An adaptive empirical likelihood test for parametric time series regression models

We propose an adaptive empirical likelihood (EL) test for a parametric regression model against a class of alternatives for weakly dependent time series observations. The test is formulated by maximizing a standardized version of the EL statistic over a set of smoothing bandwidths. It is demonstrated that the proposed test is able to distinguish the null hypothesis from a series of local alternatives at an optimal rate.     

Measuring and testing Granger causality over the spectrum: An application to European production expectation surveys

Decomposing Granger causality over the spectrum allows us to disentangle potentially different Granger causality relationships over different frequencies. This may yield new and complementary insights compared to traditional versions of Granger causality. In this paper, we compare two existing approaches in the frequency domain, proposed originally by Pierce [Pierce, D. A. (1979). R-squared measures for time series. Journal of the American Statistical Association, 74, 901–910] and Geweke [Geweke, J. (1982). Measurement of linear dependence and feedback between multiple time series. Journal of the American Statistical Association, 77, 304–324], and introduce a new testing procedure for the Pierce spectral Granger causality measure. To provide insights into the relative performance of this test, we study its power properties by means of Monte Carlo simulations. In addition, we apply the methodology in the context of the predictive value of the European production expectation surveys. This predictive content is found to vary widely with the frequency considered, illustrating the usefulness of not restricting oneself to a single overall test statistic.     

A consistent nonparametric test for nonlinear causality—Specification in time series regression

Since the pioneering work by Granger (1969), many authors have proposed tests of causality between economic time series. Most of them are concerned only with “linear causality in mean”, or if a series linearly affects the (conditional) mean of the other series. It is no doubt of primary interest, but dependence between series may be nonlinear, and/or not only through the conditional mean. Indeed conditional heteroskedastic models are widely studied recently. The purpose of this paper is to propose a nonparametric test for possibly nonlinear causality. Taking into account that dependence in higher order moments are becoming an important issue especially in financial time series, we also consider a test for causality up to the Kth conditional moment. Statistically, we can also view this test as a nonparametric omitted variable test in time series regression. A desirable property of the test is that it has nontrivial power against T^1/2-local alternatives, where T is the sample size. Also, we can form a test statistic accordingly if we have some knowledge on the alternative hypothesis. Furthermore, we show that the test statistic includes most of the omitted variable test statistics as special cases asymptotically. The null asymptotic distribution is not normal, but we can easily calculate the critical regions by simulation. Monte Carlo experiments show that the proposed test has good size and power properties.     

Testing overidentifying restrictions with many instruments and heteroskedasticity

This paper gives a test of overidentifying restrictions that is robust to many instruments and heteroskedasticity. It is based on a jackknife version of the overidentifying test statistic. Correct asymptotic critical values are derived for this statistic when the number of instruments grows large, at a rate up to the sample size. It is also shown that the test is valid when the number of instruments is fixed and there is homoskedasticity. This test improves on recently proposed tests by allowing for heteroskedasticity and by avoiding assumptions on the instrument projection matrix. This paper finds in Monte Carlo studies that the test is more accurate and less sensitive to the number of instruments than the Hausman–Sargan or GMM tests of overidentifying restrictions.     

A bootstrap-assisted spectral test of white noise under unknown dependence

To test for the white noise null hypothesis, we study the Cramér-von Mises test statistic that is based on the sample spectral distribution function. Since the critical values of the test statistic are difficult to obtain, we propose a blockwise wild bootstrap procedure to approximate its asymptotic null distribution. Using a Hilbert space approach, we establish the weak convergence of the difference between the sample spectral distribution function and the true spectral distribution function, as well as the consistency of bootstrap approximation under mild assumptions. Finite sample results from a simulation study and an empirical data analysis are also reported.     

Testing for Flexible Nonlinear Trends with an Integrated or Stationary Noise Component

This paper proposes a new test for the presence of a nonlinear deterministic trend approximated by a Fourier expansion in a univariate time series for which there is no prior knowledge as to whether the noise component is stationary or contains an autoregressive unit root. Our approach builds on the work of Perron and Yabu ( 2009a ) and is based on a Feasible Generalized Least Squares procedure that uses a super‐efficient estimator of the sum of the autoregressive coefficients α when α = 1. The resulting Wald test statistic asymptotically follows a chi‐square distribution in both the I(0) and I(1) cases. To improve the finite sample properties of the test, we use a bias‐corrected version of the OLS estimator of α proposed by Roy and Fuller ( 2001 ). We show that our procedure is substantially more powerful than currently available alternatives. We illustrate the usefulness of our method via an application to modelling the trend of global and hemispheric temperatures.     

A new fluctuation test for constant variances with applications to finance

We present a test to determine whether variances of time series are constant over time. The test statistic is a suitably standardized maximum of cumulative first and second moments. We apply the test to time series of various assets and find that the test performs well in applications. Moreover, we propose a portfolio strategy based on our test which hedges against potential financial crises and show that it works in practice.     

