Estimation of Time Series Models via Robust Wavelet Variance

Journal of the American Statistical Association, 2013

This paper presents a new estimation method for the parameters of a times series model. We consider here composite Gaussian processes that are the sum of independent Gaussian processes which in turn explain an important aspect of the time series, as is the case in engineering and natural sciences. The proposed estimation method offers an alternative to classical estimation based on the likelihood, that is straightforward to implement and often the only feasible estimation method with complex models. The estimator results as the optimization of a criterion based on a standardized distance between the sample wavelet variances (WV) estimates and the model based WV. Indeed, the WV provides a decomposition of the variance process through different scales, so that they contain the information about different features of the stochastic model. We derive the asymptotic properties of the proposed estimator for inference and perform a simulation study to compare our estimator to the MLE and the LSE with different models. We also set sufficient conditions on composite models for our estimator to be consistent, that are easy to verify. We use the new estimator to estimate the stochastic error's parameters of the sum of three first order Gauss-Markov processes by means of a sample of over 800, 000 issued from gyroscopes that compose inertial navigation systems.

Journal of Physics: Conference Series, 2013

In general, time series is modeled as summation of known information i.e. historical information components, and unknown information i.e. random component. In wavelet based model, time series is represented as linear model of wavelet coefficients. Wavelet based model captures the time series feature perfectly when the historical information components dominate the process. In other hand, it has low enforcement when the random component dominates the process. This paper proposes an effort to develop the adequateness of wavelet based model, when the random component dominated the process. By weighted summation, the data is carried to the new form which has higher dependencies. Consequently, wavelet based model will work better. Finally, it is hoped that the better prediction of wavelet based model will be carried to the original prediction in reverting process.

Control and Cybernetics, 2005

In the paper we review stochastic properties of wavelet coefficients for time series indexed by continuous or discrete time. The main emphasis is on decorrelation property and its implications for data analysis. Some new properties are developed as the rates of correlation decay for the wavelet coefficients in the case of longrange dependent processes such as the fractional Gaussian noise and the fractional autoregressive integrated moving average processes. It is proved that for such processes the within-scale covariance of the wavelet coefficients at lag k is O(k 2(H−N)−2), where H is the Hurst exponent and N is the number of vanishing moments of the wavelet employed. Some applications of decorrelation property are briefly discussed.

Journal of Economic Dynamics and Control, 2000

In this paper we apply compactly supported wavelets to the ARFIMA(p, d, q) longmemory process to develop an alternative maximum likelihood estimator of the di!erencing parameter, d, that is invariant to unknown means, model speci"cation, and contamination. We show that this class of time series have wavelet transforms whose covariance matrix is sparse when the wavelet is compactly supported. It is shown that the sparse covariance matrix can be approximated to a high level of precision by a matrix equal to the covariance matrix except with the o!-diagonal elements set equal to zero. This diagonal matrix is shown to reduce the order of calculating the likelihood function to an order smaller than those associated with the exact MLE method. We test the robustness of the wavelet MLE of the fractional di!erencing parameter to a variety of compactly supported wavelets, series length, and contamination levels by generating ARFIMA(p, d, q) processes for di!erent values of p, d, and q, and calculating the wavelet MLE using only the main diagonal elements of its covariance matrix. In our simulations we "nd the wavelet MLE to be superior to the approximate frequency MLE when estimating contaminated ARFIMA(0, d, 0), and uncontaminated ARFIMA(1, d, 0) and ARFIMA(0, d, 1) processes except when the MA parameter is close to one. We also "nd the wavelet MLE to be robust to model speci"cation and as such is an attractive alternative semiparameter estimator to the Geweke, Porter}Hudak estimator. 2000 Elsevier Science B.V. All rights reserved.

2000

As is well-known, a heteroskedasticity and autocorrelation consistent covariance matrix is proportional to a spectral density matrix at frequency zero and can be consistently estimated by such popular kernel methods as those of Andrews-Newey-West. In practice, it is difficult to estimate the spectral density matrix if it has a peak at frequency zero, which can arise when there is strong autocorrelation, as often encountered in economic and financial time series. Kernels, as a local averaging method, tend to underestimate the peak, thus leading to strong overrejection in testing and overly narrow confidence intervals in estimation. As a new mathematical tool generalizing Fourier transform, wavelet transform is a powerful tool to investigate such local properties as peaks and spikes, and thus is suitable for estimating covariance matrices. In this paper, we propose a class of wavelet estimators for the covariance matrices of econometric parameter estimators. We show the consistency of...

Annals of the Institute of Statistical Mathematics, 2005

We estimate nonlinear autoregressive models using a design-adapted wavelet estimator. We show two properties of the wavelet transform adapted to an autoregressive design. First, in an asymptotic setup, we derive the order of the threshold that removes all the noise with a probability tending to one asymptotically. Second, with this threshold, we estimate the detail coefficients by soft-thresholding the empirical detail coefficients. We show an upper bound on the l 2 -risk of these soft-thresholded detail coefficients. Finally, we illustrate the behavior of this design-adapted wavelet estimator on simulated and real data sets.

Bernoulli, 1999

We consider nonparametric estimation of the parameter functions a i (X), i 1, F F F , p, of a time-varying autoregressive process. Choosing an orthonormal wavelet basis representation of the functions a i , the empirical wavelet coef®cients are derived from the time series data as the solution of a least-squares minimization problem. In order to allow the a i to be functions of inhomogeneous regularity, we apply nonlinear thresholding to the empirical coef®cients and obtain locally smoothed estimates of the a i. We show that the resulting estimators attain the usual minimax L 2 rates up to a logarithmic factor, simultaneously in a large scale of Besov classes. The ®nite-sample behaviour of our procedure is demonstrated by application to two typical simulated examples.

1998

A new wavelet-based estimator is introduced which combines the state-space model with the wavelet transform in an effort to explore the stock market inefficiency. The new estimator possesses some superior qualities that are illustrated through its actual performance in forecasting the S&P 500.

Computational Statistics & Data Analysis, 2007

Vector autoregressive (VAR) modelling is one of the most popular approaches in multivariate time series analysis. The parameters interpretation is simple, and provide an intuitive identification of relationships and Granger causality among time series. However, the VAR modelling requires stationarity conditions which could not be valid in many practical applications. Locally stationary or time dependent modelling seem attractive generalizations, and several univariate approaches have already been proposed. In this paper we propose an estimation procedure for time-varying vector autoregressive processes, based on wavelet expansions of autoregressive coefficients. The asymptotic properties of the estimator are derived and illustrated by computer intensive simulations. We also present an application to brain connectivity identification using functional magnetic resonance imaging (fMRI) datasets.

Computational Economics, 2000

A new wavelet-based nonparametric estimator is introduced in an effort toapproximate variance functions. The new estimator possesses some superiorqualities that are illustrated through its actual performance in somesimulations.