Login Create Account

ARX Time Series Forecasting

Time Series - Intro

Forecast marketing metrics using ARX (AutoRegressive with eXogenous variables). Upload time series data, include external predictors like ad spend, and generate forecasts. This is a simplified introduction to time series - no moving average terms!

👨‍🏫 Professor Mode: Guided Learning Experience

New to time series forecasting? Enable Professor Mode for step-by-step guidance through building your first ARX model with AR patterns, differencing, and external predictors!

TEST OVERVIEW & EQUATIONS

ARX (AutoRegressive with eXogenous variables) is a simplified time series model perfect for learning forecasting basics. It combines past values (AR), differencing to remove trends, and external predictors like advertising spend or temperature.

ARX Model (regression with autoregressive errors): $$ Y_t = \sum_{j=1}^{k} \beta_j X_{j,t} + u_t \qquad\text{where}\qquad (1 - \phi_1 B - \cdots - \phi_p B^p)(1 - B)^d\, u_t = \varepsilon_t $$

where \(Y_t\) is the outcome at time \(t\), \(B\) is the backshift operator, \(d\) is the differencing order, \(\phi\) are AR coefficients, and \(\beta_j\) are coefficients for exogenous predictors \(X_j\). The persistence acts on the regression error \(u_t\), not on \(Y_t\) directly, so for \(p=1\) the one-step forecast is \(\hat{Y}_{t+1} = \beta X_{t+1} + \phi\,(Y_t - \beta X_t)\) — the predicted level plus the carried-over part of today's miss. That is not the same as \(\phi Y_t + \beta X_{t+1}\): if you reason with that second form your arithmetic will not reproduce the forecast shown below.

💡 What about MA(q)?

ARX uses only AR(p) and differencing(d). The "MA(q)" moving average component is removed to simplify learning. Once you master ARX, graduate to the ARIMAX tool to learn MA terms!

Key Concepts
  • AR (p): Autoregressive terms capture how past values influence current values.
  • Differencing (d): Removes trends to achieve stationarity (usually d=0 or d=1).
  • X (exogenous): External predictors like ad spend, promotions, or temperature.
📈 Why differencing? Stationarity in one picture

A series is stationary when its statistical behaviour — especially its mean — doesn't drift over time. Most raw business series (sales, traffic, spend) trend, so their mean keeps climbing; a model fit to them spends all its effort chasing the trend instead of the signal you care about.

Differencing (subtracting each value from the one before it, controlled by d) removes that trend, leaving fluctuations around a constant mean that the AR terms can actually learn from. Here's the same series before and after one round of differencing:

Top: the original series trends upward — non-stationary. Bottom: after differencing (d = 1) it hovers around a flat mean — stationary, and ready for forecasting.

When to Use ARX

Use ARX when you have a time series outcome (sales, traffic, conversions) influenced by external factors you can measure. Common marketing applications include:

  • Sales forecasting with advertising spend as a predictor
  • Website traffic prediction with campaign activity
  • Revenue forecasting with pricing or promotion indicators

Yes/no outcome instead of a number? (a stockout, a sell-out, a target hit) Use Dynamic Logistic Regression, which puts last period's outcome in as a predictor.

📚

Use a Case Study

Use presets to auto-load realistic marketing time series data with external predictors already configured.

📤

Upload Your Data

Upload a CSV with a date/time column, outcome column, and optional exogenous predictor columns.

Drag & Drop CSV file (.csv, .tsv, .txt, .xls, .xlsx)

First row = column headers. Include date, outcome, and predictor columns.

No file uploaded.