AR(1) and MA(1) Interactive Visualization

Generate a random AR(1) or MA(1) series and adjust the autoregressive coefficient φ or moving-average coefficient θ live to see how each parameter shapes memory, stationarity, and the resulting forecast path. The MA(1) forecast collapses to its mean after a single step, while the AR(1) forecast decays toward its long-run mean geometrically — a distinction the chart makes visible rather than merely asserted.

Interactive Teaching Tool for Understanding Autoregressive and Moving Average Models

📚 AR(1) - Autoregressive Model (Order 1)

y_t = c + φ·y_t-1 + ε_t

Key Concepts:
• Autoregressive: Current value depends on previous value(s)
• φ (Phi): Controls memory - how much past influences present
• |φ| < 1: Required for stationarity (series doesn't explode)
• φ > 0: Positive autocorrelation (smooth, trending patterns)
• φ < 0: Negative autocorrelation (oscillating patterns)
• φ → 1: Approaches random walk (long memory)
• φ → 0: Approaches white noise (no memory)

Forecasting:
• Uses past values to predict future
• Multi-step forecasts decay toward long-run mean: c/(1-φ)
• Forecast: ŷ_t+h = c(1-φ^h)/(1-φ) + φ^h·y_t

📚 MA(1) - Moving Average Model (Order 1)

y_t = μ + θ·ε_t-1 + ε_t

Key Concepts:
• Moving Average: Current value depends on past error(s), not past values
• θ (Theta): Controls how past shock affects current value
• |θ| < 1: Invertibility condition (ensures unique representation)
• θ > 0: Recent errors have positive impact
• θ < 0: Recent errors have negative impact (error correction)
• Limited Memory: MA(1) only "remembers" 1 period back

Forecasting:
• Uses past errors to predict future
• One-step-ahead: ŷ_t+1 = μ + θ·ε_t
• Beyond h=1: ŷ_t+h = μ (reverts to mean immediately!)
• Key Difference: MA forecasts converge to mean much faster than AR

📊 AR(1) & MA(1) Models Visualization

📚 AR(1) - Autoregressive Model (Order 1)

📚 MA(1) - Moving Average Model (Order 1)