Variance ratio tests of the seasonal unit root hypothesis

In this paper we develop tests of the seasonal (quarterly) unit root null hypothesis which reject in favour of stationarity for small values of certain variance ratio statistics, similar to those used by Canova and Hansen (J. Bus. Econom. Statist. 13 (1995) 237) in a different testing context. We demonstrate that our proposed statistics have pivotal limiting distributions under both the null and near seasonally integrated alternatives even when we allow for the possibility of both weak dependence and periodically heteroscedastic behaviour in the driving shocks. This is in contrast to the popular regression-based lag-augmented seasonal unit root tests of Hylleberg et al. (J. Econometrics 44 (1990) 215). A simulation study into the finite sample size and power properties of our proposed tests suggests that they display far superior size properties and, overall, broadly comparable power properties to the corresponding tests of Hylleberg et al. (J. Econometrics 44 (1990) 215), implemented with data-based lag augmentation. The results for the variance ratio tests at the seasonal harmonic frequency are particularly encouraging.     

Seasonal Unit Root Tests with Structural Breaks in Deterministic Seasonality

The seasonal root tests of Hylleberg et al (1990) are extended using the sequential approach of Zivot and Andrews (1992). This paper presents Monte Carlo evidence to support a sequential approach to estimation and critical values are estimated. It is demonstrated that non-stationary data with structurally unstable deterministic seasonality can lead to low power in standard tests for seasonal roots. The sequential tests are applied to US agricultural price data and macroeconomic data and compared with the standard tests. Seasonal roots are rejected in all series.     

Efficient tests for the presence of a pair of complex conjugate unit roots in real time series

We introduce a framework which allows us to draw a clear parallel between the test for the presence of seasonal unit roots and that for unit root at frequency 0 (or π$\pi$). It relies on the properties of the complex conjugate integrated of order one processes which are implicitly at work in the real time series. In the same framework as that of Phillips and Perron (Biometrica 75 (1988) 335), we derive tests for the presence of a pair of conjugate complex unit roots. The asymptotic distribution we obtain are formally close to those derived by these authors but expressed with complex Wiener processes. We then introduce sequences of near-integrated processes which allow us to study the local-to-unity asymptotic of the above test statistics. We state a result on the weak convergence of the partial sum of complex near-random walks which leads to complex Orstein–Uhlenbeck processes. Drawing on Elliott et al. (Econometrica 64 (1996) 813) we then study the design of point-optimal invariant test procedures and compute their envelope employing local-to-unity asymptotic approximations. This leads us to introduce new feasible and near efficient seasonal unit root tests. Their finite sample properties are investigated and compared with the different test procedures already available (J. Econometrics 44 (1991) 215; 62 (1994) 415; 85 (1998) 269) and those introduced in the first part of the paper.     

New unit root asymptotics in the presence of deterministic trends

Recent work by Phillips (Econometrica 66 (1998) 1299) has shown that stochastic trends can be validly represented in empirical regressions in terms of deterministic functions of time. These representations offer an alternative mechanism for modelling stochastic trends. It is shown here that the alternate representations affect the asymptotics of all commonly used unit root tests in the presence of trends. In particular, the critical values of unit root tests diverge when the number of deterministic regressors K→∞ as the sample size n→∞. When they are appropriately recentered and standardized, unit root limit distributions are shown to be normal as K→∞.     

Fluctuation Tests for a Change in Persistence*

In this paper, we develop a set of new persistence change tests which are similar in spirit to those of Kim [Journal of Econometrics (2000) Vol. 95, pp. 97–116], Kim et al. [Journal of Econometrics (2002) Vol. 109, pp. 389–392] and Busetti and Taylor [Journal of Econometrics (2004) Vol. 123, pp. 33–66]. While the exisiting tests are based on ratios of sub‐sample Kwiatkowski et al. [Journal of Econometrics (1992) Vol. 54, pp. 158–179]‐type statistics, our proposed tests are based on the corresponding functions of sub‐sample implementations of the well‐known maximal recursive‐estimates and re‐scaled range fluctuation statistics. Our statistics are used to test the null hypothesis that a time series displays constant trend stationarity [I(0)] behaviour against the alternative of a change in persistence either from trend stationarity to difference stationarity [I(1)], or vice versa. Representations for the limiting null distributions of the new statistics are derived and both finite‐sample and asymptotic critical values are provided. The consistency of the tests against persistence change processes is also demonstrated. Numerical evidence suggests that our proposed tests provide a useful complement to the extant persistence change tests. An application of the tests to US inflation rate data is provided.     