Inference on the cointegration rank in fractionally integrated processes

For univariate time series we suggest a new variant of efficient score tests against fractional alternatives. This test has three important merits. First, by means of simulations we observe that it is superior in terms of size and power in some situations of practical interest. Second, it is easily understood and implemented as a slight modification of the Dickey–Fuller test, although our score test has a limiting normal distribution. Third and most important, our test generalizes to multivariate cointegration tests just as the Dickey–Fuller test does. Thus it allows to determine the cointegration rank of fractionally integrated time series. It does so by solving a generalized eigenvalue problem of the type proposed by Johansen (J. Econ. Dyn. Control 12 (1988) 231). However, the limiting distribution of the corresponding trace statistic is χ², where the degrees of freedom depend only on the cointegration rank under the null hypothesis. The usefulness of the asymptotic theory for finite samples is established in a Monte Carlo experiment.     

A score test under logistic regression models based on case–control data

We propose a score statistic to test the vector of odds ratio parameters under the logistic regression model based on case–control data. The proposed score test is based on the semiparametric profile loglikelihood function under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. The proposed score statistic has an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. Moreover, we show that the proposed score test is asymptotically equivalent to the Wald test under the logistic regression model based on case–control data. In addition, we demonstrate that the proposed score statistic and its asymptotic distribution may be obtained by fitting the prospective logistic regression model to case–control data. We present some results on simulation and on the analysis of two real datasets.     

Nonparametric simultaneous testing for structural breaks

In this paper we consider a regression model with errors that are martingale differences. This modeling includes the regression of both independent and time series data. The aim is to study the appearance of structural breaks in both the mean and the variance functions, assuming that such breaks may occur simultaneously in both the functions. We develop nonparametric testing procedures that simultaneously test for structural breaks in the conditional mean and the conditional variance. The asymptotic distribution of an adaptive test statistic is established, as well as its asymptotic consistency and efficiency. Simulations illustrate the performance of the adaptive testing procedure. An application to the analysis of financial time series also demonstrates the usefulness of the proposed adaptive test in practice.     

Testing Integration of Macroeconomic Time Series in Transitional Socialist Economies. A Modification of Perron Test

The presence of structural breaks reduces the power of integration tests. A number of methods were suggested to improve the statistical properties of integration tests in the presence of structural breaks. The most known are Perron tests, which allow to test for the level of integration of time series with one structural break. Perron tests allow for two types of structural breaks: additive outlier an innovative outlier. These tests are, however, not very useful in testing the level of integration of macroeconomic time series in countries in transition from centrally-planned to market economy. In such case one should expect two structural breaks to affect the time series: one at the beginning and one at the end of the transformation process. Test that allows for two additive outlier type structural breaks in time series is developed in this paper. This test has superior power as compared to standard Dickey-Fuller and Perron tests. This paper provides asymptotic distribution as well as finite sample properties of proposed test. Therefore practitioners receive a reliable tool for analyzing macroeconomic processes in transitional economies. This revised version was published online in July 2006 with corrections to the Cover Date.     

Regime shifts in stock–flow I(2)–I(1) systems: the case of US fiscal sustainability

In the last two decades, fiscal sustainability has been tested through the use of non‐stationary time series analysis. Two different approximations can be found in the literature: first, a univariate approach that has focused on the stochastic properties of the stock of debt and, second, a multivariate one that has focused on the long‐run properties of the flows of expenditures and revenues, i.e., in the stochastic properties of the deficit. In this paper we unify these approaches considering the stock–flow system that fiscal variables configure. Our approach involves working in an I(2) stochastic processes framework. Given the possibility of the existence of regime shifts in the sustainability of US deficit that the literature has pointed out, we develop a new statistic that can be applied to test several types of I(2) cointegration and multicointegration relationships allowing for regime shifts. To test for these kinds of changing long‐run relationships we propose the use of a residual‐based Dickey–Fuller class of statistic that accounts for one structural break. We show that consistent estimates of the break fraction can be obtained through the minimization of the sum of squared residuals when there is I(2) cointegration. The finite sample performance of the proposed statistic is investigated by Monte Carlo simulations. The econometric methodology is applied to assess whether the US fiscal deficit and debt are sustainable. Copyright © 2010 John Wiley & Sons, Ltd.     

A non-parametric independence test using permutation entropy

In the present paper we construct a new, simple, consistent and powerful test for independence by using symbolic dynamics and permutation entropy as a measure of serial dependence. We also give a standard asymptotic distribution of an affine transformation of the permutation entropy under the null hypothesis of independence. The test statistic and its standard limit distribution are invariant to any monotonic transformation. The test applies to time series with discrete or continuous distributions. Eventhough the test is based on entropy measures, it avoids smoothed non-parametric estimation. An application to several daily financial time series illustrates our approach.     

