Characterization of the partial autocorrelation function of nonstationary time series. (Q1414600)

Let \(\{X(t), t\in \mathbb Z\}\) be a scalar complex valued nonstationary process with zero mean and finite second moments. The authors define the partial autocorrelation function (PACF) \(\beta(t,s)\) of \(X(t)\). In their setting \(\beta(t,t)= \text{Var}\,X(t)\) and \(|\beta(t,s)| \leq 1\) for \(t\neq s\). The domain \({\mathcal D}_{\beta}\) of the PACF is given. The constraints describing \({\mathcal D}_{\beta}\) are very simple in comparison with the autocovariance function which must be nonnegative definite. In the nonstationary case \(X(t)\) is called AR\((p)\) process if it satisfies \(\sum_{k=0}^p a_t(k) X(t-k) = \varepsilon(t)\) where \(a_t(0)=1\) and \(\{\varepsilon_t\}\) is the innovation process with zero mean and variance \(\sigma_{\varepsilon}^2 (t) \geq 0\). It is proved that \(X(t)\) is AR\((p)\) process if and only if its PACF satisfies \(\beta(t, t-k)=0\) for \(k>p\) and all \(t\) whereas \(\beta(t,t-p) \neq 0\) for some \(t\).