Cross-sectional dependence robust block bootstrap panel unit root tests

In this paper we consider the issue of unit root testing in cross-sectionally dependent panels. We consider panels that may be characterized by various forms of cross-sectional dependence including (but not exclusive to) the popular common factor framework. We consider block bootstrap versions of the group-mean (Im et al., 2003) and the pooled (Levin et al., 2002) unit root coefficient DF tests for panel data, originally proposed for a setting of no cross-sectional dependence beyond a common time effect. The tests, suited for testing for unit roots in the observed data, can be easily implemented as no specification or estimation of the dependence structure is required. Asymptotic properties of the tests are derived for T going to infinity and N finite. Asymptotic validity of the bootstrap tests is established in very general settings, including the presence of common factors and cointegration across units. Properties under the alternative hypothesis are also considered. In a Monte Carlo simulation, the bootstrap tests are found to have rejection frequencies that are much closer to nominal size than the rejection frequencies for the corresponding asymptotic tests. The power properties of the bootstrap tests appear to be similar to those of the asymptotic tests.     

Trend‐stationary GNP: evidence from a new exact pointwise most powerful invariant unit root test

There has been a substantial debate whether GNP has a unit root. However, statistical tests have had little success in distinguishing between unit‐root and trend‐reverting specifications because of poor statistical properties. This paper develops a new exact small‐sample, pointwise most powerful unit root test that is invariant to the unknown mean and scale of the time series tested, that generates exact small‐sample critical values, powers and p‐values, that has power which approximates the maximum possible power, and that is highly robust to conditional heteroscedasticity. This test decisively rejects the unit root null hypothesis when applied to annual US real GNP and US real per capita GNP series. This paper also develops a modified version of the test to address whether a time series contains a permanent, unit root process in addition to a temporary, stationary process. It shows that if these GNP series contain a unit root process in addition to the stationary process, then it is most likely very small. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical distribution functions for unit root and cointegration tests

This paper employs response surface regressions based on simulation experiments to calculate distribution functions for some well-known unit root and cointegration test statistics. The principal contributions of the paper are a set of data files that contain estimated response surface coefficients and a computer program for utilizing them. This program, which is freely available via the Internet, can easily be used to calculate both asymptotic and finite-sample critical values and P-values for any of the tests. Graphs of some of the tabulated distribution functions are provided. An empirical example deals with interest rates and inflation rates in Canada.     

Bootstrap unit root tests in panels with cross-sectional dependency

We apply bootstrap methodology to unit root tests for dependent panels with N cross-sectional units and T time series observations. More specifically, we let each panel be driven by a general linear process which may be different across cross-sectional units, and approximate it by a finite order autoregressive integrated process of order increasing with T. As we allow the dependency among the innovations generating the individual series, we construct our unit root tests from the estimation of the system of the entire N cross-sectional units. The limit distributions of the tests are derived by passing T to infinity, with N fixed. We then apply bootstrap method to the approximated autoregressions to obtain critical values for the panel unit root tests, and establish the asymptotic validity of such bootstrap panel unit root tests under general conditions. The proposed bootstrap tests are indeed quite general covering a wide class of panel models. They in particular allow for very general dynamic structures which may vary across individual units, and more importantly for the presence of arbitrary cross-sectional dependency. The finite sample performance of the bootstrap tests is examined via simulations, and compared to that of commonly used panel unit root tests. We find that our bootstrap tests perform relatively well, especially when N is small.     

Modified CADF and CIPS Panel Unit Root Statistics with Standard Chi‐squared and Normal Limiting Distributions

In an influential paper Pesaran (‘A simple panel unit root test in presence of cross‐section dependence’, Journal of Applied Econometrics, Vol. 22, pp. 265–312, 2007) proposes two unit root tests for panels with a common factor structure. These are the CADF and CIPS test statistics, which are amongst the most popular test statistics in the literature. One feature of these statistics is that their limiting distributions are highly non‐standard, making for relatively complicated implementation. In this paper, we take this feature as our starting point to develop modified CADF and CIPS test statistics that support standard chi‐squared and normal inference.     

