SARIMAX – Forecasting with External Influences

Until now, all our models depended only on past values of the same time series. That works well when history alone explains the future.

But in the real world, outcomes are rarely isolated. They are influenced by external forces.

This lesson focuses on one powerful idea:

What if we include outside information to improve forecasting?

Real-World Scenario

Imagine forecasting electricity consumption for a city.

Consumption depends on:

• Past electricity usage
• Season (summer vs winter)
• Temperature

If we ignore temperature, forecasts will be inaccurate during heatwaves or cold spells.

This is exactly where SARIMAX shines.

What SARIMAX Adds

SARIMAX is SARIMA with one extra component:

X → eXogenous variables

These are external inputs that influence the target series but are not predicted by it.

Examples of exogenous variables:

• Temperature
• Marketing spend
• Promotions
• Holidays
• Fuel prices

Model Structure

SARIMAX is written as:

SARIMAX(p, d, q)(P, D, Q, m, X)

Everything works like SARIMA, but now the model also learns:

How external variables move the series up or down

Example Data: Electricity Demand + Temperature

We’ll work with a simulated but realistic dataset:

• Electricity demand grows over time
• Seasonal pattern every year
• Higher temperature increases demand

Create the Time Series and External Variable

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(3)
time = np.arange(120)

trend = time * 0.4
seasonal = 20 * np.sin(2 * np.pi * time / 12)
temperature = 25 + 10 * np.sin(2 * np.pi * time / 12)
noise = np.random.normal(0, 4, 120)

demand = trend + seasonal + (temperature * 0.8) + noise

plt.figure(figsize=(9,4))
plt.plot(demand)
plt.title("Electricity Demand Over Time")
plt.show()

This is the actual demand pattern:

What you should notice:

• Upward long-term trend
• Clear seasonal cycles
• Stronger peaks during hotter periods

Visualizing the External Variable

Now look at the temperature pattern itself.

plt.figure(figsize=(9,4))
plt.plot(temperature)
plt.title("Temperature Over Time")
plt.show()

Temperature clearly follows a seasonal cycle. This makes it a strong candidate as an external predictor.

Why SARIMA Alone Is Not Enough

SARIMA can capture:

• Trend
• Seasonality

But it cannot answer:

How much does temperature increase demand?

SARIMAX learns that relationship directly.

How SARIMAX Uses External Information

Internally, the model:

• Fits trend and seasonality
• Adjusts predictions based on temperature values

When temperature rises, forecasts rise. When it falls, forecasts fall.

This makes forecasts more responsive to real conditions.

Conceptual Forecast Comparison

Below is how forecasts behave when external data is included.

Interpretation:

• Without temperature → smooth seasonal curve
• With temperature → peaks adapt to heat changes

Where SARIMAX Is Used

• Energy demand forecasting
• Sales forecasting with promotions
• Traffic prediction with weather
• Supply chain planning

Important Rules

• External variables must be known in advance
• They must logically influence the target
• Garbage external data worsens forecasts

Practice Questions

Key Takeaways

• SARIMAX extends SARIMA with external signals
• It improves realism and responsiveness
• Perfect for business-driven forecasting

Next, we’ll evaluate models properly using diagnostics and residual analysis.