Recursive method for ARMA model estimation. I
Let \(x(t)\) be a stationary invertible ergodic ARMA\((p_ 0,q_ 0)\) time series satisfying \[ \sum_{k=0}^{p_ o}\alpha_ k x(n- k)=\sum_{j=0}^{q_ 0}\beta_ j\varepsilon(n-j), \qquad \alpha_ 0=\beta_ 0=1, \] where \(\varepsilon(t)\) is a white noise with \(E\varepsilon(t)=0\), \(0<E\varepsilon^ 4(t)<\infty\). Then \[ \varepsilon(n)=\sum_{t=0}^ \infty \varphi_ t x(n-t), \qquad \varphi_ 0=1. \] Let \(\hat\varphi_ t\), \(t=1,\dots,K_ N\), be the estimators of \(\varphi_ t\) calculated by fitting a long least squares autoregression. Then \(p_ 0\), \(q_ 0\), \(\alpha_ k\), \(\beta_ j\) are estimated by solving the least squares problem \[ \hat\mu^ 2_{p,q}=\inf_{b_ j}\sum_{t=p+1}^ \infty (\sum_{j=0}^ q b_ j\tilde\varphi_{t-j})^ 2, \qquad b_ 0=0, \] where \(\tilde\varphi_ t=0\) for \(t<0\) or \(t>P_ N\), \(\tilde\varphi_ t=\hat\varphi_ t\) for \(1\leq t\leq P_ n\), \(\tilde\varphi_ 0=1\); \(P_ N\) is an integer. The \(\hat\mu^ 2_{q-p}\) can be minimized by three sets of recursive formulas given in the paper.
stationary invertible ergodic ARMA\((p_ 0,q_ 0)\) time series
scientific article; zbMATH DE number 3581
scientific article