Asymptotic Relative Efficiency of Goodness‐Of‐Fit Tests Based on Inverse and Ordinary Autocorrelations

Abstract

We compare the performance of the inverse and ordinary (partial) autocorrelations for time series model identification. It is found that, both in terms of Bahadur's slope and Pitman's asymptotic relative efficiency, the inverse partial autocorrelations are more efficient than the ordinary autocorrelations for identification of moving-average models. By duality, the partial autocorrelations turn out to be more powerful than the inverse autocorrelations to identify autoregressive models. Numerical experiments on both simulated and real data sets are presented to highlight the theoretical results. Copyright 2006 The Authors Journal compilation 2006 Blackwell Publishing Ltd.

Probabilistic Properties of Parametric Dual and Inverse Time Series Models Generated by ARMA Models

“…Cleveland (1972) defined the inverse autocovariance γ i (h) of (X t ) as the Fourier coefficient of the inverse of the spectral density, and defined ρ i (h) = γ i (h)/γ i (0) as the inverse autocorrelation. Many studies have shown the interest of inverse autocorrelation function for identifying and estimating autoregressive moving-average (ARMA), in interpolation problems and in the linear determinism of time series (see, e.g., Bhansali, 1980;Battaglia, 1983;Peña and Maravall, 1991;El Ghini and Francq, 2006). The inverse autocorrelation function can also be used for model specification of a multivariate time series as well as that of a Gaussian Markov random field (Bhansali and Ippoliti, 2005).…”

“…It is also easy to see that if (X t ) is a short memory process (i.e., the autocorrelations are absolutely summable) then its inverse (Z t ) is in turn a short memory process. As shown in El Ghini and Francq (2006), the inverse autocorrelation of an ARMA (p, q) process (X t ): (B)X t = θ (B) t can be interpreted as the ordinary autocorrelation of the dual process (McLeod, 1984) …”

Processus dual et inverse d'un ARMA et application à la réversibilité temporelle

“…La fonction d'autocorrélation inverse ρ i (h) = γ i (h)/γ i (0) a été largement étudiée dans la littérature des séries temporelles. Cette fonction joue un rôle important dans les problèmes d'identification, d'estimation et d'interpolation linéaire (e.g., Bhansali [3] ; El Ghini et Francq [10] ; Peña et Maravall [13]). …”