Aggregation and long memory: recent developments

It is well-known that the aggregated time series might have very different properties from those of the individual series, in particular, long memory. At the present time, aggregation has become one of the main tools for modelling of long memory processes. We review recent work on contemporaneous aggregation of random-coefficient AR(1) and related models, with particular focus on various long memory properties of the aggregated process

The role of Skorokhod space in the development of the econometric analysis of time series

This paper discusses the fundamental role played by Skorokhod space, through its underpinning of functional central limit theory, in the development of the paradigm of unit roots and co-integration. This paradigm has fundamentally affected the way economists approach economic time series as was recognized by the award of the Nobel Memorial Prize in Economic Sciences to Robert F. Engle and Clive W.J. Granger in 2003. Here, we focus on how P.C.B. Phillips and others used the Skorokhod topology to establish a limiting distribution theory that underpinned and facilitated the development of methods of estimation and testing of single equations and systems of equations with possibly integrated regressors. This approach has spawned a large body of work that can be traced back to Skorokhod's conception of fifty years ago. Much of this work is surprisingly confined to the econometrics literature.Skorokhod space, functional central limit theorems, non-stationary time series, unit roots and co-integration, Wiener functionals, econometrics.

Asymptotic statistical properties of AR spectral estimators for processes with mixed spectra

Copyright © 2002 IEEEThe influence of a point spectrum on large sample statistics of the autoregressive (AR) spectral estimator is addressed. In particular, the asymptotic distributions of the AR coefficients, the innovations variance, and the spectral density estimator of a finite-order AR(p) model to a mixed spectrum process are presented. Various asymptotic results regarding AR modeling of a regular process with a continuous spectrum are arrived at as special cases of the results for the mixed spectrum setting. Finally, numerical simulations are performed to verify the analytical resultsSoon-Sen Lau, P. J. Sherman and L. B. Whit

Rank-based estimation for all-pass time series models

An autoregressive-moving average model in which all roots of the autoregressive polynomial are reciprocals of roots of the moving average polynomial and vice versa is called an all-pass time series model. All-pass models are useful for identifying and modeling noncausal and noninvertible autoregressive-moving average processes. We establish asymptotic normality and consistency for rank-based estimators of all-pass model parameters. The estimators are obtained by minimizing the rank-based residual dispersion function given by Jaeckel [Ann. Math. Statist. 43 (1972) 1449--1458]. These estimators can have the same asymptotic efficiency as maximum likelihood estimators and are robust. The behavior of the estimators for finite samples is studied via simulation and rank estimation is used in the deconvolution of a simulated water gun seismogram.Comment: Published at http://dx.doi.org/10.1214/009053606000001316 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Quasi maximum likelihood estimation for strongly mixing state space models and multivariate L\'evy-driven CARMA processes

We consider quasi maximum likelihood (QML) estimation for general non-Gaussian discrete-ime linear state space models and equidistantly observed multivariate L\'evy-driven continuoustime autoregressive moving average (MCARMA) processes. In the discrete-time setting, we prove strong consistency and asymptotic normality of the QML estimator under standard moment assumptions and a strong-mixing condition on the output process of the state space model. In the second part of the paper, we investigate probabilistic and analytical properties of equidistantly sampled continuous-time state space models and apply our results from the discrete-time setting to derive the asymptotic properties of the QML estimator of discretely recorded MCARMA processes. Under natural identifiability conditions, the estimators are again consistent and asymptotically normally distributed for any sampling frequency. We also demonstrate the practical applicability of our method through a simulation study and a data example from econometrics

The sources and nature of long-term memory in the business cycle

This paper examines the stochastic properties of aggregate macroeconomic time series from the standpoint of fractionally integrated models, focusing on the persistence of economic shocks. We develop a simple macroeconomic model that exhibits long-range dependence, a consequence of aggregation in the presence of real business cycles. We then derive the relation between properties of fractionally integrated macroeconomic time series and those of microeconomic data and discuss how fiscal policy may alter the stochastic behavior of the former. To implement these results empirically, we employ a test for fractionally integrated time series based on the Hurst-Mandelbrot rescaled range. This test, which is robust to short-term dependence, is applied to quarterly and annual real GNP to determine the sources and nature of long-term dependence in the business cycle..Business cycles ; Time-series analysis

Spectral estimation for mixed causal-noncausal autoregressive models

This paper investigates new ways of estimating and identifying causal, noncausal, and mixed causal-noncausal autoregressive models driven by a non-Gaussian error sequence. We do not assume any parametric distribution function for the innovations. Instead, we use the information of higher-order cumulants, combining the spectrum and the bispectrum in a minimum distance estimation. We show how to circumvent the nonlinearity of the parameters and the multimodality in the noncausal and mixed models by selecting the appropriate initial values in the estimation. In addition, we propose a method of identification using a simple comparison criterion based on the global minimum of the estimation function. By means of a Monte Carlo study, we find unbiased estimated parameters and a correct identification as the data depart from normality. We propose an empirical application on eight monthly commodity prices, finding noncausal and mixed causal-noncausal dynamics