ARIMA Family: Comparison Hub

The ARIMA family is the workhorse of classical linear time-series modeling. Every member has the form
$ϕ (B) (1 - B)^{d} X_{t} = θ (B) W_{t}, W_{t} \sim WN (0, σ_{w}^{2}),$
where $ϕ$ and $θ$ are polynomials in the backshift operator, $d$ is the differencing order, and the four standard members specialize this template:
$[[2.5.2 AR(p) Process|AR(p)]]: [[2.5.3 MA(q) Process|MA(q)]]: [[2.5.4 ARMA(p,q) Process|ARMA(p, q)]]: [[2.5.7 ARIMA(p,d,q) Process|ARIMA(p, d, q)]]: ϕ (B) X_{t} = W_{t} X_{t} = θ (B) W_{t} ϕ (B) X_{t} = θ (B) W_{t} ϕ (B) (1 - B)^{d} X_{t} = θ (B) W_{t} (q = d = 0) (p = d = 0) (d = 0) (d \geq 1)$
Decision rule (the whole reason this hub exists):

Plot the series. If the mean drifts, difference until it doesn’t ( $d = 1$ usually suffices). All later steps use the differenced series.

Plot the sample ACF and PACF.

PACF cuts off at lag $p$ , ACF tails off ⇒ AR( $p$ ).

ACF cuts off at lag $q$ , PACF tails off ⇒ MA( $q$ ).

Both tail off ⇒ ARMA( $p$ , $q$ ); pick small $(p, q)$ candidates and choose by AIC/BIC.

After fitting, check the residuals: their ACF should be flat (Ljung-Box not significant). If structure remains, increase $(p, q)$ or revisit differencing.

Causality and invertibility (see 2.5.5 Causality and Invertibility) are imposed automatically by every standard fitting routine: roots of $ϕ (z)$ and $θ (z)$ must lie strictly outside the unit circle.

Intuition

Everything in the ARIMA family is built from three primitives: the backshift operator $B$ , the white-noise process $W_{t}$ , and the rational filter $θ (B) / ϕ (B)$ . The differencing factor $(1 - B)^{d}$ is itself a special case of an AR factor, with all roots at $z = 1$ (on the unit circle, hence non-stationary). This unified algebraic view is the conceptual reason a single framework, Box and Jenkins (1970), can describe forecasting models for industrial production, airline traffic, asset returns, and weather all at once.

The variants trade along two axes. The first is stationarity: ARMA assumes a stationary mean; ARIMA handles drifting means by differencing them out. The second is memory shape: AR processes give long, geometric memory through feedback; MA processes give finite, sharp memory through fixed-window averaging; ARMA mixes both. The ACF/PACF identification rules read these signatures off the data without needing to fit candidate models, which is the great economy of the Box-Jenkins methodology.

Comparison Table

The lag-domain identification table is the central diagnostic for choosing among AR, MA, and ARMA on stationary data. After differencing (if needed) the table applies to the resulting series.

Member	Equation	Stationarity req.	Invertibility req.	ACF	PACF	When to reach for it
AR(p)	$ϕ (B) X_{t} = W_{t}$	Roots of $ϕ (z)$ outside unit circle	(Always)	Tails off (geometric / oscillating)	Cuts off after lag $p$	PACF shows a clean cutoff; long, smooth-decaying ACF
MA(q)	$X_{t} = θ (B) W_{t}$	(Always stationary)	Roots of $θ (z)$ outside unit circle	Cuts off after lag $q$	Tails off	ACF shows a clean cutoff; PACF tails
ARMA(p, q)	$ϕ (B) X_{t} = θ (B) W_{t}$	$ϕ$ roots outside	$θ$ roots outside	Tails off	Tails off	Both tail off; pick small $(p, q)$ by AIC/BIC
ARIMA(p, d, q)	$ϕ (B) (1 - B)^{d} X_{t} = θ (B) W_{t}$	$ϕ$ roots outside (after differencing)	$θ$ roots outside	(After differencing) per ARMA rules	(After differencing) per ARMA rules	Series shows trend or random walk; difference first, then identify $(p, q)$

Two helper concepts are pulled out into their own siblings because they apply across all four members:

2.2.6 Partial Autocorrelation Function: the diagnostic that cuts off for AR but not for MA/ARMA.
2.5.5 Causality and Invertibility: the unit-circle root conditions on $ϕ$ and $θ$ .

When to Use Which

The decision usually unfolds in three stages.

Stage 1: stationarity check. Plot the series. A visible upward or downward drift, a unit-root behavior (variance growing with time), or a clearly trending mean all point to ARIMA with $d \geq 1$ . A constant-mean series with stable variance points to ARMA with $d = 0$ . The Augmented Dickey-Fuller test or KPSS test can confirm. Default: $d = 1$ if differencing is needed; $d = 2$ only with strong evidence (an integrated random walk).

Stage 2: order identification. On the (possibly differenced) series, plot the ACF and PACF. A PACF that drops sharply to zero at lag $p$ with an ACF that tails off geometrically points to AR( $p$ ), the cleanest case. An ACF that drops sharply to zero at lag $q$ with a PACF that tails off points to MA( $q$ ). When both tail off, neither AR nor MA alone fits, and the choice is ARMA( $p$ , $q$ ) for small candidate orders, picked by information criteria.

Stage 3: residual check. After fitting, the residuals should look like white noise. Plot their ACF (should be inside $\pm 1.96/ T$ at all lags) and run a Ljung-Box test. Significant residual autocorrelation means the order chosen was too small or the wrong family was selected. If the model fits but residuals show variance clustering, the linear ARIMA framework is not enough. Move to GARCH (volatility models) or non-linear extensions.

In practice, the modern shortcut is auto.arima (R) or pmdarima.auto_arima (Python), which automates the search over $(p, d, q)$ using BIC and a unit-root test. The hand-tuned Box-Jenkins workflow is still essential for understanding why a model fits, especially when residuals refuse to whiten and the diagnostic table is the only guide back to the root cause.

tags	stats/time-series, type/framework, level/intermediate, status/complete
aliases	ARIMA, AR/MA/ARMA/ARIMA, Box-Jenkins models

Project Alexandria

Explorer

ARIMA Models

Intuition

Comparison Table

When to Use Which

See Also

Backshift Operator

AR(p) Process

MA(q) Process

ARMA(p,q) Process

Causality and Invertibility

ARIMA(p,d,q) Process