Day 5: Dynamics in Variance

ARCH/GARCH

Robert W. Walker

2026-07-24

Dynamics in Variance

  • Thinking about heteroscedasticity as theoretically interesting
  • ARCH
  • GARCH

Heteroscedasticity as Theoretically Interesting

A Different Take on Heteroscedasticity

  • ARCH and GARCH are time series versions of models that buck the traditional approach to heteroscedastic error processes.

  • These models depart from the typical treatment of heteroscedasticity as a disease to be cured.

  • Instead, they treat heteroscedasticity as a subject of theoretical interest to be modelled.

  • In these frameworks we develop two levels of theories: theories about the mean and variance of our dependent variable.

ARCH

Origins of ARCH

The first ARCH model was proposed by Engle in a 1982 Econometrica article titled:

“Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of UK Inflation”

The name derives from AutoRegressive Conditional Heteroscedasticity.

The Standard Time Series Setup

Beginning with Harvey’s (1990) autoregressive distributed (AD) lag notation for time series data:

Y_t = \sum_{i=0}^{L} X_{t-i}\beta_i + \sum_{i=1}^{M} \phi_i Y_{t-i} + \epsilon_t

In such models we still assume:

\epsilon_t \sim N(0,\, \sigma^2)

where the absence of a subscript on \sigma^2 indicates a homoscedastic population stochastic component.

Engle’s Relaxation

Engle’s model starts from the same theoretically-driven specification but relaxes homoscedasticity by assuming instead that:

\epsilon_t \sim N(0,\, \sigma_t^2)

Where \sigma_t^2, the variance of the population stochastic component, varies across time.

Two Equations

ARCH models involve estimating two equations:

  1. Mean equation — the theoretically-derived equation with Y_t as the dependent variable.

  2. Variance equation — an equation in which \sigma_t^2 is the dependent variable.

ARCH Order

ARCH processes are described by their q order.

ARCH(1): h_t = \sigma_t^2 = \gamma_0 + \gamma_1 \epsilon_{t-1}^2

ARCH(2): h_t = \sigma_t^2 = \gamma_0 + \gamma_1 \epsilon_{t-1}^2 + \gamma_2 \epsilon_{t-2}^2

General ARCH(q): h_t = \sigma_t^2 = \gamma_0 + \sum_{j=1}^{q} \gamma_j \epsilon_{t-j}^2

Conditional Variance with Covariates

We can also allow conditional variance to depend on theoretically-chosen covariates — developing and testing theories about what makes the stochastic process larger or smaller.

In such models the variance equation becomes:

h_t = \sigma_t^2 = e^{\lambda_0 + \lambda_1 Z_{t-1}} + \gamma_1 \epsilon_{t-1}^2

where the Z variables are covariates for which we have theoretical expectations about how they affect model variance.

Notes on This Notation

h_t = \sigma_t^2 = e^{\lambda_0 + \lambda_1 Z_{t-1}} + \gamma_1 \epsilon_{t-1}^2

  • This generalises to any ARCH(q) process.
  • Z variables can follow any time process we can theoretically justify.
  • Unlike the \gamma parameters, the \lambda (lambda) parameters can be negative — meaning that as Z increases, the error variance decreases.
  • The same variable can appear as both a Z and an X in an ARCH model.
  • If \lambda_1 = 0, the model collapses to a standard ARCH(1) where \gamma_0 = e^{\lambda_0}.

GARCH

From ARCH to GARCH

  • The ARCH framework allows for conditional heteroscedasticity of a moving average flavour.

  • However, our theories about heteroscedasticity in time series may also be of an autoregressive flavour.

  • We can approximate this by making q large — but the GARCH model handles it more parsimoniously.

The GARCH Variance Equation

GARCH re-specifies the variance equation as:

h_t = \sigma_t^2 = \gamma_0 + \sum_{i=1}^{p} \delta_i h_{t-1} + \sum_{j=1}^{q} \gamma_j \epsilon_{t-j}^2

GARCH processes are described as GARCH(p, q), where p is the order of the autoregressive variance terms and q is the order of the lagged squared errors.

GARCH with Independent Variables

An additionally attractive feature of GARCH is the ability to include independent variables in the variance equation:

h_t = \sigma_t^2 = \gamma_0 + \sum_{i=1}^{p} \delta_i h_{t-1} + \sum_{j=1}^{q} \gamma_j \epsilon_{t-j}^2 + \sum_{k=1}^{m} \mu_k X_k

This allows direct modelling of which substantive factors drive the size of the stochastic process.

ARCH in Many Flavours

See the day 5 post for a guide to computing these models.

ARCH in Mean

ARCH-in-Mean Models

ARCH-in-Mean (ARCH-M) models allow heteroscedasticity to directly affect Y_t. For instance:

Y_t = \beta_0 + \beta_1 X_t + \psi\sigma_t^2 + \epsilon_t \sigma_t^2 = \gamma_0 + \gamma_1 \epsilon_{t-1}

This class of model tests two distinct theories about the heteroscedastic process:

  1. The standard ARCH/GARCH theory about the structure of heteroscedasticity (captured by \gamma_1).

  2. Whether heteroscedasticity has a direct impact \psi on the mean equation — where \psi can be either positive or negative.