Series Generating Process

Abstract

An inferential approach is proposed to identify the nature of the generating process corresponding to a real time series. This new sequential and iterative testing procedure goes beyond the Box and Jenkins methodology for the identification, estimation, and validation of linear data generating processes by investigating the probabilistic structure of non-Gaussian estimated residuals {? _t } for the possible presence of nonlinear serial dependence. The testing procedure aims at indicating the right type of dependence present in a series by means of specific inferential tests on the moments of the generating structure probability distribution. The test statistics adopted are very popular and powerful and encompass a wide range of stochastic nonlinearity alternatives. The U.S. Industrial Production Index series is used to illustrate the iterative testing procedure proposed.     

CUSUM of Squares‐Based Tests for a Change in Persistence

Abstract. Using standardized cumulative sums of squared sub‐sample residuals, we propose a new ratio‐based test of the null hypothesis that a time series exhibits no change in its persistence structure [specifically that it displays constant I(1) behaviour] against the alternative of a change in persistence from trend stationarity to difference stationarity, or vice versa. Neither the direction nor location of any possible change under the alternative hypothesis need be assumed known. A key feature of our proposed test which distinguishes it from extant tests for persistence change [certain of which test the null hypothesis of constant I(0) behaviour while others, like our proposed test, test the null hypothesis of constant I(1) behaviour] is that it displays no tendency to spuriously over‐reject when applied to series which, although not constant I(1) series, do not display a change in persistence [specifically are constant I(0) processes]. Moreover, where our ratio test correctly rejects the null of no persistence change, the tail in which the rejection occurs can also be used to identify the direction of change since, even in relatively small samples, the test almost never rejects in the right [left] tail when there is a change from I(0) to I(1) [I(1) to I(0)]. Again this useful property is not shared by existing tests. As a by‐product of our analysis, we also propose breakpoint estimators which are consistent where the timing of the change in persistence is unknown.     

Testing for parameter constancy in non‐Gaussian time series

This paper investigates testing for parameter constancy in models for non‐Gaussian time series. Models for discrete valued count time series are investigated as well as more general models with autoregressive conditional expectations. Both sup‐tests and CUSUM procedures are suggested depending on the complexity of the model being used. The asymptotic distribution of the CUSUM test is derived for a general class of conditional autoregressive models.     

Persistence Heterogeneity Testing in Panels with Interactive Fixed Effects

We consider large N,T panel data models with fixed effects, a common factor allowing for cross‐section dependence, and persistent data and shocks, which are assumed fractionally integrated. In a basic setup, the main interest is on the fractional parameter of the idiosyncratic component, which is estimated in first differences after factor removal by projection on the cross‐section average. The pooled conditional‐sum‐of‐squares estimate is consistent but the normal asymptotic distribution might not be centred, requiring the time series dimension to grow faster than the cross‐section size for correction. We develop tests of homogeneity of dynamics, including the degree of integration, that have no trivial power under local departures from the null hypothesis of a non‐negligible fraction of cross‐section units. A simulation study shows that our estimates and tests have good performance even in moderately small panels.     

PORTMANTEAU AUTOCORRELATION TESTS UNDER Q‐DEPENDENCE AND HETEROSKEDASTICITY

We propose extensions of the Box–Pierce ( 1970 ) portmanteau autocorrelation test to allow for two generalizations: (i) time series that exhibit unconditional heteroskedasticity and (ii) to test for the presence of autocorrelation only after a fixed lag q. These extensions involve a generalized quadratic form of the Box–Pierce test that uses the heteroskedasticity autocorrelation consistent‐type estimator. While we show that this modified test is robust to unconditional heteroskedasticity, the resulting power loss may be substantial. We therefore develop feasible weighted tests that make use of nonparametric estimates of the unobserved variance process. Simulation experiments show that the weighted tests have good size and superior power properties over the unweighted tests.     

