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convenient alternative. We show how the parameters of a noncausal autoregressive model can be estimated by the method of maximum likelihood and derive related test procedures. Because noncausal autoregressive models cannot be distinguished from conventional causal autoregressive models by second order properties or Gaussian likelihood, a model selection procedure is proposed. As an empirical application, we consider modeling the U.S. inflation which, according to our results, exhibits purely forward-looking dynamics. KEYWORDS: noncausal autoregression, non-Gaussian time

. This approach is parametric, so we also compare the proposed parametric approach with a semi- parametric approach. Simulation studies and applications to real time series show that this method works very well. KEYWORDS: Bayesian method, quantile function, non-Gaussian time series, simulation, parametric and semi-parametric approaches Author Notes: The author would like to thank the referee and Professor W. G. Gilchrist for their very suggestive comments which have greatly enhanced the quality and the presentation of the paper. 1 Introduction An autoregressive time

ON TIME-REVERSIBILITY AND THE UNIQUENESS OF MOVING AVERAGE REPRESENTATIONS FOR NON-GAUSSIAN STATIONARY TIME SERIES MARC HALLIN, CLAUDE LEFEVRE and MADAN L. PURI Institut de Statistique, Universite Libre de Bruxelles, B-1050 Bruxelles, Belgium Department of Mathematics, Indiana University, Bloomington, Indiana 47405, U.S.A. S U M M A R Y A recent result by Findley (1986) on the uniqueness of moving average representations for non-Gaussian time series is shown to establish a conjecture by Weiss (1975) on the time-reversibility of general linear processes

, 1987, p.175) that the prediction error A n ( h ) Xn+h — PnXn+h is normally distributed with mean zero and variance cr^(h). Assuming that the true autocovariance function of the model 7(-) is known we can calculate the best linear /i-step predictor PnXn+h and corresponding PMSE cr£(/¡,) from the innovations al- gorithm presented above and then (1 — 2a)-Gaussian prediction interval is given by ( 5 ) IG{h) = [PnXn+h - an{h), PnXn+h + $a_a an(h)], where denotes the (1 —a)-quantile of the standard normal distribution. 474 A. Zagdanski For non-Gaussian time series

samples of the random variable α with the density /(*). We will give two examples: (1) if fa) = x*-\l -χ)»-ι/Β(ν,μ) , 0 <* < 1, V = 1/2, μ = 1/2, /ι2 = 2A, Α(ν,/ι) is the Beta-function, then R(p) = exp(-a/>2)/0(/>2) where /0(/>2) is the modified Bessel function of the first kind; (2) if /(*) = λ€χ€'ιβ ~**/r(c)9 0 < jc < oo, A > 0, c> 0, then where .T(c) is the Euler gamma-function. 5. SIMULATION OF NON-GAUSSIAN TIME SERIES As mentioned above, algorithms for simulating Gaussian stationary time series (both scalar and vectorial) with an arbitrary correlation structure can

(d)), if limT→∞ ∑T τ=1 |ρx(τ, t |) = ∞, ∀t , and d is the smallest positive real number such that limT→∞ ∑T τ=1 |ρz (τ, t)| <∞, ∀t , with zt = (1− B)dxt . The parameter d that appears in this latter definition serves to quantify the memory span in the series. The previous characterization of memory in terms of the ACF is adequate for Gaussian series, since all of the dependence structure is captured by its second-order moments. With non-Gaussian time series, in particular, nonlinear time series, the ACF cannot provide a full account of the serial dependence

–253. Takahashi, M., Y. Omori, and T. Watanabe. 2009. “Estimating Stochastic Volatility Models Using Daily Returns and Realized Volatility Simultaneously.” Computational Statistics and Data Analysis 53: 2404–2426. 10.1016/j.csda.2008.07.039 Wang, J. J. J., J. S. K. Chan, and S. T. B. Choy. 2011. “Stochastic Volatility Models with Leverage and Heavy-Tailed Distributions: A Bayesian Approach Using Scale Mixtures.” Computational Statistics and Data Analysis 55: 852–862. 10.1016/j.csda.2010.07.008 Watanabe, T., and Y. Omori. 2004. “A Multi-move Sampler for Estimating Non-Gaussian

statistic observed from a given time series for: (i) i.i.d. noise, (ii) linearly filtered noise, and (iii) a static monotonic nonlinear transformation of linearly filtered noise. Some recent anal- yses of financial time series have exploited this method to provide more certain results (for example (Harrison, Yu, Lu, and George 1999) and (Kugiumtzis 2001)). However, it has been observed that for non-Gaussian time series the application of these methods can be problematic (Small and Tse 2002a) or lead to incorrect results (Kugiumtzis 2000). But it is widely accepted that

, J. O., and Ramsey, J. B. (2002): “Functional data analysis of the dynam- ics of the monthly index of nondurable goods production,” Journ. of Econo- metrics, 107, 327-344. Serroukh, A., and Walden, A. T. (1998): “The Scale Analysis of Bivariate Non- Gaussian Time Series via Wavelet Cross-Covariance,” Statistics Section Tech. Report, TR-98-02, Imperial College of Science, London. Strang, G. and Nguyen, T. (1996): Wavelets and Filter Banks, Wellesley- Cambridge Press, Cambridge. Whitcher, B. (2000): “Wavelet-based estimation for seasonal long-memory processes

optimal rate to be O(T 4/5). Recently, Deo and Hurvich (2001) have shown that the GPH estimator is also valid for some non-Gaussian time-series, and Velasco (1999) has shown that consistency extends to .5 ≤ d < 1, and asymptotic normality to .5 ≤ d < .75. The other popular semiparametric estimator is due to Robinson (1995a). The estimator is also based on the log-periodogram and solves, d̂ = argmin d R(d) (2.8) R(d) = log ( 1 m m∑ j=1 ω2dj Ij ) − 2d m m∑ j=1 ωj. (2.9) 6 Studies in Nonlinear Dynamics & Econometrics Vol. 9 [2005], No. 4, Article 1 This estimator is