Autoregressive processes: going from the particular to the general is the safest option. Simple observations are the foundation of any theory.

## Intuition

Figure 1. Electricity load in France and Great Britain for 2001 to 2006

If you have only one variable, what can you regress it on? Only on its own past values (future values are not available at any given moment). Figure 1 on electricity demand from a paper by J.W. Taylor illustrates this. A low value of electricity demand, say, in summer last year, will drive down its value in summer this year. Overall, we would expect the electricity demand now to depend on its values in the past 12 months. Another important observation from this example is that probably this time series is stationary.

## AR(p) model

We want a definition of a class of stationary models. From this example we see that excluding the time trend increases chances of obtaining a stationary process. The idea to regress the process on its own past values is realized in

(1)

Here **autoregressive process** of order

**Exercise 1**. Repeat calculations on AR(1) process to see that in case

**Question**. How does this stability condition generalize to AR(p)?

## Characteristic polynomial

Denote **lag operator** defined by

Whoever first did this wanted to solve the equation for

The identity operator is defined by

(2)

Finally, formally solving for

(3)

**Definition 1**. In **characteristic polynomial**:

(3)

**Definition 2**. We say that model (1) is **stable** if its characteristic polynomial (3) has roots outside the unit circle, that is, the roots are larger than 1 in absolute value.

Under this stability condition the passage from (2) to (3) can be justified. For AR(1) process this actually has been done.

**Example 1**. In case of a first-order process,

**Example 2**. In case of a second-order process,

**Remark**. Hamilton uses a different definition of the characteristic polynomial (linked to vector autoregressions), that's why in his definition the roots of the characteristic equation should lie inside the unit circle.

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