A note on the invertibility of nonlinear ARMA models

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Abstract

We review the concepts of local and global invertibility for a nonlinear auto-regressive moving-average (NLARMA) model. Under very general conditions, a local invertibility analysis of an NLARMA model shows the generic dichotomy that the innovation reconstruction errors either diminish geometrically fast or grow geometrically fast. We derive a simple sufficient condition for an NLARMA model to be locally invertible. The invertibility of the polynomial MA models is revisited. Moreover, we show that the threshold MA models may be globally invertible even though some component MA models are non-invertible. One novelty of our approach is its cross-fertilization with dynamical systems.

Introduction

Despite the growing literature on nonlinear time series analysis (Priestley, 1988, Tong, 1990, Franses and van Dijk, 2000, Chan and Tong, 2001, Fan and Yao, 2003, Small, 2005, Gao, 2007), the general framework makes use of nonlinear auto-regressive models. In contrast, nonlinear moving-average (NLMA) models are relatively under-explored. Part of the problem contributing to the slow development, both empirical and theoretical, on NLMA models is due to the difficulty in establishing the invertibility of an NLMA model; see De Gooijer and Brännäs (1995).

Here, we focus on nonlinear auto-regressive moving-average (NLARMA) models, and discuss two concepts of invertibility for these models. We illustrate these concepts with the polynomial MA models and the threshold MA models.

Section snippets

Nonlinear auto-regressive moving-average models

The linear moving-average (MA) model of order q is characterized by the feature that it has memory of q lags. Recall that an MA(q) process {Yt} is defined by the equation: Yt=μ+ɛt−∑i=1qθiɛt−i,where μ is the mean of Yt and the innovations {ɛt} are white noise, i.e. uncorrelated random variables of zero mean and finite (identical) variance σ2. For simplicity, we shall assume μ=0. It is well known that the auto-correlation function (ACF) of an MA(q) process has a cut-off after lag q, i.e. corr(Yt,Y

Global and local invertibility

In this section, we elaborate on the concept of invertibility. We focus on the case of an NLMA model defined by (1) for which the innovations may be estimated by the residuals defined by (2), but note that all results in this section and the next can be extended to the case of NLARMA models. On the other hand, the innovations satisfy a similar difference equation: ɛt=Yt−h(ɛt−1,…,ɛt−q;θ),so that the reconstruction errors Wt=ɛ^t−ɛt satisfy the equationWt=h(ɛt−1,…,ɛt−q;θ)−h(Wt−1+ɛt−1,…,Wt−q+ɛt−q;θ)

Dichotomy of local invertibility analysis

Recall that a linear MA(q) model is invertible if and only if (or iff for short) all the roots of the characteristic equation lie outside the unit circle. This result follows from a stability analysis of the difference equation for the reconstruction errors which satisfy the equation (with F being a companion matrix whose first row equals (θ1,θ2,…,θq)): Wt=FWt−1=FtW0,and the fact that the asymptotic behavior of Ft depends solely on the largest eigenvalue of F in magnitude which is smaller than

Threshold MA model revisited

Ling and Tong (2005) studied the threshold MA (TMA) model, which is a piecewise linear MA model. For simplicity, consider the simple case of the two regimes:Yt=ɛt−I(Yt−d≤r)∑j=1qθ1,jɛt−j−I(Yt−d>r)∑j=1qθ2,jɛt−j,where the θ's are parameters and r the unknown threshold parameter and d is a positive integer known as the delay parameter. Intuitively, the transition of a TMA process switches between two MA processes where the MA process indexed by the parameter vector (θ1,1,…,θ1,q)T is in operation if

Conclusion

In this note, we illustrate the use of the subadditive ergodic theory for investigating the local invertibility of an NLARMA model, after linking the problem with the stability of an attractor in a dynamical system. This approach yields simple sufficient conditions for invertibility for the threshold MA models. We conjecture that more sophisticated analysis via this approach may yield weaker conditions for invertibility. Moreover, it is of interest to develop empirical approaches to assess

Acknowledgements

We gratefully acknowledge partial support from the US National Science Foundation (CMG-0620789 for K.S.C.), the Hong Kong Research Grants Committee (GRF HKU 7036/06P for H.T.) and the University of Hong Kong (distinguished visiting professorship for H.T.).

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