ts1115

ts1115

Define

\(\hat{x}_{t-1} = E(x_{t-1}|Z_{t-1}=z_{t-1})\)

\(P_{t-1} = E[(x_{t-1} - \hat{x}_{t-1})(x_{t-1} - \hat{x}_{t-1})^T|Z_{t-1}=z_{t-1}]\)

\(\hat{x}_{t}^- = E(x_{t}|Z_{t-1}=z_{t-1})\)

\(P_{t}^- = E[(x_{t} - \hat{x}_{t}^-)(x_{t} - \hat{x}_{t}^-)^T|Z_{t-1}=z_{t-1}]\)

\(\omega_{t-1}\) is independent of \(x_{t-1}\) ,
\(v_t\) is independent of \(x_t\)

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\(P_{t}^- = E[(x_{t} - \hat{x}_{t}^-)(x_{t} - \hat{x}_{t}^-)^T|Z_{t-1}=z_{t-1}]\)

\(~~~~~= AP_{t-1}A^T+2E[A(x_{t-1} - \hat{x}_{t-1})\omega_{t-1}^T|Z_{t-1}=z_{t-1}] + E[\omega_{t-1} \omega_{t-1}^T |Z_{t-1}=z_{t-1}]\)

1. \(E[A(x_{t-1} - \hat{x}_{t-1})\omega_{t-1}^T|Z_{t-1}=z_{t-1}]\)

\(E[A(x_{t-1} - \hat{x}_{t-1})\omega_{t-1}^T|Z_{t-1}=z_{t-1}]\)

\(=E(Ax_{t-1}\omega_{t-1}^T|Z_{t-1}=z_{t-1}) - E(A\hat{x}_{t-1}\omega_{t-1}^T|Z_{t-1}=z_{t-1})\)

\(=A~E(x_{t-1}|Z_{t-1}=z_{t-1})~E(\omega_{t-1}^T|Z_{t-1}=z_{t-1}) - A~\hat{x}_{t-1}~E(\omega_{t-1}^T|Z_{t-1}=z_{t-1})\)

\(=A~\hat{x}_{t-1}~E(\omega_{t-1}^T|Z_{t-1}=z_{t-1}) - A~\hat{x}_{t-1}~E(\omega_{t-1}^T|Z_{t-1}=z_{t-1})\)

\(=0\)

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2. \(E[\omega_{t-1} \omega_{t-1}^T |Z_{t-1}=z_{t-1}]\)

\(Var(\omega_{t-1}|Z_{t-1})\) \(=E(\omega_{t-1}\omega_{t-1}^T|Z_{t-1}) - [E(\omega_{t-1}|Z_{t-1})]^2\)

\(............\)

\(\omega_{t-1}|z_{t-1} \sim N(~\hat{x}_t - A\hat{x}_{t-1}~,~Var(\omega_{t-1}|z_{t-1})~)\)

\(............\)