Fitted Model
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Current Setup

About This App

by 4kasters: Kevaan Chirag Buch, Kevin Joseph, Eshan Hussain, Will Wang, and Amin Amir

Purpose

This app simulates a time series with a linear trend and ARMA errors. To get started, go to the "Explorer" tab and use the controls on the left to choose the data-generating process, fit a possibly different ARMA model, and inspect how well it captures the underlying structure through the plots and residuals.

What to Look For

If the fitted model is doing a good job, the residual plot should not show a strong pattern over time. If the residuals still show visible structure, the fitted model may be missing part of the time-series behaviour.

How to Use

  • Set the fitted model order to match the true process. Residuals should look like white noise.
  • Overfit by using a larger order than the true process. Watch what changes.
  • Underfit by using a smaller order. See if residual structure remains.
  • Increase sample size to see how estimation improves with more data.

Features

  • Data-Generating Process: Set AR coefficients (φ), MA coefficients (θ), sample size (n), innovation SD (σ), trend slope (b₁) and intercept (b₀), and random seed.
  • Fitted Model: Choose AR order (p) and MA order (q) for the model to fit. Use 'Use generated order' to match the true process.
  • Series Display: Toggle between observed data (y) and error process (e). Check 'Show fitted overlay' to visualize the model fit.
  • Diagnostic Plots: Main series plot, residual time plot, ACF and PACF of residuals for checking model adequacy.
  • Estimated Coefficients: View the fitted φ, θ, b₀, b₁ values and σ̂ estimate.

Key Terms

  • AR (Auto Regressive): current value depends on its own past values.
  • MA (Moving Average): current value depends on past error terms.
  • ARMA(p, q): p AR lags and q MA lags combined.
  • Residuals: what's left after the model is subtracted from the data. A good fit leaves no visible pattern.
  • Stationarity: A process is stationary if its characteristic polynomial roots lie outside the unit circle. The app checks this and displays a stationarity badge.