New international evidence on the cyclical behaviour of output: Kuznets swings reconsidered

Spectral analysis is a particularly valuable method for seeking dependences expressed as lags between different magnitudes. Its use in this article was first determined by the search for maximum objectivity in the observation of time series. The possibility of applying it to a large number of series was then examined. This twin requirement resulted from a desire to avoid the criticism generally levelled at statistical studies concerning cyclical movements of the economy. Spectral analysis is based on the theory of stochastic processes. It starts with the core hypothesis that a given time series consists of a large number of sinusoidal components with different frequencies (univariate spectral analysis). It makes it possible to divide a particular category of records into a set of oscillations of different frequencies and then to show the links between the components with the same frequency in the various series examined (cross-spectral or bivariate spectral analysis). It has had limited applications in cliometrics to date. It is used here to determine the frequency of GDP series of several OECD countries. A reminder of the method Sect. 2 is followed by successive examination of the various series chosen, the treatment of these series and the results of spectral analysis Sect. 3. It is then possible as a conclusion to show the prospects of this type of approach and to synthesise a completely new major result for understanding economic dynamics in nineteenth and twentieth centuries, that is to say the existence of a single intermediate cycle with 15–20-year frequency that calls into question or even partially contradicts previous work on economic cycles.      

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。     

Granger-causality in multiple time series

The concept of Granger-causality is formulated for a finite-dimensional multiple time series. Special attention is given to causality patterns in autoregressive series, and it is shown how these patterns can be tested under quite general assumptions using a χ² statistic. The power of the test is discussed, and it is shown that the χ² statistic results from a Lagrange multiplier test in the Gaussian case. The causality test is tried both on artificial data and some economic time series. Finally we consider the problem of constrained estimation in models with a known causality structure.     

A robust version of the KPSS test based on indicators

This paper proposes a test of the null hypothesis of stationarity that is robust to the presence of fat-tailed errors. The test statistic is a modified version of the so-called KPSS statistic. The modified statistic uses the “sign” of the data minus the sample median, whereas KPSS used deviations from means. This “indicator” KPSS statistic has the same limit distribution as the standard KPSS statistic under the null, without relying on assumptions about moments, but a different limit distribution under unit root alternatives. The indicator test has lower power than standard KPSS when tails are thin, but higher power when tails are fat.     

Goodness-of-fit tests in linear EV regression with replications

This paper proposes a goodness-of-fit test for checking the adequacy of parametric forms of the regression error density functions in linear errors-in-variables regression models. Instead of assuming the distribution of the measurement error to be known, we assume that replications of the surrogates of the latent variables are available. The test statistic is based upon a weighted integrated squared distance between a nonparametric estimator and a semi-parametric estimator of the density functions of certain residuals. Under the null hypothesis, the test statistic is shown to be asymptotically normal. Consistency and local power results of the proposed test under fixed alternatives and local alternatives are also established. Finite sample performance of the proposed test is evaluated via simulation studies. A real data example is also included to demonstrate an application of the proposed test.     

Housing prices in Metropolitan Toronto: An empirical analysis

This paper examines short-run fluctuations in real house prices in Metropolitan Toronto. We hypothesize that the average time that a house has been on the market before it is sold provides information on the expectation of future movements in real house prices. The paper combines the use of cross spectral analysis (in the frequency domain) with regression analysis (in the time domain) to examine the relationship between monthly real house prices and the average waiting times. In particular, we use a Hannan estimator to form a distributed lag function from the spectral analysis and use these results as an input to the regression model. The empirical findings support our use of waiting times as a proxy of future real house price movements.     

Fitting dynamic factor models to non-stationary time series

Factor modelling of a large time series panel has widely proven useful to reduce its cross-sectional dimensionality. This is done by explaining common co-movements in the panel through the existence of a small number of common components, up to some idiosyncratic behaviour of each individual series. To capture serial correlation in the common components, a dynamic structure is used as in traditional (uni- or multivariate) time series analysis of second order structure, i.e. allowing for infinite-length filtering of the factors via dynamic loadings. In this paper, motivated from economic data observed over long time periods which show smooth transitions over time in their covariance structure, we allow the dynamic structure of the factor model to be non-stationary over time by proposing a deterministic time variation of its loadings. In this respect we generalize the existing recent work on static factor models with time-varying loadings as well as the classical, i.e. stationary, dynamic approximate factor model. Motivated from the stationary case, we estimate the common components of our dynamic factor model by the eigenvectors of a consistent estimator of the now time-varying spectral density matrix of the underlying data-generating process. This can be seen as a time-varying principal components approach in the frequency domain. We derive consistency of this estimator in a “double-asymptotic” framework of both cross-section and time dimension tending to infinity. The performance of the estimators is illustrated by a simulation study and an application to a macroeconomic data set.