General Linear Process

Definition

A process ${Z_{t}}$ is a general linear process if it can be represented as a weighted linear combination of current and past white noise shocks: $Z_{t} = a_{t} + ψ_{1} a_{t - 1} + ψ_{2} a_{t - 2} + \dots = \sum_{j = 0}^{\infty} ψ_{j} a_{t - j}$ where $ψ_{0} = 1$ and ${a_{t}}$ is white noise.

Stationarity Condition

For a general linear process to be stationary, the weights must satisfy: $\sum_{j = 0}^{\infty} ψ_{j}^{2} < \infty$ This ensures the variance of the process is finite.

Example: Representing AR(1) as GLP

Let ${Z_{t}}$ be defined as

Z_{t} = a_{t} + ϕ Z_{t - 1}

where ${a_{t}}$ is white noise.

Expanding:

Z_{t} = a_{t} + ϕ Z_{t - 1} = a_{t} + ϕ (a_{t - 1} + ϕ Z_{t - 2}) = a_{t} + ϕ a_{t - 1} + ϕ^{2} Z_{t - 2} = a_{t} + ϕ a_{t - 1} + ϕ^{2} (a_{t - 2} + ϕ Z_{t - 3}) = a_{t} + ϕ a_{t - 1} + ϕ^{2} a_{t - 2} + ϕ^{3} Z_{t - 3} ⋮ = j = 0 \sum \infty ϕ^{j} a_{t - j}

The process above can be represented as GLM with $ψ_{j} = ϕ^{j}$

Example: Representing AR(2) as GLP

Let ${Z_{t}}$ be defined as

Z_{t} = a_{t} + ϕ_{1} Z_{t - 1} + ϕ_{2} Z_{t - 2}

where ${a_{t}}$ is white noise.

Expanding:

Z_{t} = a_{t} + ϕ_{1} Z_{t - 1} + ϕ_{2} Z_{t - 2} = a_{t} + ϕ_{1} (a_{t - 1} + ϕ_{1} Z_{t - 2} + ϕ_{2} Z_{t - 3}) + ϕ_{2} (a_{t - 2} + ϕ_{1} Z_{t - 3} + ϕ_{2} Z_{t - 4}) = a_{t} + [ϕ_{1} a_{t - 1} + ϕ_{1}^{2} Z_{t - 2} + ϕ_{1} ϕ_{2} Z_{t - 3}] + [ϕ_{2} a_{t - 2} + ϕ_{1} ϕ_{2} Z_{t - 3} + ϕ_{2}^{2} Z_{t - 4}] = a_{t} + ϕ_{1} a_{t - 1} + ϕ_{2} a_{t - 2} + ϕ_{1}^{2} Z_{t - 2} + 2 ϕ_{1} ϕ_{2} Z_{t - 3} + ϕ_{2}^{2} Z_{t - 4} = a_{t} + ϕ_{1} a_{t - 1} + ϕ_{2} a_{t - 2} + ϕ_{1}^{2} (a_{t - 2} + ϕ_{1} Z_{t - 3} + ϕ_{2} Z_{t - 4}) + 2 ϕ_{1} ϕ_{2} Z_{t - 3} + ϕ_{2}^{2} Z_{t - 4} = a_{t} + ϕ_{1} a_{t - 1} + ϕ_{2} a_{t - 2} + ϕ_{1}^{2} a_{t - 2} + ϕ_{1}^{3} Z_{t - 3} + ϕ_{1}^{2} ϕ_{2} Z_{t - 4} + 2 ϕ_{1} ϕ_{2} Z_{t - 3} + ϕ_{2}^{2} Z_{t - 4} = a_{t} + ϕ_{1} a_{t - 1} + (ϕ_{1}^{2} + ϕ_{2}) a_{t - 2} + (ϕ_{1}^{3} + 2 ϕ_{1} ϕ_{2}) Z_{t - 3} + (ϕ_{1}^{2} ϕ_{2} + ϕ_{2}^{2}) Z_{t - 4} ⋮

Notice that the equation above can be written as $Z_{t} = \sum_{j = 0}^{\infty} a_{t - j} ψ_{j}$ , where

ψ_{0} ψ_{1} ψ_{2} ψ_{3} ψ_{j} = 1 = ϕ_{1} = ϕ_{1}^{2} + ϕ_{2} = ϕ_{1}^{3} + 2 ϕ_{1} ϕ_{2} ⋮ = ϕ_{1} ψ_{j - 1} + ϕ_{2} ψ_{j - 2} = ϕ_{1} (ϕ_{1}) + ϕ_{2} (1) = ϕ_{1} (ϕ_{1}^{2} + ϕ_{2}) + ϕ_{2} (ϕ_{1})

$∴$ So, AR(2) process can be represented as GLM as:

Z_{t} = j = 0 \sum \infty ψ_{j} a_{t - j}, ψ_{j} = ϕ_{1} ψ_{j - 1} + ϕ_{2} ψ_{j - 2}, j \geq 2

General Linear Process

Table of Contents

Table of Contents

General Linear Process

Definition

Stationarity Condition

Example: Representing AR(1) as GLP

Example: Representing AR(2) as GLP

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