ARCH: Autoregressive Conditional Heteroskedasticity

ARCH is a statistical model used to analyze time series data where the variance changes over time. It was introduced by Robert Engle in 1982 mainly for financial data like stock returns. ARCH models help capture the fact that big market moves tend to be followed by more big moves and calm periods by low volatility a pattern called volatility clustering.

Key Components

1. Mean Equation

r_t =μ+ϵ _t

r_t: observed value at time t
μ: mean (can be constant or modeled as AR process)
\\epsilon_t : error term or residual (difference between actual and predicted)

2. Error Term (Residuals)

ϵ_t=σ_tz_t

z_t : white noise (random shocks), usually assumed z _t ∼N(0,1)
\sigma_t : conditional standard deviation at time t, changes over time

3. Conditional Variance Equation

\sigma_t^2 = \alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \cdots + \alpha_q \epsilon_{t-q}^2

\sigma_t^2 : expected variance given past data
\alpha_0 : constant (must be > 0)
\alpha_1 \epsilon_{t-1}^2 : coefficients (must be ≥ 0 for all i)
\alpha_q \epsilon_{t-q}^2 : past squared residuals captures volatility clustering

4. Lag Order (q)

Defines how many past squared errors are used.
Larger q means the model looks further into the past to estimate current volatility.

5. Assumptions

Residuals have zero mean and time varying variance
No autocorrelation in residuals
Stationarity of variance

ARCH vs GARCH

Let us now compare ARCH and GARCH:

Feature	ARCH (Autoregressive Conditional Heteroskedasticity)	GARCH (Generalized ARCH)
Definition	Models conditional variance using past squared errors	Extends ARCH by including past conditional variances along with past squared errors
Variance Equation	\sigma_t^2 = \alpha_0 + \sum_{i=1}^{q} \alpha_i \epsilon_{t-i}^2	\sigma_t^2 = \alpha_0 + \sum_{i=1}^{q} \alpha_i \epsilon_{t-i}^2 + \sum_{j=1}^{p} \beta_j \sigma_{t-j}^2
Memory	Short memory and depends only on past squared errors	Longer memory and depends on both past errors and past variances
Parameters	Typically only \alpha_i coefficients	Both \alpha_i and \beta_j coefficients
Flexibility	Less flexible in capturing volatility clustering	More flexible, better captures persistence in volatility
Typical Use	Early model for volatility, less used alone now	Widely used for modeling financial time series volatility

Applications

Financial Market Volatility: ARCH is widely used to model stock market returns where volatility tends to cluster periods of high and low volatility come together.
Risk Management: Used to estimate Value at Risk (VaR) by modeling conditional variance which is important for financial institutions managing portfolios.
Option Pricing: Volatility is a key input in models like Black Scholes. ARCH helps in generating more accurate, time dependent volatility estimates.
Macroeconomic Forecasting: ARCH can be applied to inflation rates, GDP growth, interest rates which also exhibit time varying uncertainty.