Estimation of time-varying ARMA models with Markovian changes in regime

Large sample properties of parameter least squares estimates for time‐varying arma models

“…The intercept c t ( θ , θ 0 ) can be written in the form For convenience we write ψ t , i ( θ , θ 0 ) = 0 and ϕ t , i ( θ , s t , …, s t − i +1 ) = 0 for i ≥ t . Explicit forms of the ψ t , i ( θ , θ 0 )s and c t ( θ , θ 0 )s can be found in Francq and Gautier (2003). For instance, in the time‐varying MA(1) case, we have In the time‐varying AR(1) case, we have …”

“…These lemmas provide ergodic‐type results for general non‐stationary processes, and can therefore be of interest outside the framework of this paper. Their proofs can be found in Francq and Gautier (2003). They rely on the standard ergodic theorem for stationary processes and the independence assumption between ( η t ) and ( S t ).…”

“…Proofs and technical results are relegated to the appendix. An extended version of this paper (Francq and Gautier, 2003) provides detailed calculations and numerical illustrations using models of the form (1) in which ( s t ) is generated by a Markov chain.…”

Testing Instantaneous Causality in Presence of Nonconstant Unconditional Covariance

“…Kokoszka and Leipus (2000) and Dahlhaus and Rao (2006) studied ARCH processes with non constant unconditional variance. Robinson (1987), Hansen (1995), Francq and Gautier (2004) or Xu and Phillips (2008) among other references investigated univariate linear models allowing for a non constant variance. Stȃricȃ (2003) considered a deterministic non constant specification for the unconditional variance of stock returns, and noted that such an approach can perform as well as the stationary GARCH(1,1) model.…”

Structure and estimation of a class of nonstationary yet nonexplosive GARCH models

“…GARCH models with time‐varying coefficients have been considered by Polzehl and Spokoiny (2006). Time‐series models in which the coefficients are subordinated to an exogenous process have been recently proposed and analysed for the conditional mean by Bibi and Francq (2003) and Francq and Gautier (2004a,b). They consider ARMA models with time‐varying coefficients driven by an observed process and prove asymptotic properties for least squares and generalized least squares estimators.…”