A computationally convenient unit root test with covariates,conditional heteroskedasticity and efficient detrending

When testing for a unit root in a time series, in spite of the well‐known power problem of univariate tests, it is quite common to use only the information regarding the autoregressive behaviour contained in that series. In a series of influential papers, Elliott et al. (Efficient tests for an autoregressive unit root, Econometrica 64, 813–836, 1996), Hansen (Rethinking the univariate approach to unit root testing: using covariates to increase power, Econometric Theory 11, 1148–1171, 1995a) and Seo (Distribution theory for unit root tests with conditional heteroskedasticity, Journal of Econometrics 91, 113–144, 1999) showed that this practice can be rather costly and that the inclusion of the extraneous information contained in the near‐integratedness of many economic variables, their heteroskedasticity and their correlation with other covariates can lead to substantial power gains. In this article, we show how these information sets can be combined into a single unit root test.     

Comparison of unit root tests for time series with level shifts

Unit root tests are considered for time series which have a level shift at a known point in time. The shift can have a very general nonlinear form, and additional deterministic mean and trend terms are allowed for. Prior to the tests, the deterministic parts and other nuisance parameters of the data generation process are estimated in a first step. Then, the series are adjusted for these terms and unit root tests of the Dickey–Fuller type are applied to the adjusted series. The properties of previously suggested tests of this sort are analysed and modifications are proposed which take into account estimation errors in the nuisance parameters. An important result is that estimation under the null hypothesis is preferable to estimation under local alternatives. This contrasts with results obtained by other authors for time series without level shifts.     

Unit root bootstrap tests under infinite variance

This article presents a family of new tests for unit roots based on M‐estimators. Their robustness makes them very appealing when working with distributions that have infinite variance or heavy tails. These tests are completely automatic regardless of the complex distributions of this kind of estimators because the critical values are approximated using bootstrap, no additional parameter has to be estimated and the results obtained are very good in small samples. An exhaustive Monte Carlo study shows the high performance of these tests compared with others proposed in the literature when the variance is infinite.     

Time‐Transformed Unit Root Tests for Models with Non‐Stationary Volatility

Abstract. Conventional unit root tests are known to be unreliable in the presence of permanent volatility shifts. In this paper, we propose a new approach to unit root testing which is valid in the presence of a quite general class of permanent variance changes which includes single and multiple (abrupt and smooth transition) volatility change processes as special cases. The new tests are based on a time transformation of the series of interest which automatically corrects their form for the presence of non‐stationary volatility without the need to specify any parametric model for the volatility process. Despite their generality, the new tests perform well even in small samples. We also propose a class of tests for the null hypothesis of stationary volatility in (near‐) integrated time‐series processes.     

Unit root testing with stationary covariates and a structural break in the trend function

The issue of testing for a unit root allowing for a structural break in the trend function is considered. The focus is on the construction of more powerful tests using the information in relevant multi‐variate data sets. The proposed test adopts the generalized least squares detrending approach and uses correlated stationary covariates to improve power. As it is standard in the literature, the break date is treated as unknown. Asymptotic distributions are derived, and a set of asymptotic and finite sample critical values are tabulated. Asymptotic local power functions show that power gains can be large. Finite sample results show that the test exhibits small‐size distortions and power that can be far beyond what is achievable by univariate tests.     

Unit‐root testing against the alternative hypothesis of up to m structural breaks

Abstract. In this paper we provide tests for the unit‐root hypothesis against the occurrence of an unspecified number of breaks which may be larger than 2 but smaller that the maximum number of breaks allowed, m, in univariate time‐series models. The advocated procedure is considerably less computationally intensive than those widely used in the literature. We provide critical values for the test and examine its small sample properties through Monte Carlo experiments.     

On the Robustness of Unit Root Tests in the Presence of Double Unit Roots

We examine some of the consequences on commonly used unit root tests when the underlying series is integrated of order two rather than of order one. It turns out that standard augmented Dickey–Fuller type of tests for a single unit root have excessive density in the explosive region of the distribution. The lower (stationary) tail, however, will be virtually unaffected in the presence of double unit roots. On the other hand, the Phillips–Perron class of semi-parametric tests is shown to diverge to plus infinity asymptotically and thus favouring the explosive alternative. Numerical simulations are used to demonstrate the analytical results and some of the implications in finite samples.     

