Estimating an AR Model with Exogenous Driver

In this paper, we introduce an autoregressive model which has an evolution that is driven by an exogenous pilot signal. This model shares some properties with TAR (Threshold Auto Regressive) models and STAR (Smooth Transition Auto Regressive) models. This text de nes the model, it presents an estimator for this model, and an estimator for the variance of the innovation, which is not constant in this model. An exact computation of the likelihood of this driven autoregressive model is then presented. Two appendices present a state-space realization of this model and the expression of a Kalman lter for such a model. Résumé Dans cet article, nous introduisons un modèle auto régressif dont l'évolution est pilotée par un signal exogène. Ce modèle présente des analogies avec les modèles autorégressifs à seuil (TAR, Threshold AutoRegression) ainsi que les modèles STAR (Smooth Transition Autoregressive). Le texte présente le modèle, puis un estimateur pour ce modèle ainsi que pour la variance de son innovation, qui n'est pas constante dans ce modèle. Un calcul exact de la vraisemblance du modèle autorégressif sera ensuite présenté. Deux annexes montreront la réalisation de ce type de modèle autorégressif sous forme de modèle d'état ainsi que l'expression du ltre de Kalman reposant sur ce modèle. 1 ha l-0 08 75 06 4, v er si on 1 21 O ct 2 01 3 1 Autoregressive model with exogenous pilot 1.1 Model de nition Let us consider a signal yt, t ∈ [0, T ], and a representation of this signal as a non stationary autoregressive process with coe cients ai(t), its innovation being denoted et: yt + a1(t− 1)yt−1 + · · ·+ ap(t− p)yt−p = et We assume that the innovation is a non-stationary white noise, which is centered (E(et) = 0), with variance σ t : E(etet′) = δt,t′σ t We also assume that we observe a second signal xt, that is taken as deterministic; this signal drives the coe cients ai(t) of the autoregressive model of the rst signal yt: ai(t) = gi(xt) where gi(x), i ∈ [1, p] are nonlinear functions of x; for instance, we could assume that these functions have a simple parametric representation only depending on a set of functions fm(.) such as: ai(t) = M ∑ m=0 aimfm(xt) Most often, we will take fm(x) = (x/α) m where α will be a normalization coe cient applied to the signal xt. The autoregressive model which is written: yt + p ∑ i=1 ai(t− i)yt−i = et may be rewritten (if we omit the normalization parameter α) as follows: