Autoregressive Model

Study guide: Exam MAS-II

An autoregressive model of order $p$ , AR( $p$ ), regresses a Time Series on its own past $p$ values. It is the "AR" in ARIMA and the natural model for a series that reverts to a mean at a rate set by $\phi$ .

$Y_t = c + \phi_1 Y_{t-1} + \cdots + \phi_p Y_{t-p} + \varepsilon_t$

$\phi(B)Y_t = c + \varepsilon_t, \qquad \phi(B) = 1 - \phi_1 B - \cdots - \phi_p B^{p}$

AR(1): $\mathrm{Corr}(Y_t, Y_{t-k}) = \phi^{k}$ , mean $\mu = c/(1-\phi)$ , variance $\sigma^2/(1-\phi^2)$
Stationarity requires all roots of $\phi(B) = 0$ outside the unit circle; for AR(1) that is $\lvert\phi\rvert < 1$ , and $\phi = 1$ is a Random Walk
Identification: the ACF tails off geometrically, the PACF cuts off after lag $p$
$\phi$ near 1 means slow mean reversion and long memory; $\phi$ near 0 means the series is nearly White Noise. Negative $\phi$ gives an alternating, oscillating series
Forecasts decay geometrically back to the mean: $\hat Y_{t+h} = \mu + \phi^{h}(Y_t - \mu)$

AR(1) Mean, Variance and Two-Step Forecast An annual loss ratio follows $Y_t = 0.24 + 0.70Y_{t-1} + \varepsilon_t$ with $\sigma = 0.05$ , and $Y_{20} = 0.92$ . Find the long-run mean, the stationary variance, and the two-step forecast.

Answer $\mu = \frac{c}{1-\phi} = \frac{0.24}{0.30} = 0.80 \qquad \mathrm{Var}(Y_t) = \frac{0.05^{2}}{1 - 0.49} = 0.0049$ $\hat Y_{21} = 0.24 + 0.70(0.92) = 0.884$ $\hat Y_{22} = 0.24 + 0.70(0.884) = 0.859$ Equivalently $\hat Y_{22} = 0.80 + 0.70^{2}(0.92 - 0.80) = 0.80 + 0.0588 = 0.859$ ✓ — the deviation from the mean shrinks by a factor $\phi$ each step.