= Fibonacci-coefficient autoregression
{c}
{title2=$\psi_j=F_{j+1}\alpha^j$}
For $X_t=\alpha X_{t-1}+\alpha^2X_{t-2}+\varepsilon_t$, let $\varphi=(1+\sqrt5)/2$. The autoregressive <polynomial> factors as $(1-\alpha\varphi z)(1+\alpha z/\varphi)$. Its roots lie outside the closed <unit disk> exactly when $|\alpha|<1/\varphi$. In that causal region,
$$
X_t=\sum_{j\geq0}F_{j+1}\alpha^j\varepsilon_{t-j},\qquad
F_{j+1}=\frac{\varphi^{j+1}-(-1/\varphi)^{j+1}}{\sqrt5}.
$$
The coefficients follow from $\psi_0=1,\psi_1=\alpha$ and $\psi_j=\alpha\psi_{j-1}+\alpha^2\psi_{j-2}$. They are absolutely summable in the stated region. Causality makes future <white noise> orthogonal to the observation past, giving forecasts $\alpha X_T+\alpha^2X_{T-1}$ and $2\alpha^2X_T+\alpha^3X_{T-1}$ at horizons one and two.
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