COMMON FEATURES IN TIME SERIES WITH BOTH DETERMINISTIC AND STOCHASTIC SEASONALITY

This paper extends the notions of common cycles and common seasonal features to time series having deterministic and stochastic seasonality at different frequencies. The conditions under which quarterly time series with these characteristics have common features are investigated, various representations are presented and statistical inference is discussed. Finally, the analysis is applied to study comovements between different components of consumption and income using UK data.     

Testing for cointegration at any frequency using spectral methods

Summary In this paper a new simple test for cointegration at any frequency is presented. This method can thus be applied to test for cointegration both at the zero and at the seasonal frequencies. It requires the estimation of the coherency spectrum of weakly stationary processes, therefore only standard spectral theory is involved. The testing procedure is similar to the one suggested by Phillips and Ouliaris (1988) and recently generalized by Joyeux (1992) to frequencies different from zero, but it does not suffer of some problems connected with the use of principal components methods in the frequency domain. Invited paper at the Conference held in Bologna, Italy, 27–28 May 1993, on ?Statistical Tests: Methodology and Econometric Applications?.     

A vector of quarters representation for bivariate time series

In this paper it is shown that several models for a bivariate nonstationary quarterly time series are nested in a vector autoregression with cointegration restrictions for the eight annual series of quarterly observations. Or, the Granger Representation Theorem is extended to incorporate, e.g., seasonal and periodic cointegration.     

A vector of quarters representation for bivariate time series

In this paper it is shown that several models for a bivariate nonstationary quarterly time series are nested in a vector autoregression with cointegration restrictions for the eight annual series of quarterly observations. Or, the Granger Representation Theorem is extended to incorporate, e.g., seasonal and periodic cointegration.     

A bivariate fractionally cointegrated relationship in the context of cyclical structures

This paper deals with the analysis of cointegration in a bivariate system. However, we depart from the classic concept of cointegration in two aspects. First, we permit fractional degrees of integration in both the parent series and in their linear combination. Second, instead of assuming that the pole or singularity in the spectrum takes places at the zero frequency, we consider the case where the singularity occurs at a frequency λ in the interval (0, π]. We use a procedure that follows the same lines as the two-step testing strategy of R.F. Engle, and C.W.J. Granger, [Cointegration and error correction model. Representation, estimation and testing, Econometrica 55 (1987), pp. 251–276]. Thus, we test first the order of integration in the individual series, which are specified in terms of the Gegenbauer polynomials. Then, if the two series share the same degree of integration at a given frequency, we test the null hypothesis of no cointegration against the alternative of fractional cyclical cointegration, by testing the order of integration on the estimated residuals from the cointegrating regression. Finite sample critical values are obtained, and the power properties of the test are examined. An empirical application is also carried out at the end of the article.     

Panel Seasonal Unit Root Test: Further Simulation Results and An Application to Unemployment Data

Summary: In this paper the seasonal unit root test of Hylleberg et al. (1990) is generalized to cover a heterogenous panel. The procedure follows the work of Im, Pesaran and Shin (2002) and is independently proposed by Otero et al. (2004). Test statistics are given and critical values are obtained by simulation. Moreover, the properties of the tests are analyzed for different deterministic and dynamic specifications. Evidence is presented that for a small time series dimension the power is low even for increasing cross section dimension. Therefore, it seems necessary to have a higher time series dimension than cross section dimension. The test is applied to unemployment data in industrialized countries. In some cases seasonal unit roots are detected. However, the null hypotheses of panel seasonal unit roots are rejected. The null hypothesis of a unit root at the zero frequency is not rejected, thereby supporting the presence of hysteresis effects. * The research of this paper was supported by the Deutsche Forschungsgemeinschaft. The paper was presented at the workshop “Unit roots and cointegration in panel data” in Frankfurt, October 2004 and in the poster-session at the EC2 meeting in Marseille, December 2004. We are grateful to the participants of the workshops and an anonymous referee for their helpful comments.     

Bayesian Inference in Cointegrated I (2) Systems: A Generalization of the Triangular Model

This paper generalizes the cointegrating model of Phillips (1991) to allow for I (0), I (1) and I (2) processes. The model has a simple form that permits a wider range of I (2) processes than are usually considered, including a more flexible form of polynomial cointegration. Further, the specification relaxes restrictions identified by Phillips (1991) on the I (1) and I (2) cointegrating vectors and restrictions on how the stochastic trends enter the system. To date there has been little work on Bayesian I (2) analysis and so this paper attempts to address this gap in the literature. A method of Bayesian inference in potentially I (2) processes is presented with application to Australian money demand using a Jeffreys prior and a shrinkage prior.     

Critical values for unit root tests in seasonal time series

SUMMARY In this paper, we present tables with critical values for a variety of tests for seasonal and non-seasonal unit roots in seasonal time series. We consider (extensions of) the Hylleberg et al. and Osborn et al. test procedures. These extensions concern time series with increasing seasonal variation and time series with structural breaks in the seasonal means. For each case, we give the appropriate auxiliary test regression, the test statistics, and the corresponding critical values for a selected set of sample sizes. We also illustrate the practical use of the auxiliary regressions for quarterly new car sales in the Netherlands. Supplementary to this paper, we provide Gauss programs with which one can generate critical values for particular seasonal frequencies and sample sizes.     