Truncated Product Methods for Panel Unit Root Tests*

This paper proposes two new panel unit root tests based on Zaykin et al. (2002) ’s truncated product method. The first one assumes constant correlation between P‐values and the second one uses sieve bootstrap to allow for general forms of cross‐section dependence in the panel units. Monte Carlo simulation shows that both tests have reasonably good size and are powerful in cases of some very large P‐values. The proposed tests are applied to a panel of real GDP and inflation density forecasts, resulting in evidence that professional forecasters may not update their forecast precision in an optimal Bayesian way.     

Spurious Rejections by Perron Tests in the Presence of a Break

In this paper, we concentrate on the case of an exogeneously chosen break date, but entertain the possibility that an incorrect choice is made. In fact, the Perron test statistics considered are invariant to any break in the generating process at the assumed break date. Our results therefore apply equally to the case of a generating process with two breaks, only one of which is specifically accounted for in the analysis. As in Leybourne et al . (1998), we find that a neglected relatively early break can lead to spurious rejections of the unit root null hypothesis. Moreover, for all but one of the tests analyzed, spurious rejections now also arise if a true break occurs relatively soon after the assumed break date.     

A class of simple distribution-free rank-based unit root tests

We propose a class of distribution-free rank-based tests for the null hypothesis of a unit root. This class is indexed by the choice of a reference densityg, which need not coincide with the unknown actual innovation density f. The validity of these tests, in terms of exact finite-sample size, is guaranteed, irrespective of the actual underlying density, by distribution-freeness. Those tests are locally and asymptotically optimal under a particular asymptotic scheme, for which we provide a complete analysis of asymptotic relative efficiencies. Rather than stressing asymptotic optimality, however, we emphasize finite-sample performances, which also depend, quite heavily, on initial values. It appears that our rank-based tests significantly outperform the traditional Dickey-Fuller tests, as well as the more recent procedures proposed by Elliott et al. (1996), Ng and Perron (2001), and Elliott and Müller (2006), for a broad range of initial values and for heavy-tailed innovation densities. Thus, they provide a useful complement to existing techniques.     

HEGY Tests in the Presence of Moving Averages*

We analyze the asymptotic distributions associated with the seasonal unit root tests of Hylleberg et al. (1990) for quarterly data when the innovations follow a moving average process. Although both the t‐ and F‐type tests suffer from scale and shift effects compared with the presumed null distributions when a fixed order of autoregressive augmentation is applied, these effects disappear when the order of augmentation is sufficiently large. However, as found by Burridge and Taylor (2001) for the autoregressive case, individual t‐ratio tests at the semi‐annual frequency are not pivotal even with high orders of augmentation, although the corresponding joint F‐type statistic is pivotal. Monte Carlo simulations verify the importance of the order of augmentation for finite samples generated by seasonally integrated moving average processes.     

基于Sieve Bootstrap方法的季节时间序列模型检验问题研究

研究目标：完善季节时间序列模型建模理论,解决建模过程烦琐、各类检验方法的结论差异大以及模型误设定问题。研究方法：基于对各季节时间序列模型的数理分析及比较,提出合理的模型检验程序;再运用Sieve Bootstrap方法,给出季节性单位根检验及确定性季节过程检验的统计量的临界值,并比较基于Sieve Bootstrap的检验方法与HEGY检验、BT检验的异同。研究发现：本文提出的检验程序能有效识别模型,检验统计量有限样本性质优良;实证分析表明,本文提出的检验程序及方法能更有效地识别中国宏观经济数据中的季节性。研究创新：将Sieve Bootstrap方法应用于季节时间序列的平稳性检验及趋势性检验中。研究价值：提出季节时间序列模型检验程序及检验方法,促进其在季节性经济数据中的应用。     

