A note on the invertibility of nonlinear ARMA models

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 a NLARMA model admits the generic dichotomy that the innovation reconstruction errors either diminish geometrically fast or grow geometrically fast. We derive a simple sufficient condition for a 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.

Non Linear Bilinear Type Models to Forecast Critical Dry Spell Lengths in Anamaduwa, Sri Lanka

“…Also model (8), NBLX (2,2,0,3) has slight higher RMSE than that in model (12), it also satisfies the randomness and the contant variance of the errors. Thus of the four models those two can be considered as most suiatble models.…”

“…[9], [7], [3] and [2]) One of these nonlinear models, proposed by [8]) is called a Bilinear model, denoted by BN(p,q,m,k) and given by equation (1). (1) where {e t } is a sequence of iid random variables, p= order of MA term, q=order of AR term, m=order of bilinear term (Y) and k= order of bilinear term (e).…”

On the Ergodicity of First‐Order Threshold Autoregressive Moving‐Average Processes

“…A TARMA(1,1) model is said to be invertible if the preceding reconstruction condition holds. It can be checked that under any of the conditions (i)–(iii) in (C3), the TARMA(1,1) model is non‐invertible, see Chan and Tong (). Hence (C3) holds for any invertible TARMA(1,1) model, even though some non‐invertible TARMA(1,1) models satisfy (C3), e.g., ( φ 1,1 + θ 1,1 )( φ 2,1 + θ 2,1 ) > 0 and θ i ,1 ≥ 1, i = 1,2.…”

Are neural networks able to forecast nonlinear time series with moving average components?

“…Este hecho se debe, en parte, a la dificultad que se presenta para establecer la propiedad de invertibilidad del modelo [21]; dicha propiedad se refiere a la posibilidad de realizar la reconstrucción de las innovaciones a partir de las observaciones , suponiendo que el verdadero modelo es conocido. Sin embargo, Chan y Tong [22] establecieron que el modelo NLMA puede llegar a ser localmente invertible; es decir, que se pueden establecer condiciones iniciales que permiten reconstruir asintóticamente las innovaciones a partir de las observaciones.…”