Asymmetric adjustment and smooth transitions: a combination of some unit root tests

Abstract.  Conventional Dickey–Fuller unit root tests have been generalized to allow for nonlinearity under the alternative hypothesis by Enders and Granger [ Journal of Business Economics and Statistics , 16 (1998) 304] (EG) and Leybourne, Newbold and Vougas [ Journal of Time Series Analysis , 19 (1998) 83] (LNV). EG focus on the case of asymmetric adjustment modelled as threshold autoregression, while LNV extend the concept of trend stationarity to that of stationarity around a smooth transition between deterministic linear trends. In this study, the EG and LNV methodologies are combined to develop tests of the null hypothesis of a unit root, that under the alternative hypothesis allow for stationary asymmetric adjustment around a smooth transition between deterministic linear trends. The empirical power of the combined tests is briefly investigated and an empirical application to time series on aggregate industrial production in the UK and the US is considered.     

Testing Monthly Seasonal Unit Roots With Monthly and Quarterly Information

Abstract. This paper proposes a method for testing seasonal unit roots that combines monthly and quarterly Hylleberg, Engle, Granger and Yoo (HEGY) tests. The new approach is more powerful than the method that does not use quarterly information, i.e. the monthly HEGY test. An empirical illustration of the proposed approach is given for monthly US Industrial Production.     

Long-Memory Errors in Time Series Regressions with a Unit Root

This paper is concerned with estimation and inference in univariate time series regression with a unit root when the error sequence exhibits long-range temporal dependence. We consider generating mechanisms for the unit root process which include models with or without a drift term and we study the limit behavior of least squares statistics in regression models without drift and trend, with drift but no time trend, and with drift and time trend. We derive the limit distribution and rate of convergence of the ordinary least squares (OLS) estimator of the unit root, the intercept and the time trend in the three regression models and for the two different data-generating processes. The limiting distributions for the OLS estimator differ from those obtained under the hypothesis of weakly dependent errors not only in terms of the limiting process involved but also in terms of functional form. Further, we characterize the asymptotic behavior of both the t statistics for testing the unit root hypothesis and the t statistic for the intercept and time trend coefficients. We find that t ratios either diverge to infinity or collapse to zero. The limiting behavior of Phillips's Z _α and Z _t semiparametric corrections is also analyzed and found to be similar to that of standard Dickey– Fuller tests. Our results indicate that misspecification of the temporal dependence features of the error sequence produces major effects on the asymptotic distribution of estimators and t ratios and suggest that alternative approaches might be more suited to testing for a unit root in time series regression.     

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.     

Testing for a unit root under errors with just barely infinite variance

Abstract. This article investigates the problem of testing for a unit root in the case that the error, {u_t}, of the model is a strictly stationary, mixing process with just barely infinite variance. Such errors have the property that for every δ such that 0 ≤ δ < 2, the moments E|u_t|^δ are finite. Under some additional restrictions on the rate of decay of the mixing rates, these errors belong to the domain of the non‐normal attraction of the normal law and obey the invariance principle. This in turn implies that there might be conditions under which the usual Phillips‐type test statistics for unit roots may still converge to the corresponding Dickey–Fuller distributions. In such a case, the unit‐root hypothesis can be tested within an infinite‐variance framework without any modifications to either the tests themselves or the critical values employed. This article derives a necessary and sufficient condition for convergence of the standard test statistics to the Dickey–Fuller distributions. By means of Monte Carlo simulations, the article also shows that this condition is likely to hold in the case that {u_t} is a serially correlated, integrated generalized autoregressive conditionally heteroskedastic (IGARCH) process and the standard unit‐root tests work well.     

