The ARIMA(p, d, q) model

An ARIMA model combines three components:

• AR(p): the current value depends on $p$ past values.
• I(d): the series is differenced $d$ times to achieve stationarity.
• MA(q): the current value depends on $q$ past forecast errors.

The general form after differencing:

\[\phi(L)\,\Delta^d y_t = \theta(L)\,\varepsilon_t, \qquad \varepsilon_t \sim \text{WN}(0, \sigma^2)\]

where $L$ is the lag operator, $\phi(L) = 1 - \phi_1 L - \cdots - \phi_p L^p$ is the AR polynomial, and $\theta(L) = 1 + \theta_1 L + \cdots + \theta_q L^q$ is the MA polynomial.

Stationarity requirement

The AR polynomial $\phi(L)$ must have all roots outside the unit circle for the process to be (weakly) stationary — a direct analogy to stable poles in a digital filter. This is why ARIMA and signal processing share so much mathematics.

Identification via ACF and PACF

Pattern	Suggests
ACF cuts off at lag $q$, PACF decays	MA(q)
ACF decays, PACF cuts off at lag $p$	AR(p)
Both decay slowly	ARMA — use information criteria

Connection to spectral analysis

The power spectral density of a stationary ARMA process is:

\[S(\omega) = \sigma^2 \left|\frac{\theta(e^{-i\omega})}{\phi(e^{-i\omega})}\right|^2\]

This connects time-series modelling directly to the Fourier transform and signal processing. An AR(1) process with $

\phi_1

< 1$ is a low-pass filter; MA processes shape the spectrum via their zero structure.