Josu Arteche

Semiparametric Inference in Seasonal and Cyclical Long Memory Processes

Several semiparametric estimates of the memory parameter in standard long memory time series are now available. They consider only local behaviour of the spectrum near zero frequency, about which the spectrum is symmetric. However long‐range dependence can appear as a spectral pole at any Nyqvist frequency (reflecting seasonal or cyclical long‐memory), where the spectrum need display no such symmetry. We introduce Seasonal/Cyclical Asymmetric Long Memory (SCALM) processes that allow differing rates of increase on either side of such a pole. To estimate the two consequent memory parameters we extend two semiparametric methods that were proposed for the standard case of a spectrum diverging at the origin, namely the log‐periodogram and Gaussian or Whittle methods. We also provide three tests of symmetry. Monte Carlo analysis of finite sample behaviour and an empirical application to UK inflation data are included. Our models and methods allow also for the possibility of negative dependence, described by a possibly asymmetric spectral zero.

Gaussian semiparametric estimation in long memory in stochastic volatility and signal plus noise models

This paper considers the persistence found in the volatility of many financial time series by means of a local Long Memory in Stochastic Volatility model and analyzes the performance of the Gaussian semiparametric or local Whittle estimator of the memory parameter in a long memory signal plus noise model which includes the Long Memory in Stochastic Volatility as a particular case. It is proved that this estimate preserves the consistency and asymptotic normality encountered in observable long memory series and under milder conditions it is more efficient than the estimator based on a log-periodogram regression. Although the asymptotic properties do not depend on the signal-to-noise ratio the finite sample performance relies upon this magnitude and an appropriate choice of the bandwidth is important to minimize the influence of the added noise. I analyze the effect of the bandwidth via Monte Carlo. An application to a Spanish stock index is finally included.

Semiparametric Robust Tests on Seasonal or Cyclical Long Memory Time Series

The concept of SCLM (seasonal or cyclical long memory) implies the existence of one or more spectral poles or zeros. The processes traditionally used to model such a behaviour assume the same persistence across different frequencies. In this paper, we propose semiparametric Wald and Lagrange multiplier (LM) tests of the equality of memory parameters at different frequencies (extendable to other linear restrictions) which are standard in the sense that they have well known χ2 distributions under the null hypothesis – although Gaussianity is nowhere assumed – and are consistent against constant and local alternatives. They have also the advantage of being robust against misspecification at frequencies distant from those of interest. Their finite sample performance is compared with the asymptotically locally efficient Robinsons tests (1994). An empirical application to a UK inflation series is also included.

Fractional integration analysis and its implications on profitability: The case of the mackerel market in the Basque Country

This paper analyses weekly prices for mackerel landed by the inshore fleet at the ports of the Basque Country in 1995-2008, using new econometric techniques never before applied to the fishing market. The idea is to learn to what extent fishermen can pass on the effects of negative shocks (e.g. fuel price increases) to their ex-vessel prices. This will give an idea of the profitability of the fishery in question. To that end, a cyclical ARFIMA model is adjusted to the series analysed, then the impulse-response function is constructed. Among other things, the behaviour of this function shows that possible increases in production costs are not being passed on to prices, which lowers the profitability of fishing. In view of these results, it is suggested that fishermen need to be able to pass the shocks that they suffer on to prices if the profitability of this fleet is to be assured.

SEMIPARAMETRIC INFERENCE IN CORRELATED LONG MEMORY SIGNAL PLUS NOISE MODELS

This article proposes an extension of the log periodogram regression in perturbed long memory series that accounts for the added noise, while also allowing for correlation between signal and noise, a common situation in many economic and financial series. Consistency (for d < 1) and asymptotic normality (for d < 3/4) are shown with the same bandwidth restriction as required for the original log periodogram regression in a fully observable series, with the corresponding gain in asymptotic efficiency and faster convergence over competitors. Local Wald, Lagrange Multiplier, and Hausman type tests of the hypothesis of no correlation between the latent signal and noise are also proposed.

