Autoregressive Integrated Moving Average (ARIMA)

Key takeaways
* ARIMA is a statistical model for analyzing and forecasting univariate time series.
* It combines autoregression (AR), differencing to achieve stationarity (I), and moving averages (MA).
* The model is specified by three integers: p (AR order), d (degree of differencing), and q (MA order).
* ARIMA is effective for short-term forecasting but less reliable for long horizons or predicting turning points.
* Building an ARIMA model involves testing for stationarity, choosing d, and selecting p and q from autocorrelation patterns.

What is ARIMA?

ARIMA (Autoregressive Integrated Moving Average) models relationships in a time series by expressing current values in terms of past values and past forecast errors. Rather than modeling raw levels, ARIMA often models differences between observations to remove trends and stabilize variance, enabling better forecasting of sequential data such as economic indicators, sales, or asset prices.

Core components

• Autoregressive (AR) — current value regressed on its own lagged values (e.g., AR(1) uses one lag).
• Integrated (I) — differencing the series d times to remove trends and achieve stationarity.
• Moving Average (MA) — models the dependency between an observation and past forecast errors.

Key parameters (p, d, q)

• p: number of lagged observations (AR order).
• d: number of differences required to make the series stationary.
• q: number of lagged forecast errors in the prediction equation (MA order).

A zero for any parameter means that component is omitted (for example, ARIMA(1,0,0) is an AR(1) model).

Stationarity and differencing

Stationarity means the statistical properties of the series (mean, variance, autocorrelation) remain constant over time. Most economic and market series display trends or seasonality, so differencing (the I part) is used to remove these patterns. Choosing the minimal d that yields stationarity avoids over-differencing. Note that integrated models can transmit shocks forward for a long time, so past events may have prolonged effects on forecasts.

Building an ARIMA model — step by step

1. Collect and visualize the time series; check for trends and seasonality.
2. Test for stationarity (e.g., ADF or KPSS tests) and inspect autocorrelations.
3. Apply differencing until the series appears stationary; this determines d.
4. Examine the autocorrelation function (ACF) and partial autocorrelation function (PACF) to suggest q and p:
5. PACF cut-off after lag p suggests AR(p).
6. ACF cut-off after lag q suggests MA(q).
7. Fit candidate ARIMA(p,d,q) models and compare with information criteria (AIC, BIC) and residual diagnostics.
8. Validate forecasts on holdout data and refine as needed.

How ARIMA forecasting works

Statistical software automates identification and estimation: it tests stationarity, recommends differencing, and searches for p and q that minimize an information criterion. Forecasts are generated from the fitted ARIMA equations and typically accompanied by prediction intervals reflecting model uncertainty.

Pros and cons

Pros
* Requires only historical series values (no exogenous inputs required).
* Handles nonstationary series via differencing.
* Well understood and interpretable for short-term forecasting.

Cons
* Generally best for short horizons; performance degrades for long-term forecasts.
* Poor at anticipating structural breaks or turning points not present in historical data.
* Model selection can be subjective; computationally intensive for large model searches.
* Assumes linear relationships and may miss nonlinear dynamics.

AR vs MA — a quick comparison

• Autoregressive (AR): current value depends on prior values of the series.
• Moving average (MA): current value depends on past forecast errors (the noise terms).
  Combining them (ARIMA) captures both momentum from past values and structure in past shocks.

Practical tips

• Detrend or remove seasonality separately (or use seasonal ARIMA, SARIMA, when appropriate).
• Always check residuals for autocorrelation and heteroskedasticity.
• Compare ARIMA forecasts with simpler benchmarks (random walk, exponential smoothing).
• Consider combining ARIMA with other models or including exogenous variables if relevant.

Bottom line

ARIMA is a cornerstone technique for time series forecasting, particularly useful for short-term prediction when patterns are reasonably stable and driven by past values. Its transparency and solid theoretical basis make it a useful starting point, but practitioners should be cautious about long-horizon forecasts, structural changes, and nonlinearity—complement ARIMA with other methods where appropriate.