Panicca: Panic on Cross‐Section Averages

The cross‐section average (CA) augmentation approach of Pesaran (A simple panel unit root test in presence of cross‐section dependence. Journal of Applied Econometrics 2007; 22 : 265–312) and Pesaran et al. (Panel unit root test in the presence of a multifactor error structure. Journal of Econometrics 2013; 175 : 94–115), and the principal components‐based panel analysis of non‐stationarity in idiosyncratic and common components (PANIC) of Bai and Ng (A PANIC attack on unit roots and cointegration. Econometrica 2004; 72 : 1127–1177; Panel unit root tests with cross‐section dependence: a further investigation. Econometric Theory 2010; 26 : 1088–1114) are among the most popular ‘second‐generation’ approaches for cross‐section correlated panels. One feature of these approaches is that they have different strengths and weaknesses. The purpose of the current paper is to develop PANICCA, a combined approach that exploits the strengths of both CA and PANIC. Copyright © 2015 John Wiley & Sons, Ltd.     

Multivariate saddlepoint tests on the mean direction of the von Mises–Fisher distribution

This article provides P values for two new tests on the mean direction of the von Mises–Fisher distribution. The test statistics are obtained from the exponent of the saddlepoint approximation to the density of M-estimators, as suggested by Robinson et al. (Ann Stat 31:1154–1169, 2003). These test statistics are chi-square distributed with asymptotically small relative errors. Despite the high dimensionality of the problem, the proposed P values are accurate and simple to compute. The numerical precision of the P values of the new tests is illustrated by some simulation studies.     

Unit root testing under a local break in trend

Recent approaches to testing for a unit root when uncertainty exists over the presence and timing of a trend break employ break detection methods, so that a with-break unit root test is used only if a break is detected by some auxiliary statistic. While these methods achieve near asymptotic efficiency in both fixed trend break and no trend break environments, in finite samples pronounced “valleys” in the power functions of the tests (when mapped as functions of the break magnitude) are observed, with power initially high for very small breaks, then decreasing as the break magnitude increases, before increasing again. In response to this problem, we propose two practical solutions, based either on the use of a with-break unit root test but with adaptive critical values, or on a union of rejections principle taken across with-break and without-break unit root tests. These new procedures are shown to offer improved reliability in terms of finite sample power. We also develop local limiting distribution theory for both the extant and the newly proposed unit root statistics, treating the trend break magnitude as local-to-zero. We show that this framework allows the asymptotic analysis to closely approximate the finite sample power valley phenomenon, thereby providing useful analytical insights.     

Asymmetry and nonstationarity for a seasonal time series model

Tests for symmetry and seasonal unit roots are developed for an extended model of Hylleberg et al. (1990. Seasonal integration and cointegration. Journal Econometrics 44, 215–238.) which can represent both partial seasonal unit roots and threshold effects. Methods based on ordinary least squares (OLS) estimation and instrumental variable (IV) estimation are proposed and compared. For adjusting mean functions, ordinary mean adjustment and recursive mean adjustment are both considered. Several tests are constructed from various combination of estimation schemes and mean adjustment schemes. Among the tests, the tests based on IV-estimation are recommended because they have very simple limiting null distributions and have finite sample power properties comparable to those based on the OLSE. The recommended tests are applied to a US unemployment rate data set and find evidences for both nonstationarities associated with zero frequency and threshold effects.     

Testing for unit roots in the possible presence of multiple trend breaks using minimum Dickey–Fuller statistics

Trend breaks appear to be prevalent in macroeconomic time series, and unit root tests therefore need to make allowance for these if they are to avoid the serious effects that unmodelled trend breaks have on power. Carrion-i-Silvestre et al. (2009) propose a pre-test-based approach which delivers near asymptotically efficient unit root inference both when breaks do not occur and where multiple breaks occur, provided the break magnitudes are fixed. Unfortunately, however, the fixed magnitude trend break asymptotic theory does not predict well the finite sample power functions of these tests, and power can be very low for the magnitudes of trend breaks typically observed in practice. In response to this problem we propose a unit root test that allows for multiple breaks in trend, obtained by taking the infimum of the sequence (across all candidate break points in a trimmed range) of local GLS detrended augmented Dickey–Fuller-type statistics. We show that this procedure has power that is robust to the magnitude of any trend breaks, thereby retaining good finite sample power in the presence of plausibly-sized breaks. We also demonstrate that, unlike the OLS detrended infimum tests of Zivot and Andrews (1992), these tests display no tendency to spuriously reject in the limit when fixed magnitude trend breaks occur under the unit root null.