Using the bootstrap for finite sample confidence intervals of the log periodogram regression

Log periodogram regression is widely applied in empirical applications to estimate the memory parameter, d, of long memory time series. This estimator is consistent for d<1 and pivotal asymptotically normal for d<3/4. However, the asymptotic distribution is a poor approximation of the (unknown) finite sample distribution if the sample size is small. Finite sample improvements in the construction of confidence intervals can be achieved by different nonparametric bootstrap procedures based on the residuals of log periodogram regression. In addition to the basic residual bootstrap, the local and block bootstraps seem adequate for replicating the structure that may arise in the errors of the regression when the series shows weak dependence in addition to long memory. The performances of different bias correcting bootstrap techniques and a bias reduced log periodogram regression are also analyzed with a view to adjusting the bias caused by that structure. Finally, an application to the Nelson and Plosser US macroeconomic data is included.

Bootstrap-based bandwidth choice for log-periodogram regression

The choice of the bandwidth in the local log-periodogram regression is of crucial importance for estimation of the memory parameter of a long memory time series. Different choices may give rise to completely different estimates, which may lead to contradictory conclusions, for example about the stationarity of the series. We propose here a data-driven bandwidth selection strategy that is based on minimizing a bootstrap approximation of the mean-squared error (MSE). Its behaviour is compared with other existing techniques for optimal bandwidth selection in a MSE sense, revealing its better performance in a wider class of models. The empirical applicability of the proposed strategy is shown with two examples: the widely analysed in a long memory context Nile river annual minimum levels and the input gas rate series of Box and Jenkins. Copyright 2009 Blackwell Publishing Ltd

SIGNAL EXTRACTION IN LONG MEMORY STOCHASTIC VOLATILITY

Long memory in stochastic volatility (LMSV) models are flexible tools for the modeling of persistent dynamic volatility, which is a typical characteristic of financial time series. However, their empirical applicability is limited because of the complications inherent in the estimation of the model and in the extraction of the volatility component. This paper proposes a new technique for volatility extraction, based on a semiparametric version of the optimal Wiener–Kolmogorov filter in the frequency domain. Its main characteristics are its simplicity and generality, because no parametric specification is needed for the volatility component and it remains valid for both stationary and nonstationary signals. The applicability of the proposal is shown in a Monte Carlo and in a daily series of returns from the Dow Jones Industrial index.

Doubly fractional models for dynamic heteroscedastic cycles

Strong cyclical persistence is a common phenomenon that has been documented not only in the levels but also in the volatility of many time series, specially in astronomical or business cycle data. The class of doubly fractional models is extended to include the possibility of long memory in cyclical (non-zero) frequencies in both levels and volatility, and a new model, the GARMA-GARMASV (Gegenbauer AutoRegressive Moving Average-Gegenbauer AutoRegressive Moving Average Stochastic Volatility), is introduced. A sequential estimation strategy based on the Whittle approximation to maximum likelihood is proposed and its finite sample performance is evaluated with a Monte Carlo analysis. Finally, a trifactorial in the mean and bifactorial in the volatility version of the model is proved to successfully fit the well-known sunspot index.

A bootstrap approximation for the distribution of the Local Whittle estimator

The asymptotic properties of the Local Whittle estimator of the memory parameter d have been widely analysed and its consistency and asymptotic distribution have been obtained for values of d ź ( - 1 / 2 , 1 in a wide range of situations. However, the asymptotic distribution may be a poor approximation of the exact one in several cases, e.g. with small sample sizes or even with larger samples when d 0.75 . In other situations the asymptotic distribution is unknown, as for example in a noninvertible context or in some nonlinear transformations of long memory processes, where only consistency is obtained. For all these cases a bootstrap strategy based on resampling a (perhaps locally) standardised periodogram is proposed. A Monte Carlo analysis shows that this strategy leads to a good approximation of the exact distribution of the Local Whittle estimator in those situations where the asymptotic distribution is not reliable.