ARIMA

ARIMA (AutoRegressive Integrated Moving Average)

ARIMA is a classical statistical model for forecasting a time series from its own history. Unlike a normal table where row order may not matter, time-series observations arrive in sequence, so yesterday, last week, or last month can contain useful information about what comes next.

A Tiny Example First

Suppose monthly demand is:

The level is rising, but the changes are only +8 and +9. ARIMA can model a series after differencing and combine that with lagged values and lagged errors.

Common time-series examples include sales, demand, traffic, sensor readings, financial quantities, website activity, and other measurements collected repeatedly over time. ARIMA is most appropriate when the series can be represented reasonably well by a linear non-seasonal ARIMA structure; strongly seasonal series usually need seasonal terms such as SARIMA.

What Does ARIMA(p, d, q) Mean?

The Three Parts of ARIMA

1. AR — AutoRegression

The AR part relates the current value to earlier values of the series. An AR(p) model can be written as:

2. I — Integrated / Differencing

Differencing replaces the original levels with changes between observations. First differencing is:

3. MA — Moving Average

In ARIMA, Moving Average does not mean a rolling average of recent observations. The MA part models the current value using previous innovation/error terms.

Stationarity and Why d Matters

A stationary process has statistical behavior that does not systematically drift over time. In practical ARIMA modeling, we often difference a non-stationary level series so that the remaining process is more suitable for ARMA-style modeling.

A Practical ARIMA Workflow

ADF Test — What the p-value Actually Means

The Augmented Dickey-Fuller test has a null hypothesis that the series contains a unit root. A small p-value, such as below a chosen 5% significance level, is evidence for rejecting that null under the test specification. It is better to say this than to say “p < 0.05 proves stationarity.”

ACF and PACF

These are identification heuristics, not mechanical rules that always reveal the correct p and q for a mixed ARIMA model.

AIC — Useful, but Not a Forecast Guarantee

AIC balances model fit with model complexity. When several candidate models are fitted to the same data under comparable assumptions, a lower AIC is preferred by that criterion. But the lowest AIC does not guarantee the best future forecast, so chronological validation still matters.

Python: A Self-Contained ARIMA Example

This example creates a reproducible non-seasonal series, keeps the final 12 months as an untouched time-ordered test period, checks the effect of first differencing, compares a small set of ARIMA orders by AIC, and evaluates the forecast.

import itertools
import warnings
import numpy as np
import pandas as pd
from statsmodels.tsa.arima.model import ARIMA
from statsmodels.tsa.stattools import adfuller

# 1. Create a reproducible non-seasonal time series
rng = np.random.default_rng(42)
n = 120
noise = rng.normal(0, 1.2, n)

values = np.zeros(n)
values[0] = 50
change = 0.0

for t in range(1, n):
    change = 0.55 * change + 0.25 + noise[t]
    values[t] = values[t - 1] + change

series = pd.Series(
    values,
    index=pd.date_range("2015-01-01", periods=n, freq="MS"),
    name="value"
)

# 2. Chronological split: last 12 months are test data
train = series.iloc[:-12]
test = series.iloc[-12:]

# 3. ADF before and after first differencing
adf_level = adfuller(train, autolag="AIC")
adf_diff = adfuller(train.diff().dropna(), autolag="AIC")

print("Level ADF p-value:", round(adf_level[1], 4))
print("Differenced ADF p-value:", round(adf_diff[1], 4))

# 4. Small teaching search over p and q, keeping d=1
candidates = list(itertools.product(range(3), [1], range(3)))

best_aic = np.inf
best_order = None

for order in candidates:
    try:
        # Some deliberately tested candidates may emit start-value/convergence warnings.
        # We compare the models that successfully fit.
        with warnings.catch_warnings():
            warnings.simplefilter("ignore")
            fitted = ARIMA(train, order=order).fit()

        if fitted.aic < best_aic:
            best_aic = fitted.aic
            best_order = order
    except Exception:
        pass

print("Best order by AIC:", best_order)
print("Best AIC:", round(best_aic, 2))

# 5. Fit selected model and forecast the test period
model = ARIMA(train, order=best_order).fit()
forecast = model.forecast(steps=len(test))

mae = np.mean(np.abs(test - forecast))
rmse = np.sqrt(np.mean((test - forecast) ** 2))

print("First 3 forecasts:", forecast.iloc[:3].round(2).tolist())
print("MAE:", round(mae, 2))
print("RMSE:", round(rmse, 2))

ARIMA, SARIMA and Exogenous Variables

ARIMA vs Other Forecasting Approaches

Avoid treating one family as universally better. ARIMA is a useful interpretable baseline for many linear time-series problems, while other statistical or machine-learning approaches may be stronger when seasonality, nonlinear relationships, many external predictors, multiple related series, or large-scale data become central.

Common ARIMA Mistakes

Quick FAQ