A Power Variance Test for Nonstationarity in Complex-Valued Signals

We propose a novel algorithm for testing the hypothesis of nonstationarity in complex-valued signals. The implementation uses both the bootstrap and the Fast Fourier Transform such that the algorithm can be efficiently implemented in O(NlogN) time, where N is the length of the observed signal. The test procedure examines the second-order structure and contrasts the observed power variance - i.e. the variability of the instantaneous variance over time - with the expected characteristics of stationary signals generated via the bootstrap method. Our algorithmic procedure is capable of learning different types of nonstationarity, such as jumps or strong sinusoidal components. We illustrate the utility of our test and algorithm through application to turbulent flow data from fluid dynamics

A fractional Dickey-Fuller test for unit roots

This paper presents a new test for fractionally integrated (FI) processes. In particular, it proposes a testing procedure in the time domain that extends the well-known Dickey-Fuller approach. Monte-Carlo simulations support the analytical results derived in the paper and show that proposed tests fare very well, both in terms of power and size, when compared with others available in the literature. The paper ends with two empirical applications.Publicad

Bootstrapping frequency domain tests in multivariate time series with an application to comparing spectral densities

We propose a general bootstrap procedure to approximate the null distribution of nonparametric frequency domain tests about the spectral density matrix of a multivariate time series. Under a set of easy to verify conditions, we establish asymptotic validity of the proposed bootstrap procedure. We apply a version of this procedure together with a new statistic in order to test the hypothesis that the spectral densities of not necessarily independent time series are equal. The test statistic proposed is based on a L2-distance between the nonparametrically estimated individual spectral densities and an overall, 'pooled' spectral density, the later being obtained using the whole set of m time series considered. The effects of the dependence between the time series on the power behavior of the test are investigated. Some simulations are presented and a real-life data example is discussed. --

Persistence and cycles in US hours worked

This paper analyses monthly hours worked in the US over the sample period 1939m1 – 2011m10 using a cyclical long memory model; this is based on Gegenbauer processes and characterised by autocorrelations decaying to zero cyclically and at a hyperbolic rate along with a spectral density that is unbounded at a non-zero frequency. The reason for choosing this specification is that the periodogram of the hours worked series has a peak at a frequency away from zero. The empirical results confirm that this model works extremely well for hours worked, and it is then employed to analyse their relationship with technology shocks. It is found that hours worked increase on impact in response to a technology shock (though the effect dies away rapidly), consistently with Real Business Cycle (RBC) models.This study is partly funded by the the Ministry of Education of Spain (ECO2011-2014 ECON Y FINANZAS, Spain) and from a Jeronimo de Ayanz project of the Government of Navarra

Modelling long-run trends and cycles in financial time series data

Copyright @ 2012 Wiley Publishing Ltd. This is the accepted version of the following article: "Modelling long-run trends and cycles in financial time series data", Journal of Time Series Analysis, 34(3), 405-421, 2013, which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1111/jtsa.12010/abstract.This article proposes a general time series framework to capture the long-run behaviour of financial series. The suggested approach includes linear and segmented time trends, and stationary and non-stationary processes based on integer and/or fractional degrees of differentiation. Moreover, the spectrum is allowed to contain more than a single pole or singularity, occurring at both zero but non-zero (cyclical) frequencies. This framework is used to analyse five annual time series with a long span, namely dividends, earnings, interest rates, stock prices and long-term government bond yields. The results based on several likelihood criteria indicate that the five series exhibit fractional integration with one or two poles in the spectrum, and are quite stable over the sample period examined.Ministerio de Ciencia y Tecnologia and Jeronimo de Ayanz project of the Government of Navarra

The weekly structure of US stock prices

In this paper we use fractional integration techniques to examine the degree of integration of four US stock market indices, namely the Standard and Poor, Dow Jones, Nasdaq and NYSE, at a daily frequency from January 2005 till December 2009. We analyse the weekly structure of the series and investigate their characteristics depending on the specific day of the week. The results indicate that the four series are highly persistent; a small degree of mean reversion (i.e., orders of integration strictly smaller than 1) is found in some cases for S&P and the Dow Jones indices. The most interesting findings are the differences in the degree of dependence for different days of the week. Specifically, lower orders of integration are systematically observed for Mondays and Fridays, consistently with the “day of the week” effect frequently found in financial data.The second-named author gratefully acknowledges financial support from the the Ministerio de Ciencia y Tecnología (ECO2008-03035 ECON Y FINANZAS, Spain) and from a PIUNA Project from the University of Navarra

Residual Log-Periodogram Inference for Long-Run-Relationships

We assume that some consistent estimator of an equilibrium relation between non-stationary series integrated of order d E (0:5; 1:5) is used to compute residuals ˆut = yt - xt (or differences there of). We propose to apply the semiparametric log-periodogram regression to the (differenced) residuals in order to estimate or test the degree of persistence ± of the equilibrium deviation ut. Provided converges fast enough, we describe simple semiparametric conditions around zero frequency that guarantee consistent estimation of ±. At the same time limiting normality is derived, which allows to construct approximate confidence intervals to test hypotheses on ±. This requires that d ¡ ± > 0:5 for superconsistent b¯, so the residuals can be good proxies of true cointegrating errors. Our assumptions allow for stationary deviations with long memory, 0 · ± < 0:5, as well as for non-stationary but transitory equilibrium errors, 0:5 < ± < 1. In particular, if xt contains several series we consider the joint estimation of d and ±. Wald statistics to test for parameter restrictions of the system have a limiting Â2 distribution. We also analyze the benefits of a pooled version of the estimate. The empirical applicability of our general cointegration test is investigated by means of Monte Carlo experiments and illustrated with a study of exchange rate dynamics

Residual log-periodogram inference for long-run relationships

We assume that some consistent estimator of an equilibrium relation between non-stationary series integrated of order d(0.5,1.5) is used to compute residuals (or differences thereof). We propose to apply the semiparametric log-periodogram regression to the (differenced) residuals in order to estimate or test the degree of persistence δ of the equilibrium deviation ut. Provided converges fast enough, we describe simple semiparametric conditions around zero frequency that guarantee consistent estimation of δ. At the same time limiting normality is derived, which allows to construct approximate confidence intervals to test hypotheses on δ. This requires that d-δ>0.5 for superconsistent , so the residuals can be good proxies of true cointegrating errors. Our assumptions allow for stationary deviations with long memory, 0δ<0.5, as well as for non-stationary but transitory equilibrium errors, 0.5<δ<1. In particular, if xt contains several series we consider the joint estimation of d and δ. Wald statistics to test for parameter restrictions of the system have a limiting χ2 distribution. We also analyse the benefits of a pooled version of the estimate. The empirical applicability of our general cointegration test is investigated by means of Monte Carlo experiments and illustrated with a study of exchange rate dynamics.Publicad