Asymptotic self‐similarity and wavelet estimation for long‐range dependent fractional autoregressive integrated moving average time series with stable innovations

Abstract. Methods for parameter estimation in the presence of long‐range dependence and heavy tails are scarce. Fractional autoregressive integrated moving average (FARIMA) time series for positive values of the fractional differencing exponent d can be used to model long‐range dependence in the case of heavy‐tailed distributions. In this paper, we focus on the estimation of the Hurst parameter H = d + 1/α for long‐range dependent FARIMA time series with symmetric α‐stable (1 < α < 2) innovations. We establish the consistency and the asymptotic normality of two types of wavelet estimators of the parameter H. We do so by exploiting the fact that the integrated series is asymptotically self‐similar with parameter H. When the parameter α is known, we also obtain consistent and asymptotically normal estimators for the fractional differencing exponent d = H ? 1/α. Our results hold for a larger class of causal linear processes with stable symmetric innovations. As the wavelet‐based estimation method used here is semi‐parametric, it allows for a more robust treatment of long‐range dependent data than parametric methods.     

A bootstrap test for additive outliers in non‐stationary time series

In this paper we propose a new procedure for detecting additive outliers in a univariate time series based on a bootstrap implementation of the test of Perron and Rodríguez (2003, Journal of Time Series Analysis 24, 193‐220). This procedure is used to test the null hypothesis that a time series is uncontaminated by additive outliers against the alternative that one or more additive outliers are present. We demonstrate that the existing tests of, inter alia, Vogelsang (1999, Journal of Time Series Analysis 20, 237–52) Perron and Rodríguez (2003) and Burridge and Taylor (2006, Journal of Time Series Analysis 27, 685–701) are unable to strike a balance between size and power when the order of integration of a time series is unknown and the time series is driven by innovations drawn from an unknown distribution. We show that the proposed bootstrap testing procedure is able to control size to such an extent that its size properties are comparable with the robust test of Burridge and Taylor (2006) when the distribution of the innovations is not assumed known, whilst maintaining power in the Gaussian environment close to that of the test of Perron and Rodríguez (2003).     

Scheduling of crude oil operations under demand uncertainty: A robust optimization framework coupled with global optimization

Scheduling of crude oil operations is an important component of overall refinery operations, because crude oil costs account for about 80% of the refinery turnover. The mathematical modeling of blending different crudes in storage tanks results in many bilinear terms, which transform the problem into a challenging, nonconvex, mixed‐integer nonlinear programming (MINLP) optimization model. In practice, uncertainties are unavoidable and include demand fluctuations, ship arrival delays, equipment malfunction, and tank unavailability. In the presence of these uncertainties, an optimal schedule generated using nominal parameter values may often be suboptimal or even become infeasible. In this article, the robust optimization framework proposed by Lin et al. and Janak et al. is extended to develop a deterministic robust counterpart optimization model for demand uncertainty. The recently proposed branch and bound global optimization algorithm with piecewise‐linear underestimation of bilinear terms by Li et al. is also extended to solve the nonconvex MINLP deterministic robust counterpart optimization model and generate robust schedules. Two examples are used to illustrate the capability of the proposed robust optimization approach, and the extended branch and bound global optimization algorithm for demand uncertainty. The computational results demonstrate that the obtained schedules are robust in the presence of demand uncertainty. © 2011 American Institute of Chemical Engineers AIChE J, 58: 2373–2396, 2012     

Robust and powerful serial correlation tests with new robust estimates in ARX models

Abstract. We consider robust serial correlation tests in autoregressive models with exogenous variables (ARX). Since the least squares estimators are not robust when outliers are present, a new family of estimators is introduced, called residual autocovariances for ARX (RA‐ARX). They provide resistant estimators that are less sensible to abnormal observations in the output variable of the dynamic model. Such ‘bad’ observations could be due to unexpected phenomena such as economic crisis or equipment failure in engineering, among others. We show that the new robust estimators are consistent and we can consider robust and powerful tests of serial correlation in ARX models based on these estimators. The new one‐sided tests of serial correlation are obtained in extending Hong's (1996) approach in a framework resistant to outliers. They are based on a weighted sum of robust squared residual autocorrelations and on any robust and n^1/2‐consistent estimators. Our approach generalizes Li's (1988) test statistic, that can be interpreted as a test using the truncated uniform kernel. However, many kernels deliver a higher power. This is confirmed in a simulation study, where we investigate the finite sample properties of the new robust serial correlation tests in comparison to some commonly used robust and non‐robust tests.