Lesson 2.1: Autoregressive (AR) Models

Modeling how past values of a series influence its present value.

Part 1: The Core Idea - Regressing on the Past

An Autoregressive (AR) model is a regression of a time series on its own past values, called **lags**. It formalizes the idea of "memory" or momentum.

The Core Analogy: Driving by Looking in the Rear-View Mirror

An AR(1) model is like saying, "My speed right now ( $Y_t$ ) is some fraction ( $\phi_1$ ) of my speed one second ago ( $Y_{t-1}$ ), plus a random jolt ( $\epsilon_t$ )." The order `p` in AR(p) tells you how many lags are included.

Part 2: The AR(p) Model Specification

The AR(p) Model

The value of the series $Y_t$ is a linear function of its own $p$ past values.

Y_t = c + \phi_1 Y_{t-1} + \phi_2 Y_{t-2} + \dots + \phi_p Y_{t-p} + \epsilon_t

$\phi_i$ are the autoregressive coefficients.
$\epsilon_t$ is a white noise error term.

Part 3: Model Identification

The Signature of an AR(p) Process

The **ACF plot** will show a pattern of **gradual decay**.
The **PACF plot** will **cut off sharply** after lag $p$ .

The PACF plot is the primary tool for identifying the order `p` of an AR model.

What's Next? Modeling Shocks

AR models capture the memory of past *values*. But what about the memory of past *shocks* or forecast errors?

In the next lesson, we will explore the complementary **Moving Average (MA) Model**, which models the present value as a function of past error terms.

ACF and PACF

Moving Average (MA) Models