Bootstrap LR tests of stationarity,common trends and cointegration

This paper considers the likelihood ratio (LR) tests of stationarity, common trends and cointegration for multivariate time series. As the distribution of these tests is not known, a bootstrap version is proposed via a state- space representation. The bootstrap samples are obtained from the Kalman filter innovations under the null hypothesis. Monte Carlo simulations for the Gaussian univariate random walk plus noise model show that the bootstrap LR test achieves higher power for medium-sized deviations from the null hypothesis than a locally optimal and one-sided Lagrange Multiplier (LM) test that has a known asymptotic distribution. The power gains of the bootstrap LR test are significantly larger for testing the hypothesis of common trends and cointegration in multivariate time series, as the alternative asymptotic procedure – obtained as an extension of the LM test of stationarity – does not possess properties of optimality. Finally, it is shown that the (pseudo-)LR tests maintain good size and power properties also for the non-Gaussian series. An empirical illustration is provided.     

Robust Stationarity Tests in Seasonal Time Series Processes

This article builds on the existing literature on (stationarity) tests of the null hypothesis of deterministic seasonality in a univariate time series process against the alternative of unit root behavior at some or all of the zero and seasonal frequencies. This article considers the case where, in testing for unit roots at some proper subset of the zero and seasonal frequencies, there are unattended unit roots among the remaining frequencies. Monte Carlo results are presented that demonstrate that in this case, the stationarity tests tend to distort below nominal size under the null and display an associated (often very large) loss of power under the alternative. A modification to the existing tests, based on data prefiltering, that eliminates the problem asymptotically is suggested. Monte Carlo evidence suggests that this procedure works well in practice, even at relatively small sample sizes. Applications of the robustified statistics to various seasonally unadjusted time series measures of U.K. consumers' expenditure are considered; these yield considerably more evidence of seasonal unit roots than do the existing stationarity tests.     

Testing nested and non-nested periodically integrated autoregressive models

Periodically integrated time series require a periodic differencing filter to remove the stochastic trend. A non-periodic integrated time series needs the first-difference filter for similar reasons. When the changing sea- sonal fluctuations for the non-periodic integrated series can be described by seasonal dummy variables for which the corresponding parameters are not constant within the sampie, such a series may not be easily & stinguished from a periodically integrated time series. In this paper, nested and non-nested testing procedures are proposed to distinguish between these two alternative stochastic and non-stochastic seasonal processes, When it is assumed there is a single unknown structural break in the seasonal dummy parameters. Several empirical examples using quarterly real macroeconomic time series for the United Kingdom illustrate the nested and non-nested approaches.     

基于LSTAR模型的非线性协整检验

大量的经济理论和实践都表明,宏观经济时间序列经常会出现非平稳和非线性特征,因而在统计分析时,需要进行非线性协整检验。基于逻辑平滑转换自回归(LSTAR)模型将传统的线性协整表述方法拓展为非线性形式,构造实用的检验程序及合适的统计量,利用软件R进行蒙特卡洛模拟给出非线性协整检验统计量的临界值,并通过实际数据分析购买力平价动态系统的非线性协整关系,说明方法的有效性。     

Testing for climate warming in Sweden during 1850-1999, using wavelets analysis

This paper describes an alternative approach for testing for the existence of trend among time series. The test method has been constructed using wavelet analysis which has the ability of decomposing a time series into low frequencies (trend) and high-frequency (noise) components. Under the normality assumption, the test is distributed as F. However, using generated empirical critical values, the properties of the test statistic have been investigated under different conditions and different types of wavelet. The Harr wavelet has shown to exhibit the highest power among the other wavelet types.

The methodology here has been applied to real temperature data in Sweden for the period 1850-1999. The results indicate a significant increasing trend which agrees with the 'global warming' hypothesis during the last 100 years.     

The effects of additive outliers on the seasonal KPSS test: a Monte Carlo analysis

This article builds on the test proposed by Lyhagen [The seasonal KPSS statistic, Econom. Bull. 3 (2006), pp. 1–9] for seasonal time series and having the null hypothesis of level stationarity against the alternative of unit root behaviour at some or all of the zero and seasonal frequencies. This new test is qualified as seasonal-frequency Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test and it is not originally supported by a regression framework.

The purpose of this paper is twofold. Firstly, we propose a model-based regression method and provide a clear illustration of Lyhagen's test and we establish its asymptotic theory in the time domain. Secondly, we use the Monte Carlo method to study the finite-sample performance of the seasonal KPSS test in the presence of additive outliers. Our simulation analysis shows that this test is robust to the magnitude and the number of outliers and the statistical results obtained cast an overall good performance of the test finite-sample properties.     

