What does the I in ARIMA stand for and how is it implemented in the model equation?

Conditions

• The context is the ARIMA time series model.
• The explanation refers to the specific ARIMA(1,1,1) example shown on the board.

Reasoning, step by step

1. Identify the acronym ARIMA as Auto Regressive Integrated Moving Average.
2. Clarify that 'Integrated' does not mean calculus integration.
3. Explain that it involves predicting differences of the time series from one timestamp to the previous timestamp.
4. Show that in the equation $Z_t = \phi_1 Z_{t-1} + \theta_1 \epsilon_{t-1} + \epsilon_t$, the variable $Z_t$ is already the difference $q_{t+1} - q_t$.
5. Conclude that the I part is encoded by modeling the differenced series $Z_t$ instead of the original series $q_t$.

Example

The speaker states, "integrated in this context just means that instead of predicting the time series itself, you're going to be predicting differences of the time series from one timestamp to the previous timestamp." The board shows $Z_t = q_{t+1} - q_t$ and the equation uses $Z_t$.

Common misconceptions

• Confusing the 'Integrated' component with integral calculus.
• Thinking the I adds a separate term to the right-hand side of the equation.
• Believing the model predicts the original level series directly.

Watch the explanation

Explore next

• How is the differenced series $z_t$ defined from the original series?
• Why does differencing help when the original time series has a linear trend?
• What is the equation for ARIMA(1,1,1)?