Lesson 2.4: ARIMA Models
Using differencing to model non-stationary series like stock prices.
Part 1: The Problem of the Random Walk
Most financial data (like stock prices) is non-stationary. The most common model for such behavior is the **Random Walk**, which is an AR(1) process with a coefficient of 1 (a "unit root").
Our ARMA framework fails on non-stationary data. The solution is beautifully simple: **differencing**.
By modeling the *change* in the series (the returns) instead of its *level*, we transform a non-stationary problem into a stationary one.
Part 2: The ARIMA(p,d,q) Model
The ARIMA(p,d,q) Model
An Autoregressive Integrated Moving Average model is simply an ARMA(p,q) model applied to a differenced time series.
- `p`: The order of the Autoregressive component.
- `d`: The degree of differencing required to make the series stationary.
- `q`: The order of the Moving Average component.
Part 3: The Modeling Workflow
The Box-Jenkins Workflow Summary
- Identification: Use the ADF test to find `d`. Plot ACF/PACF of the differenced series to guess `p` and `q`.
- Estimation: Fit candidate ARIMA(p,d,q) models and select the best one using AIC/BIC.
- Diagnostic Checking: Ensure the residuals of the best model are white noise.
- Forecasting: Use the validated model to predict future values.
What's Next? A Systematic Process
We have now assembled all the components of the ARIMA framework. In the next lesson, we will do a deep dive into the **Box-Jenkins Methodology**, providing a rigorous, step-by-step checklist for building robust ARIMA models in practice.