On consistent testing for serial correlation in seasonal time series models

The author considers serial correlation testing in seasonal time series models. He proposes a test statistic based on a spectral approach. Many tests of this type rely on kernel-based spectral density estimators that assign larger weights to low order lags than to high ones. Under seasonality, however, large autocorrelations may occur at seasonal lags that classical kernel estimators cannot take into account. The author thus proposes a test statistic that relies on the spectral density estimator of Shin (2004), whose weighting scheme is more adapted to this context. The distribution of his test statistic is derived under the null hypothesis and he studies its behaviour under fixed and local alternatives. He establishes the consistency of the test under a general fixed alternative. He also makes recommendations for the choice of the smoothing parameters. His simulation results suggest that his test is more powerful against seasonality than alternative procedures based on classical weighting schemes. He illustrates his procedure with monthly statistics on employment among young Americans.     

No-cointegration test based on fractional differencing: Some Monte Carlo results

This paper examines the use of the t-statistic in the Geweke–Porter-Hudak regression for the estimation of the fractional differencing parameter as a test for cointegration. The critical values of the test statistic are estimated using Monte Carlo methods. The results confirm that the test will over-reject the null hypothesis of no cointegration if the standard-normal critical values are used. The estimated critical values are generally robust to the nuisance parameters in the autoregressive or moving average specification of the error process of the component time series. Exceptions occur when the dependent variable in the cointegration regression follows an autoregressive process with a large positive parameter or a moving average process with a large negative parameter.     

An approximate entropy test for randomness

We develop an entropy-based test for randomness of binary time series of finite length. The test uses the frequencies of contiguous blocks of different lengths. A simple condition ib the block lengths and the length of the time series enables one to estimate the entropy rate for the data, and this information is used to develop a statistic to test the hypothesis of randomness. This static measures the deviation of the estimated entropy of the observed data from the theoretical maximum under the randomness hypothesis. This test offers a real alternative to the conventional runs test. Critical percentage points, based on simulations, are provided for testing the hypothesis of randomness. Power calculations using dependent data show that the proposed test has higher power against the runs test for short series, and it is similar to the runs test for long series. The test is applied to two published data sets that wree investigated by others with respect to their randomness.     

The Estimation and Testing of the Cointegration Order Based on the Frequency Domain

ABSTRACT

This article proposes a method to estimate the degree of cointegration in bivariate series and suggests a test statistic for testing noncointegration based on the determinant of the spectral density matrix for the frequencies close to zero. In the study, series are assumed to be I(d), 0 < d ? 1, with parameter d supposed to be known. In this context, the order of integration of the error series is I(d ? b), b ∈ [0, d]. Besides, the determinant of the spectral density matrix for the dth difference series is a power function of b. The proposed estimator for b is obtained here performing a regression of logged determinant on a set of logged Fourier frequencies. Under the null hypothesis of noncointegration, the expressions for the bias and variance of the estimator were derived and its consistency property was also obtained. The asymptotic normality of the estimator, under Gaussian and non-Gaussian innovations, was also established. A Monte Carlo study was performed and showed that the suggested test possesses correct size and good power for moderate sample sizes, when compared with other proposals in the literature. An advantage of the method proposed here, over the standard methods, is that it allows to know the order of integration of the error series without estimating a regression equation. An application was conducted to exemplify the method in a real context.     

DATA-DRIVEN NONPARAMETRIC SPECTRAL DENSITY ESTIMATORS FOR ECONOMIC TIME SERIES: A MONTE CARLO STUDY

Spectral analysis at frequencies other than zero plays an increasingly important role in econometrics. A number of alternative automated data-driven procedures for nonparametric spectral density estimation have been suggested in the literature, but little is known about their finite-sample accuracy. We compare five such procedures in terms of their mean-squared percentage error across frequencies. Our data generating processes (DGP) include autoregressive-moving average (ARMA) models, fractionally integrated ARMA models and nonparametric models based on 16 commonly used macroeconomic time series. We find that for both quarterly and monthly data the autoregressive sieve estimator is the most reliable method overall.     

On the usability of the fluctuation test statistic to identify multiple cointegration break points

The fluctuation test suggested by Hansen and Johansen [Some tests for parameter constancy in cointegrated VAR models, Econometrics J. 2 (1999), pp. 306–333] intends to distinguish between the presence of zero and one break in cointegration relations. In this article, we provide evidence by Monte Carlo simulations that it also serves as a graphical device to detect even multiple break locations. It suffices to consider a simplified and easy-to-implement version of the original fluctuation test. Its break detection performance depends on the sign of change in cointegration parameters and the break height. The sign issue can be approached successfully by a backward application of the test statistic. If breaks are observable, the break locations are detected at the true location on average. We apply the graphical procedure to assess the cointegration of bond yields of Spain, Italy and Portugal with German yields for the period 1995–2013 which is surprisingly supported by the trace test. However, the recursive cointegration approach shows that a stable relationship with German yields is only present for sub-periods between the introduction of the Euro and the global financial crisis which is in line with expectations. The statistical robustness of these results is supported by a forward and backward application of the cointegration breakdown test by Andrews and Kim [Tests for cointegration breakdown over a short time period, J. Bus. Econom. Stat. 24 (2006), pp. 379–394].