Lecture 3 summary · January 29, 2027

ARMA models and recoverable innovations

The economic question

Observed memory can arise in the level recursion and the innovation sequence. Stationarity and invertibility answer different questions.

(1-\phi L)Y_t=(1+\theta L)\varepsilon_t,

\psi_0=1,

\psi_j=\phi^{j-1}(\phi+\theta)

Persistence is 0.6, moving-average parameter 0.2, and innovation variance one. Find the first three response weights and two-period error variance.

Worked reasoning

The response weights are one, 0.8, and 0.48. Two-period variance is one plus 0.64, or 1.64. Unobserved future innovations remain uncertain.

A short-sample ACF is treated as a unique verdict or an unstable inverse recursion is used to recover shocks.

Check polynomial roots, sign conventions, conditional initialization, and chronological diagnostics. Identify the estimator as a declared grid.