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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!

OVERVIEW & WHAT THIS TOOL SOLVES

When you forecast weekly sales or daily store traffic, two forces are at work simultaneously: controllable drivers (like ad spend, price promotions, or temperature) and time-series momentum (where a surprisingly strong week carries over into the next). ARX Time Series Forecasting provides a clean, approachable introduction to time-series forecasting by combining external marketing predictors (X), trend-removing differencing (d), and AutoRegressive persistence (AR)—without the added complexity of Moving Average terms.

🎯 Questions It Answers

  • How strongly do our external marketing drivers (ad spend, promos, weather) move sales once we account for week-to-week persistence?
  • Does our series have an upward or downward trend that requires differencing (d = 1) to become stationary?
  • What is our multi-period forecast when we combine planned future marketing inputs with carried-over momentum?

📂 What Data You Need

  • Time-Ordered CSV: Chronological rows (daily, weekly, or monthly) with no gaps.
  • 1 Continuous Outcome (Yt): Numeric series you want to forecast (for 0/1 yes-or-no outcomes, use Dynamic Logit).
  • External Predictors (Xj,t): Complete numeric or 0/1 indicator columns for controllable drivers.

💡 What You'll Discover

  • Driver Slopes & AR Persistence: Exact impact of each external lever alongside autoregressive carryover weights.
  • Interactive Stationarity & ACF Diagnostics: Visual proof of how differencing removes drift and whether residuals are clean noise.
  • Scenario Forecast Planner: Multi-step ahead predictions with 95% confidence bands under custom future input plans.
🔍 Key Nuances: Why Differencing Matters (Stationarity in One Picture) & ARX vs. ARIMAX

A series is stationary when its statistical behavior—especially its mean—doesn't drift over time. Most raw business series (sales, traffic, spend) trend upward or downward; differencing (subtracting each value from the one before it, controlled by d) removes that drift so the AR terms can learn from fluctuations around a flat baseline:

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

  • What About MA(q)? ARX uses only AR(p) and differencing(d) so you can master core time-series mechanics first. Once comfortable, graduate to the ARIMAX Tool to add Moving Average terms.
  • Yes/No Outcome Instead of a Number? For binary time series (stockout, sell-out, target hit), use Dynamic Logistic Regression.
📐 Under the Hood: Model Equations & Variable Definitions
1. ARX Model (the outcome on its own lags plus the predictors) & One-Step Forecast
\[ Y_t = c + \phi_1 Y_{t-1} + \cdots + \phi_p Y_{t-p} + \sum_{j=1}^{k} \beta_j X_{j,t} + \varepsilon_t \] \[ \text{For } p = 1,\, d = 0\text{: } \quad \hat{Y}_{t+1} = c + \phi\,Y_t + \beta X_{t+1}, \qquad \text{long-run effect of } X = \frac{\beta}{1 - \phi_1 - \cdots - \phi_p} \]
💬 Read it in plain English: This period's outcome is a share of last period's (φ, the carry-over) plus what this period's predictors add (β, the same-period effect). Because the outcome carries over, a lasting change in a predictor keeps paying in later periods until it settles at β ÷ (1 − Σφ). With differencing (d ≥ 1) the outcome and the predictors are both differenced first. The model is fitted by least squares.
Yt, Xj,t
Outcome Series & Exogenous Predictors YOUR DATA ⓘ
Outcome at time t (Yt) and external marketing or environmental predictors (Xj,t).
βj
Same-Period Driver Coefficients ESTIMATED ⓘ
Expected change in Yt in the same period per 1-unit change in external predictor Xj,t; the long-run effect is βj ÷ (1 − Σφ).
φ1 … φp
Autoregressive (AR) Carry-Over Weights ESTIMATED ⓘ
How much of last period's outcome (φ1, ar.L1) and earlier periods' carries into this one.
c, d, εt
Constant, Differencing & Noise OUTPUT ⓘ
The constant c (const), the differencing order d applied to the outcome and the predictors, and the unexplained one-step error εt.

DATA SOURCE

📚

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.