Testing for Unit Roots in Monthly Time Series

This paper is concerned with tests for seasonal roots in monthly univariate time series processes. The paper extends the procedures and tables of critical values due to Beaulieu and Miron (Seasonal unit roots in aggregate U.S. data. J. Economet . 55 (1993), 305–28) to obtain tests which are similar ( exactly and a symptotically ) with respect to both the initial values of the process and the possibility of seasonal drifts under the seasonal unit root null hypothesis. We also develop test statistics which test simultaneously for a unit root at each frequency and for a unit root at each of the seasonal frequencies. Representations are derived for the limiting distributions of each of the test statistics proposed in this paper. We illustrate the practical usefulness of the proposed test statistics by a series of empirical applications     

Testing for Cointegration with Temporally Aggregated and Mixed‐Frequency Time Series

We examine the effects of mixed sampling frequencies and temporal aggregation on the size of commonly used tests for cointegration, and we find that these effects may be severe. Matching sampling schemes of all series generally reduces size distortion, and the nominal size is obtained asymptotically only when all series are skip sampled in the same way – for example, end‐of‐period sampling. We propose and analyse mixed‐frequency versions of the cointegration tests in order to control the size when some high‐frequency data are available. Otherwise, when no high‐frequency data are available, we discuss controlling size using bootstrapped critical values. We test stock prices and dividends for cointegration as an empirical demonstration.     

Effect of temporal aggregation on multiple time series in the frequency domain

The effect of temporal aggregation on bivariate spectral measures is investigated. First, the low‐frequency regression coefficient turns out to be invariant under aggregation irrespective of differencing, with the exception of when the aggregation of flow and stock variables is combined. Second, the long‐run squared coherency is invariant with respect to aggregation irrespective of differencing. Third, for frequencies different from zero, limiting results for a growing aggregation level m are obtained equal to those at frequency 0 of the underlying basic series. Hence, all frequency domain information is distorted by aggregation apart from the long‐run one. This also holds true for the phase angle that always approaches zero with growing aggregation level m. The sole exception to these findings is the case of the skip sampling stationary series. Moreover, for finite aggregation level, one may exactly quantify the aggregational effect on each cycle of interest. Numerical examples illustrate our results.     

Implementing Residual‐Based KPSS Tests for Cointegration with Data Subject to Temporal Aggregation and Mixed Sampling Frequencies

We show how different data types (stocks and flows) and temporal aggregation affect the size and power of the dynamic ordinary least squares residual‐based Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test of the null of cointegration. Size may be more effectively controlled by setting the minimum number of leads equal to one – as opposed to zero – when selecting the lag/lead order of the dynamic ordinary least squares regression using aggregated data, but at a cost to power. If high‐frequency data for one or more series are available – that is, the model has mixed sampling frequencies – we show how to effectively utilize the high‐frequency data to increase power while controlling size.     

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.     

Limiting distributions of unconditional maximum likelihood unit root test statistics in seasonal time–series models

Abstract.  The likelihood function of a seasonal model, Y _t  =  ρ Y _{t − d}  +  e _t as implemented in computer algorithms under the assumption of stationary initial conditions is a function of ρ which is zero at the point ρ  = 1. It is a smooth function for ρ in the above seasonal model with a well-defined maximum regardless of the data-generating mechanism. Gonzalez-Farias (PhD Thesis, North Carolina State University, 1992) proposed tests for unit roots based on maximizing the stationary likelihood function in nonseasonal time series. We extend it to seasonal time series. The limiting distribution of seasonal unit root test statistics based on the unconditional maximum likelihood estimators are shown. Models having a single mean, seasonal means, and a single-trend variable across the seasons are considered.     

BISPECTRAL ANALYSIS OF RANDOMLY SAMPLED DATA

Abstract. A continuous time series is often observed or sampled at discrete intervals. Most literature has dealt with the case when the sampling intervals are equally spaced. For irregularly sampled data, most existing literature is concerned with second-order moments or anti-aliasing spectral estimations. We study the estimation of higher-order spectral density functions with the emphasis on the bispectral estimate when the continuous time series is sampled by a random point process. Estimates under the Poisson sampling scheme are studied in detail. Asymptotic bias and covariances are obtained. In particular, it is shown explicitly how the information of the sampling process comes into play in obtaining a consistent estimate of the bispectral density function of a continuous time series. In contrast to the second-order spectral density function estimation where the Poisson sampling scheme results in a constant correction term, a consistent bispectral density function estimate results in a nonlinear correction term even in the Poisson sampling scheme. A simple simulation example is presented for illustration.     

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.     

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.     

Locally Optimal Tests Against Unit Roots in Seasonal Time Series Processes

This paper builds on the existing literature on tests of the null hypothesis of deterministic seasonality in a univariate time-series process. Under the assumption of independent Gaussian errors, we derive the class of locally weighted mean most powerful invariant tests against unit roots at the zero and/or seasonal frequencies in a seasonally observed process. Representations for the limiting distributions of the proposed test statistics under sequences of local alternatives are derived, and the relationship with tests for corresponding moving average unit roots is explored. We also propose nonparametric modifications of these test statistics designed to have limit distributions which are free of nuisance parameters under weaker conditions on the errors. Our tests are shown to contain existing stationarity tests as special cases and to extend these tests in a number of useful directions.     

Fixed‐b asymptotic approximation of the sampling behaviour of nonparametric spectral density estimators

Abstract. We propose a new asymptotic approximation for the sampling behaviour of nonparametric estimators of the spectral density of a covariance stationary time series. According to the standard approach, the truncation lag grows more slowly than the sample size. We derive first‐order limiting distributions under the alternative assumption that the truncation lag is a fixed proportion of the sample size. Our results extend the approach of Neave (1970) , who derived a formula for the asymptotic variance of spectral density estimators under the same truncation lag assumption. We show that the limiting distribution of zero‐frequency spectral density estimators depends on how the mean is estimated and removed. The implications of our zero‐frequency results are consistent with exact results for bias and variance computed by Ng and Perron (1996) . Finite sample simulations indicate that the new asymptotics provides a better approximation than the standard one.     

