Guy Melard

Rank-based tests for randomness against first-order serial dependence

Abstract Optimal rank-based procedures were derived in Hallin, Ingenbleek, and Puri (1985, 1987) and Hallin and Puri (1988) for some fundamental testing problems arising in time series analysis. The optimality properties of these procedures are of an asymptotic nature, however, whereas much of the attractiveness of rank-based methods lies in their small-sample applicability and robustness features. Accordingly, the objective of this article is twofold: (a) a study of the finite-sample behavior of the asymptotically optimal tests for randomness against first-order autoregressive moving average dependence proposed in Hallin et al. (1985), both under the null hypothesis (tables of critical values) and under alternatives of serial dependence (evaluation of the power function), and (b) an (heuristic) investigation of the robustness properties of the proposed procedures (with emphasis on the identification problem in the presence of “outliers”). We begin (Sec. 2) with a brief description of the rank-based measu...

The information matrix of multiple-input single-output time series models

Expressions are given for the information matrix of the parameters of the multiple-input single-output time series model for correlated and uncorrelated inputs, allowing lags between inputs. The model under consideration is a generalization of the multiple-regression model with autocorrelated errors, the transfer function model and the autoregressive moving average exogenous (ARMAX) model. The elements of the Fisher matrix are evaluated using algorithms developed for the univariate ARMA model.

On a fast algorithm for the exact information matrix of a Gaussian ARMA time series

The paper is devoted to a new algorithm for the computation of the exact Fisher information matrix of a Gaussian autoregressive-moving average time series. The number of operations is an order of magnitude smaller than the fastest existing procedure. The algorithm is based on a set of new recursions for the covariance matrix of the derivatives of the state vector with respect to the parameters, combined with the Chandrasekhar recursions used in the evaluation of the likelihood function. >

Software for time series analysis

This paper is devoted to the presentation of (a) a general approach in univariate time series analysis, and (b) the corresponding computer software called ANSECH. The methodology is similar to that of Box and Jenkins(1970) but the class of models is wider. The main algorithms for estimation and forecasting are briefly described.

The exact quasi-likelihood of time-dependent ARMA models☆

The purpose of the paper is to propose a simple and efficient algorithm to evaluate the exact quasi-likelihood of (possibly marginally heteroscedastic) ARMA models with time-dependent coefficients. The algorithm is based on the Kalman filter and is therefore simpler than a previous algorithm based on a Cholesky factorisation. Computational efficiency is obtained by taking the ARMA structure into account. Empirical evidence is given.

The exact Gaussian likelihood estimation of time-dependent VARMA models

An algorithm for the evaluation of the exact Gaussian likelihood of an r -dimensional vector autoregressive-moving average (VARMA) process of order ( p , q ), with time-dependent coefficients, including a time dependent innovation covariance matrix, is proposed. The elements of the matrices of coefficients and those of the innovation covariance matrix are deterministic functions of time and assumed to depend on a finite number of parameters. These parameters are estimated by maximizing the Gaussian likelihood function. The advantages of that approach is that the Gaussian likelihood function can be computed exactly and efficiently. The algorithm is based on the Cholesky decomposition method for block-band matrices. It is shown that the number of operations as a function of p , q and n , the size of the series, is barely doubled with respect to a VARMA model with constant coefficients. A detailed description of the algorithm followed by a data example is provided. We consider VARMA models with time-dependent coefficients.These coefficients and the error variance can be deterministic functions of time.We present an efficient algorithm for computing their exact Gaussian likelihood.It is based on an algorithm for traditional VARMA models.An illustration on financial data is provided.

On-line estimation of ARMA models using Fisher-scoring

Recursive estimation methods for time series models usually make use of recurrences for the vector of parameters, the model error and its derivatives with respect to the parameters, plus a recurrence for the Hessian of the model error. An alternative method is proposed in the case of an autoregressive-moving average model, where the Hessian is not updated but is replaced, at each time, by the inverse of the Fisher information matrix evaluated at the current parameter. The asymptotic properties, consistency and asymptotic normality, of the new estimator are obtained. Monte Carlo experiments indicate that the estimates may converge faster to the true values of the parameters than when the Hessian is updated. The paper is illustrated by an example on forecasting the speed of wind.

On Fast Algorithms for Several Problems in Time Series Models

In a recent paper the author has given a Fortran program for the computation of the likelihood function of an ARMA process. The algorithm is extended here to handle exactly and efficiently four related problems : (a) the computation of forecasts (b) the generation of artificial time series according to a given ARMA model (c) the computation of the likelihood function of a transfer function model or a regression model with ARMA disturbances (d) the computation of the first-order derivatives of the log-likelihood of an ARMA(p,q) process with respect to the coefficients.

A New Recursive Estimation Method for Single Input Single Output Models

This article is devoted to a new recursive estimation method for dynamic time series models, more precisely for single input single output models. In that method, the recurrence for updating the Hessian is avoided, but the recurrence for updating the estimator makes use of the Fisher information matrix. The asymptotic properties, consistency and asymptotic normality, of the new estimator are obtained under weak assumptions. Monte Carlo experiments and examples indicate that the estimates converge well, comparatively with alternative methods.