Linear Time Series Models for Stationary data - Feweb

AR(∞) representation
A simple idea for linear forecasting is to (auto)regress yt on its own
lags, employing all the (partial) linear correlations of yt with its past.
Infinite order Autoregressive representation:
yt = E[yt|Yt−1] + εt
E[yt|Yt−1] = α + π1yt−1 + π2yt−2 + . . . (7.2)
E[(yt − µ)|Yt−1] = π1(yt−1 − µ) + π2(yt−2 − µ) + . . . + εt
where α is a constant so that
(1 − πk) −1 α = µ is the constant perfectly predictable
deterministic part,
π1(yt−1 − µ) + (π2 − µ)yt−2 + . . . is the (linearly) predictable
stochastic part, and
εt is the unpredictable innovation part or prediction error, satisfying
the White Noise condition.
Exercise (3): Using stationarity of yt, show E(yt) = µ = α
1− πk .
Chapter 7.1 Heij et al, TI Econometrics II 2006/2007 – p. 15/24