Bayesian Prevision of Time Series by Transfer Function Models

Given the time series {yt{, t∊T≡{l,2,…,n}, which is assumed to be caused by the series {xt}, t∊T, according to the transfer function model (1.1) $${{y}_{t}} = {{\delta }^{{ - 1}}}\left( B \right)\omega \left( B \right){{x}_{t}} + {{\phi }^{{ - 1}}}\left( B \right)\theta \left( B \right){{u}_{t}} $$ where B is the usual back-shift operator, and ω (B)=ωo-ω1B+…-ωsBs, δ(B)=1-δ1B+...-δrBr, θ (B)=1-ϕ1B+...-θqBq, ϕ(B)=1-ϕ1B+... -ϕpBp where {xt} is assumed to be generated by the ARMA process (1.2) $${{\phi }_{x}}\left( B \right){{x}_{t}} = {{\theta }_{x}}\left( B \right){{a}_{t}} $$ with $$ {{\rm{f}}_{\rm{X}}}({\rm{B}}) = 1 - {{\rm{f}}_{{\rm{X1}}}}{\rm{B}} + \ldots - {{\rm{f}}_{{\rm{Xp}}}}_{_{\rm{X}}}{{\rm{B}}^{{{\rm{P}}_{\rm{X}}}}},{{\rm{\theta }}_{\rm{X}}}({\rm{B}}) = 1 - {{\rm{\theta }}_{{\rm{X1}}}}{\rm{B}} + \ldots - {{\rm{\theta }}_{{\rm{X}}{{\rm{q}}_{\rm{X}}}}}{{\rm{B}}^{{{\rm{q}}_{\rm{X}}}}} $$ and given a set H information regarding the unknown values of ỹn+τ (°°) and $${{{\tilde{y}}}_{{n + \tau }}}\left( {^{{0 0}}} \right) $$ , in this work previsions of these values are determined in de Finetti’s sense, so that both the sample y n=(yl,Y2,…, Yn)′, x n= (x1,X2,...,xn)′:, and the information H are taken into account. In the case of economic time series, this information may concern, for instance, either the causal relationship from the variable xt to yt, or the period of the business cycle which influences the autoregressive schemes in submodels (1.1) and (1.2), or even the ARMA schemes on residuals. Furthermore, information H may concern the orders (s,r,p,q) and (px,qx) of submodels according to the opinion and experience of the model builder. Taking this information into account, “previsions” in de Finetti’s (1974) subjective meaning rather than “projections” in the traditional sense are formulated.