Identification of the dynamic shock-error model with autocorrelated errors
An extension of \textit{A. Maravall}, Identification in dynamic shock-error models. (1979; Zbl 0396.62086), is given. The structural equation of the model is: \(\sum^{p}_{\tau =0}\gamma_{\tau}\eta_{t- \tau}=\sum^{m}_{l=1}\sum^{q_ l}_{\tau =0}\alpha_{l\tau}\xi_{l,t-\tau}+\epsilon_ t,\) \(t\in {\mathbb{Z}}\), with observed variables \(x_{lt}=\xi_{lt}+v_{lt}\), \(l=1,...,m\), \(y_ t=\eta_ t+w_ t\), \(t\in {\mathbb{Z}}\), where \((\xi_{lt})\), \((\epsilon_ t)\), \((v_{lt})\), \((w_ t)\) are ARMA processes with continuous spectral density functions. The AR- and MA-orders are denoted by \((k_{\xi_ l},n_{\xi_ l})\), etc. The identifiability is closely related to the invertibility of h:s\(\mapsto d\) where: \[ s=(\gamma_{\tau},\alpha_{1\tau},...,\alpha_{m\tau},\sigma_{\xi_ 1},...,\sigma_{\xi_ m},\sigma_{\epsilon},\sigma_ v,\sigma_ w), \] \[ d=(\sigma_{yx_ 1},...,\sigma_{yx_ m},\sigma_{x_ 1},...,\sigma_{x_ m},\sigma_ y) \] (\(\sigma\) for covariance functions). Conditions of identifiability are given using the different AR- and MA-orders. The proofs of the theorems use basically the Fourier transforms \(\sigma\).(z) of the \(\sigma\).(j) and the fact that \(\sigma\).(z) are rational functions in z for ARMA processes. An interesting finding is that increasing AR-orders (resp. MA-) are favorable (resp. disfavorable) to identifiability.
ARMA processes with continuous spectral density
scientific article; zbMATH DE number 3886948
scientific article