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.     

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.     

On the Size Properties of Phillips–Perron Tests

The unit root tests of Phillips and Perron (Testing for a unit root in time series regression. Biometrika 75 (1988), 335–46) are frequently employed in applied econometric research. The chief attraction of these tests is that they are non-parametric, so it is unnecessary to specify a model, even as an approximation to the underlying generating process. The continued popularity of these tests is perhaps surprising since, beginning with Schwert (Tests for unit roots: a Monte Carlo investigation. J. Bus. Econ. Stat. 7 (1989), 147–59), simulation evidence has frequently suggested severe size distortions in specific low persistence generating models. It has been assumed that this size distortion is due to the requirement to estimate short- and long-run variances in practical implementation of the tests, and various alternative estimators have been analysed. However, the analysis of this paper suggests that part of the cause of the size distortion is more fundamental. Even in the 'idealistic case' where actual values of the true variances are used, serious size distortions remain for two simple generating models in sample sizes common in economic applications. We explore in detail the sources of this phenomenon for the ARIMA(0, 1, 1) generating model.     

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.     

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.     

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.     

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.     

ALTERNATIVE ESTIMATORS AND UNIT ROOT TESTS FOR THE AUTOREGRESSIVE PROCESS

Abstract. We compare several estimators for the second-order autoregressive process and compare the associated tests for a unit root. Monte Carlo results are reported for the ordinary least squares estimator, the simple symmetric least squares estimator and the weighted symmetric least squares estimator. The weighted symmetric least squares estimator of the autoregressive parameters generally has smaller mean square error than that of the ordinary least squares estimator, particularly when one root is close to one in absolute value. For the second-order model with known zero intercept, the one-sided ordinary least squares test for a unit root is more powerful than the symmetric tests. For the model with an estimated intercept, the one-sided weighted symmetric least squares test is the most powerful test.     

Using the HEGY Procedure When Not All Roots Are Present

Abstract. Empirical studies have shown little evidence to support the presence of all unit roots present in the Δ₄ filter in quarterly seasonal time series. This paper analyses the performance of the Hylleberg, Engle, Granger and Yoo [Journal of Econometrics (1990) Vol. 44, pp. 215–238] (HEGY) procedure when the roots under the null are not all present. We exploit the vector of quarters representation and cointegration relationship between the quarters when factors (1 − L), (1 + L), (1 + L²), (1 − L²) and (1 + L + L² + L³) are a source of nonstationarity in a process in order to obtain the distribution of tests of the HEGY procedure when the underlying processes have a root at the zero, Nyquist frequency, two complex conjugates of frequency π/2 and two combinations of the previous cases. We show both theoretically and through a Monte Carlo analysis that the t‐ratios t and t and the F‐type tests used in the HEGY procedure have the same distribution as under the null of a seasonal random walk when the root(s) is (are) present, although this is not the case for the t‐ratio tests associated with unit roots at frequency π/2.     

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.     

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.     

Nonparametric Lag Selection for Time Series

A nonparametric version of the Final Prediction Error (FPE) is analysed for lag selection in nonlinear autoregressive time series under very general conditions including heteroskedasticity. We prove consistency and derive probabilities of incorrect selections that have been previously unavailable. Since it is more likely to overfit (have too many lags) than to underfit (miss some lags), a correction factor is proposed to reduce overfitting and hence increase correct fitting. For the FPE calculation, the local linear estimator is introduced in addition to the Nadaraya-Watson estimator in order to cover a very broad class of processes. To achieve faster computation, a plug-in band-width is suggested for the local linear estimator. Our Monte-Carlo study corroborates that the correction factor generally improves the probability of correct lag selection for both linear and nonlinear processes and that the plug-in bandwidth works at least as well as its commonly used competitor. The proposed methods are applied to the Canadian lynx data and daily returns of DM/US-Dollar exchange rates.     

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.     

TESTS FOR SEASONAL COINTEGRATION USING PRINCIPAL COMPONENTS

Abstract. Precise definitions of integrations at frequencies other than zero are given. New cointegration tests, which do not present the problems encountered in existing tests, are developed. They are a generalization of the latent root tests developed by Phillips and Ouliaris.     

Power of a Unit‐Root Test and the Initial Condition

Abstract. It is now well known that how the initial observation is generated can have a significant effect on the power of a unit‐root test. In this article, we show that by taking a simple data‐dependent weighted average of the initial condition‐robust test of Elliott and Müller [Journal of Econometrics (2006), forthcoming] and the standard augmented Dickey–Fuller test, we are able to produce a new unit‐root test that can improve power, both asymptotically and in finite samples, over a wide range of possibilities governing the generation of the initial observation.