ARIMA

Study guide: Exam MAS-II

An ARIMA $(p, d, q)$ model (Autoregressive Integrated Moving Average) is the standard parametric framework for Time Series modeling. It combines autoregressive (AR), integrated (I) (differencing for stationarity), and moving average (MA) components.

$\phi(B)\nabla^d Y_t = \theta(B)\varepsilon_t$

$\phi(B) = 1 - \phi_1 B - \cdots - \phi_p B^p$

$\theta(B) = 1 + \theta_1 B + \cdots + \theta_q B^q$

$\varepsilon_t \stackrel{\text{iid}}{\sim} N(0, \sigma^2)$

$B Y_t = Y_{t-1} \text{ (backshift operator)}$

Components:

Parameter	Meaning
$p$	Order of autoregressive component — uses $p$ lags of $Y_t$
$d$	Degree of differencing — applied $d$ times to achieve stationarity
$q$	Order of moving average component — uses $q$ lags of the error $\varepsilon_t$

Special cases:

Model	Meaning
ARIMA $(p,0,0)$	Pure AR $(p)$ : $Y_t = \phi_1 Y_{t-1} + \cdots + \phi_p Y_{t-p} + \varepsilon_t$
ARIMA $(0,0,q)$	Pure MA $(q)$ : $Y_t = \varepsilon_t + \theta_1\varepsilon_{t-1} + \cdots + \theta_q\varepsilon_{t-q}$
ARIMA $(0,1,0)$	Random walk: $Y_t = Y_{t-1} + \varepsilon_t$

Model identification using ACF/PACF:

AR $(p)$ : ACF tails off, PACF cuts off after lag $p$
MA $(q)$ : ACF cuts off after lag $q$ , PACF tails off
ARMA: both tail off

Seasonal ARIMA $(p,d,q)(P,D,Q)_s$ adds seasonal AR, differencing, and MA terms at lag $s$ .

AR(1) Forecast An AR(1) model is fitted: $Y_t = 10 + 0.7 Y_{t-1} + \varepsilon_t$ . The most recent observation is $Y_{100} = 15$ . Compute the one-step-ahead forecast $\hat{Y}_{101}$ .

Answer $\hat{Y}_{101} = 10 + 0.7(15) = 10 + 10.5 = 20.5$

Identifying an ARIMA Model from ACF/PACF After first-differencing an insurance loss series, the ACF shows a significant spike at lag 1 and cuts off, while the PACF tails off gradually. What ARIMA model is suggested?

Answer ACF cuts off after lag 1 → MA(1). PACF tails off → no pure AR. $d = 1$ (one difference was needed). The identified model is ARIMA $(0, 1, 1)$ .