Discussion
Given two time series,
\(X_{t} = 0.3\,X_{t-1} - 0.2\,X_{t-2} + Z_{t}\)
\(Y_{t} - 0.4\,Y_{t-1} = Z_{t} + 1.2\,Z_{t-1}\)
where \(\{Z_{t}\}\) iid WN(0, \(\sigma^{2}\)), which of the following statements is true ?
Answer: C
The true statement is that \(\{Y_{t}\}\) is causal, but not invertible.
\(Y_{t}\) is ARMA(1, 1) with AR polynomial \(1 - 0.4z\), whose root \(z = 2.5\) lies outside the unit circle, so it is causal. Its MA polynomial \(1 + 1.2z\) has root \(z = -1/1.2 \approx -0.83\), inside the unit circle, so it is not invertible.
\(X_{t}\) is AR(2) with polynomial \(1 - 0.3z + 0.2z^{2}\). Its roots are complex with \(|z|^{2} = 1/0.2 = 5\), so \(|z| = \sqrt{5} \gt 1\) and \(X_{t}\) is causal; a pure AR process is always invertible. So B and D are false, and A has Y's properties reversed.
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