Chapter 02 · Autoregressive moving average

ARMA, when the past is the model

Past values, past errors, or both. ARMA models a stationary series as a linear combination of its own history. The right order falls out of the autocorrelation pattern, and the wrong tool falls out the moment the series is no longer stationary.

Lag operator notation

Lag$L y_t = y_{t-1}$, so $L^k y_t = y_{t-k}$

AR(p)$(1 - \phi_1 L - \cdots - \phi_p L^p)\,y_t = \varepsilon_t$

MA(q)$y_t = (1 + \theta_1 L + \cdots + \theta_q L^q)\,\varepsilon_t$

ARMA(p,q)$\phi(L)\,y_t = \theta(L)\,\varepsilon_t$, both polynomials applied to the same series

Stationaryonly when the roots of $\phi(L)$ lie outside the unit circle. ARMA assumes this. ARIMA exists because not every series obliges.

Pick a stationary series, pick an order

Eight scenarios chosen to make the diagnostic plots talk. Watch the autocorrelation cut off, and let it tell you which order to fit.

Series Model order (p, q)

Pick a series and an order

Selection appears here

ObservedFittedForecast

ACF

PACF

AIC and BIC across candidate orders

Order (p, q)	AIC	BIC	RMSE

Verdict

Pick a series and a model order to see the verdict.

Five things worth knowing

Click to reveal the diagnostics behind the diagnostics.

How does PACF tell you the AR order?

The partial autocorrelation at lag k measures the correlation between y_t and y_{t−k} after stripping out everything in between. For a true AR(p) process, the PACF cuts off sharply after lag p, while the ACF tails off geometrically. So a PACF that drops to zero at lag 2 says AR(1) is right and you stop there. Trust the cutoff, not your hopes for a richer model.

How does ACF tell you the MA order?

The roles flip. For a true MA(q) process, the ACF cuts off sharply after lag q, while the PACF tails off. ACF that drops to zero at lag 2 says MA(1). If neither plot cuts off cleanly, you are looking at ARMA, and AIC will pick the order for you.

What if both ACF and PACF tail off?

That is the textbook ARMA signature. Neither AR nor MA alone fits; you need both. Try ARMA(1,1) first, compare AIC against ARMA(2,1) and ARMA(1,2). Tailing off in both plots is your invitation to the mixed model.

Why can AIC be compared here, but not in ARIMA?

AIC is comparable across models fitted to the same data, in the same units, on the same scale. ARMA models all fit the original series, so the likelihoods are on equal footing. ARIMA models fit different orders of differenced series, so their likelihoods are computed on different data. That comparability is the gift, and the curse.

What does Bartlett's confidence band mean?

The dashed lines on the ACF and PACF plots at ±1.96 / √n are the rough 95% range under the null hypothesis of white noise. Bars inside the band are statistically indistinguishable from random sampling fluctuation; bars outside are signal. Read everything against the band.