Seasonal Unit Root Tests Based on Forward and Reverse Estimation

In this paper, we suggest a new set of regression-based statistics for testing the seasonal unit root null hypothesis. These tests are based on combining conventional Hylleberg et al . (1990 ) -type seasonal unit root test statistics calculated from both forward and reverse estimation of the auxiliary regression equation. We derive the asymptotic distributions of the new test statistics under the seasonal unit root null hypothesis. We provide finite sample critical values appropriate for the case of quarterly data together with asymptotic critical values, the latter appropriate for any seasonal aspect. Monte Carlo simulation of the finite-sample size and power properties of the new tests reveals that, overall, they perform rather better than extant tests of the seasonal unit root hypothesis.     

On the use of Sub‐sample Unit Root Tests to Detect Changes in Persistence

Abstract. We investigate the behaviour of rolling and recursive augmented Dickey–Fuller (ADF) tests against processes which display changes in persistence. We show that the power of the tests depend crucially on the window width and warm up parameter for the rolling and recursive procedures respectively, on whether forward or reverse recursive sequences of tests are computed, and on the persistence change process generating the data. To ameliorate these dependencies we extend the available critical values for these tests, and propose a number of new sub‐sample unit root tests for which finite sample and asymptotic critical values are also provided. An empirical illustration on OECD real output data is also provided.     

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.     

Unit‐root testing: on the asymptotic equivalence of Dickey–Fuller with the log–log slope of a fitted autoregressive spectrum

In this article we consider the problem of testing for the presence of a unit root against autoregressive alternatives. In this context we prove the asymptotic equivalence of the well‐known (augmented) Dickey–Fuller test with a test based on an appropriate parametric modification of the technique of log‐periodogram regression. This modification consists of considering, close to the origin, the slope (in log–log coordinates) of an autoregressively fitted spectral density. This provides a new interpretation of the Dickey–Fuller test and closes the gap between it and log‐periodogram regression. This equivalence is based on monotonicity arguments and holds on the null as well as on the alternative. Finally, a simulation study provides indications of the finite‐sample behaviour of this asymptotic equivalence.     

Joint Hypothesis Tests for a Unit Root When There is a Break in the Innovation Variance

Abstract. We develop extensions of the Dickey–Fuller F‐statistics for the joint null hypothesis of a unit root that allows for a break in the innovation variance. Our statistics are based on the modified generalized least squares (GLS) strategy outlined in Kim, Leybourne and Newbold [Journal of Econometrics (2002) Vol. 109, pp. 365–387] that requires estimation of the break‐date and corresponding pre‐break and post‐break variances. We derive the asymptotic distribution of the new F‐statistics, tabulate their finite sample and asymptotic critical values, and present finite sample simulation evidence regarding their size and power.     

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) .     

Median‐unbiased Estimation and Exact Inference Methods for First‐order Autoregressive Models with Conditional Heteroscedasticity of Unknown Form

Abstract. Consider the first‐order autoregressive model y_t = φy_t?1 + ?_t, t = 1,…, T, with arbitrary initial non‐zero value y₀. Assuming that the error terms ?_t are independently distributed according to median‐zero distributions [ Zieliński (1999) Journal of Time Series Analysis, Vol. 20, p. 477] shows that the estimator conjectured by Hurwicz (1950) Statistical Inference in Dynamic Economic Models. New York, NY: Wiley – the median of the consecutive ratios y_t/y_t?1– is an exactly median‐unbiased estimator of the autoregressive parameter φ. This paper shows that the Hurwicz estimator remains median‐unbiased under more general distributional assumptions, without assuming statistical independence. In particular, no restrictions are placed on the degree of heterogeneity and dependence of the conditional variance process. A computationally efficient method is also proposed to build exact confidence intervals for the autoregressive parameter which are valid in finite samples for any value of φ on the real line.     

含介晶基元的聚氨酯改性环氧树脂复合材料的研究

以联苯二酚与2-氯乙醇合成的介晶二醇为原料，与环氧树脂熔融共混后原位聚合，生成含刚性聚氨酯的环氧树脂复合材料。用红外光谱表征了聚氨酯的原位聚合反应，探讨了介晶二醇用量、反应时间、固化剂用量等对复合材料力学性能的影响。用扫描电镜观察了复合材料的冲击断面。结果表明：含介晶基元的聚氨酯改性环氧树脂，既能提高环氧树脂的韧性，也能提高环氧树脂的力学模量。