SOME SIMPLE TESTS OF THE MOVING-AVERAGE UNIT ROOT HYPOTHESIS

Abstract. This paper deals with three test statistics for a moving-average (MA) unit root. The spectral test is based on the estimate of the spectral density at frequency zero. The variance difference statistic compares the sample variance of the integrated series with the estimated variance imposing the MA unit root constraint. Furthermore, Tanaka's score type test statistic is modified to improve the power in higher order models. The asymptotic power of the tests is considered and Monte Carlo experiments are performed to investigate the small sample properties of the tests. Finally, the tests are applied to a number of economic time series to determine the degree of integration.     

Continuous manufacturing: Is the process mean stationary?

The statistical framework to systematically detect mean stationarity in the context of continuous manufacturing is described in this article. The methods presented in this article use econometric and financial time‐series analysis concepts in the form of unit‐root and stationarity hypothesis tests. The tests under discussion are the augmented Dickey‐Fuller, Philips‐Perron, Leybourne‐McCabe, and Kwiatkowski‐Phillips‐Schmidt‐Shin. These hypothesis tests are evaluated on data generated by a focused‐beam reflectance measurement sensor implemented on‐line in a continuous plug‐flow crystallizer. This contribution has shown that the hypothesis tests can be used to detect steady‐state conditions on‐line in a plug‐flow crystallizer. Furthermore, this econometric framework can be used as a mean stationarity “certificate” of collected samples to document that the process was mean stationary during the sampling. The statistical framework described in this article can be applied to any continuously operated unit operation or sensor measurement. © 2018 American Institute of Chemical Engineers AIChE J, 64: 2426–2437, 2018     

The Calculation of Some Limiting Distributions Arising in Near‐Integrated Models with GLS Detrending

Many unit root test statistics are based on detrended data, with the method of generalized least squares (GLS) detrending being popular in the setting of a near‐integrated model. This article determines the properties of some associated limiting distributions when the GLS detrending is based on a linear time trend. A fundamental result for the moment generating function of two key functionals of the relevant stochastic process is provided and used to compute probability density functions and cumulative distribution functions, as well as means and variances, of the limiting distributions of some statistics of interest. The exact moments and percentiles of some of these distributions are compared with those obtained by simulations, and it is found that, even with a large number of replications and a large sample size, the errors resulting from the simulation methods are not negligible. Some further applications, including a comparison of limiting power functions of different unit root test statistics and the consideration of a more complicated statistic, are also provided.     

A test for second‐order stationarity of a time series based on the discrete Fourier transform

We consider a zero mean discrete time series, and define its discrete Fourier transform (DFT) at the canonical frequencies. It can be shown that the DFT is asymptotically uncorrelated at the canonical frequencies if and only if the time series is second‐order stationary. Exploiting this important property, we construct a Portmanteau type test statistic for testing stationarity of the time series. It is shown that under the null of stationarity, the test statistic has approximately a chi‐square distribution. To examine the power of the test statistic, the asymptotic distribution under the locally stationary alternative is established. It is shown to be a generalized non‐central chi‐square, where the non‐centrality parameter measures the deviation from stationarity. The test is illustrated with simulations, where is it shown to have good power.     

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.     

The restricted likelihood ratio test for autoregressive processes

The restricted likelihood is known to produce estimates with significantly less bias in AR(p) models with intercept and/or trend. In AR(1) models, the corresponding restricted likelihood ratio test (RLRT), unlike the t‐statistic or the usual LRT, has been recently shown to be well approximated by the chi‐square distribution even close to the unit root, thus yielding confidence intervals with good coverage properties. In this article, we extend this result to AR(p) processes of arbitrary order p by obtaining the expansion of the RLRT distribution around that of the limiting chi‐squared and showing that the error is bounded even as the unit root is approached. Next, we investigate the correspondence between the AR coefficients and the partial autocorrelations, which is well known in the stationary region, and extend to the more general situation of potentially multiple unit roots. In the case of one positive unit root, which is of most practical interest, the resulting parameter space is shown to be the bounded p‐dimensional hypercube (?1, 1] × (?1, 1)^p?1. This simple parameter space, in addition with a stable algorithm that we provide for computing the restricted likelihood, allows its easy computation and optimization as well as construction of confidence intervals for the sum of the AR coefficients. In simulations, the RLRT intervals are shown to have not only near exact coverage in keeping with our theoretical results, but also shorter lengths and significantly higher power against stationary alternatives than other competing interval procedures. An application to the well‐known Nelson–Plosser data yields RLRT based intervals that can be markedly different from those in the literature.     

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.     

Time Scale Estimation by Tracking Parameter Variation

A quasi-periodic time series is sampled at a varying but unknown rate. An autoregressive moving-average model is fitted to the resulting discrete series and the time variation of its parameters is estimated. The functional dependence of the parameters on the sampling rate is then used to estimate this rate and to reconstruct the true time scale.     

Bootstrapping unit root tests for integrated processes

Abstract. In this paper, we consider two bootstrap algorithms for testing unit roots under the condition that the observed process is unit root integrated. The first method consists of generating the resampled data after fitting an autoregressive model to the first differences of the observations. The second method consists of applying the stationary bootstrap to the first differences. Both procedures are shown to give methods that approach the correct asymptotic distribution under the null hypothesis of a unit root. We also present a Monte-Carlo study comparing the two methods for some ARIMA models.