Common mistake 1 with ARIMA / SARIMA

Over-differencing. If the ACF of the differenced series is strongly negative at lag 1, you differenced once too often.

Common mistake 2 with ARIMA / SARIMA

Choosing the model with the best in-sample fit. More parameters always fit the past better and usually forecast worse - use the Forecast accuracy module.

Common mistake 3 with ARIMA / SARIMA

Forecasting far beyond the data. ARIMA has no economics in it; it just extrapolates a pattern.

Common mistake 4 with ARIMA / SARIMA

Comparing AIC across models fitted to different differencing orders. They are fitted to different data, so the numbers are not comparable and the smaller one is not the better model.

EconStatsCraft › ARIMA / SARIMA

ARIMA / SARIMA

Box–Jenkins modelling with optional seasonal terms, automatic order selection by BIC, residual white-noise tests and a forecast with intervals. SARIMA is the same method: set the seasonal period S below - 4 for quarterly, 12 for monthly - or leave the differencing box ticked and it is detected from your data.

What it needs

Series

The intuition

ARIMA says the future is a weighted blend of where the series has recently been (AR) and the surprises it has recently had (MA), after differencing away any wandering trend (I). It contains no economics whatsoever - which is its weakness and, as a forecasting benchmark, exactly its strength.

When to use it

Forecasting a single series from its own history, with no explanatory variables. It is the benchmark every other forecast is judged against.

Where the estimator comes from

Using the lag operator L, write \phi(L)(1-L)^d y_t=\theta(L)\varepsilon_t where \phi(L)=1-\phi_1 L-\dots-\phi_p L^p and \theta(L)=1+\theta_1 L+\dots+\theta_q L^q. Estimation maximises the Gaussian likelihood, usually via the Kalman filter on a state-space form, which handles missing values and exact initialisation cleanly. Stationarity requires the roots of \phi(L) to lie outside the unit circle; invertibility requires the same of \theta(L).

The model

\phi(L)\,(1-L)^{d}\,y_t=\theta(L)\,\varepsilon_t,\qquad \phi(L)=1-\phi_1L-\dots-\phi_pL^{p},\quad \theta(L)=1+\theta_1L+\dots+\theta_qL^{q}

What it assumes

After differencing d times, the series is stationary.
The residuals are white noise - this is what Ljung–Box checks.
The structure is stable over the sample.
The series has no deterministic breaks. ARIMA fits one set of parameters to the whole sample, so a policy change or a regime shift is absorbed into the error term and the forecast averages across both regimes.
Outliers have been dealt with deliberately. A single extreme observation moves the estimated MA terms substantially, and the model has no way to say it was a one-off.

Statistical properties

ML estimates are consistent and asymptotically Normal for a correctly specified, stationary and invertible model.
AIC ≈ −2logL + 2k targets forecast accuracy; BIC ≈ −2logL + k·log(n) targets finding the true model and is more parsimonious.
Every ARMA has an infinite MA representation, which is what generates the impulse response.
Over-differencing is not harmless: it inflates the variance and shows up as a large negative MA(1) coefficient.
Forecast intervals assume the parameters are known and the errors Normal, so they are systematically too narrow in practice.

How to read the output

AIC and BIC compare models: lower is better. BIC punishes extra parameters harder, so it picks simpler models.
Ljung–Box on residuals: a LARGE p-value is good - it means nothing predictable is left.
Forecast intervals widen with the horizon because uncertainty compounds.
An ARIMA with d = 1 forecasts towards a flat line; with a constant and d = 0 it forecasts towards the mean.
The constant means different things at different d. At d = 0 it is the series mean; at d = 1 it is the drift per period; at d = 2 there is usually none at all, because a quadratic trend extrapolated is rarely wanted.
Confidence intervals widen with the horizon in a shape determined by the model, and they assume the model is correct. They are the uncertainty WITHIN the model, not the uncertainty about it - which is usually larger.
AIC and BIC are comparable only across models fitted to the same differenced series. An ARIMA(1,1,1) and an ARIMA(1,0,1) are fitted to different data and their information criteria cannot be compared at all.
A root close to the unit circle - an AR coefficient near 1 or an MA coefficient near -1 - means the differencing choice is on a knife edge, and the forecast will change character if it tips.

A worked reading

On the quarterly growth series the unit-root step finds it already stationary in levels, so d = 0, and the BIC search over 22 orders selects ARIMA(1,0,0)×(0,0,0)[4]: ar.L1 = 0.8158 (SE 0.0426, z = 19.1) with a constant of 0.3696, BIC = 203.885. Ljung-Box gives p = 0.4841 and Jarque-Bera p = 0.9025, so nothing is left in the residuals to model. Because d = 0 the forecasts converge back to the series mean - 2.38 at the first step, easing towards it thereafter - rather than continuing in a straight line. Had the tests chosen d = 1 instead, the same forecasts would have converged to a constant GROWTH RATE and the picture would look quite different; which of the two you get is decided by the unit-root step, not by the ARIMA.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Ljung–Box on residuals rejects	Structure left unmodelled	Increase p or q; check for seasonality
Large negative MA(1) after differencing	Over-differenced	Reduce d by one
AR root very close to 1	Under-differenced	Increase d, or test for a unit root again
Residual ACF spikes at lag 12 (monthly)	Seasonality not modelled	Add seasonal terms with S = 12
Forecast is a flat line	Normal for d ≥ 1 with no drift	Add a trend if the series genuinely drifts
Model fits beautifully but forecasts badly	Over-fitting	Prefer BIC; verify with the Forecast accuracy module

Common mistakes

Over-differencing. If the ACF of the differenced series is strongly negative at lag 1, you differenced once too often.
Choosing the model with the best in-sample fit. More parameters always fit the past better and usually forecast worse - use the Forecast accuracy module.
Forecasting far beyond the data. ARIMA has no economics in it; it just extrapolates a pattern.
Comparing AIC across models fitted to different differencing orders. They are fitted to different data, so the numbers are not comparable and the smaller one is not the better model.

Where to read this properly

Box & Jenkins (1970)
Hamilton (1994), Time Series Analysis, Princeton University Press, chapters 4-5
Hyndman and Athanasopoulos (2021), Forecasting: Principles and Practice, 3rd ed., OTexts, chapter 9 (ARIMA models)

Related Time series - one series

Time-varying coefficients (Kalman) · Unit root tests · Correlogram (ACF / PACF) · Rolling-origin cross-validation · Theta method · Unobserved components · STL forecasting · Forecast combination · Markov regime switching · Change frequency (resample) · ARCH / GARCH volatility · Decompose & filter · Spectral analysis · Exponential smoothing · Forecast accuracy

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ARIMA / SARIMA

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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