Autoregressive conditional heteroskedasticity

Autoregressive Conditional Heteroskedasticity (ARCH)

Autoregressive Conditional Heteroskedasticity (ARCH) is a statistical model used to describe time series data that exhibit time-varying volatility, often observed in financial markets. The ARCH model was introduced by economist Robert F. Engle in 1982, for which he was awarded the Nobel Prize in Economics in 2003. The model is particularly useful for modeling and forecasting the volatility of returns, which is crucial for risk management and derivative pricing.

Overview

ARCH models are designed to capture the phenomenon where the variance of the current error term, or innovation, is a function of the variances of previous time periods' error terms. This is particularly useful in financial time series where periods of high volatility tend to cluster together.

Mathematical Formulation

The basic ARCH model can be described as follows:

\[ Y_t = \mu + \epsilon_t \]

where \(Y_t\) is the time series, \(\mu\) is the mean of the series, and \(\epsilon_t\) is the error term or innovation at time \(t\). The key feature of the ARCH model is that the variance of \(\epsilon_t\) is not constant, but rather:

\[ \epsilon_t = \sigma_t Z_t \]

where \(Z_t\) is a white noise process with zero mean and unit variance, and \(\sigma_t^2\) is the conditional variance given by:

\[ \sigma_t^2 = \alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \alpha_2 \epsilon_{t-2}^2 + \cdots + \alpha_q \epsilon_{t-q}^2 \]

Here, \(\alpha_0 > 0\) and \(\alpha_i \geq 0\) for \(i = 1, 2, \ldots, q\). The order \(q\) of the ARCH model indicates how many lagged squared residuals are included.

Extensions

The ARCH model has been extended in various ways to capture more complex volatility patterns:

• GARCH (Generalized ARCH): Introduced by Tim Bollerslev in 1986, the GARCH model generalizes ARCH by allowing past conditional variances to also affect the current conditional variance.
• EGARCH (Exponential GARCH): This model allows for asymmetric effects of positive and negative shocks on volatility.
• TARCH (Threshold ARCH): This model captures leverage effects by allowing different responses to positive and negative shocks.

Applications

ARCH models are widely used in finance for:

• Risk Management: Estimating Value at Risk (VaR) and other risk measures.
• Option Pricing: Modeling the volatility of underlying assets to price derivatives.
• Portfolio Optimization: Understanding the volatility of asset returns to optimize asset allocation.

Also see

• Time series analysis
• Volatility (finance)
• GARCH
• Econometrics
• Financial risk management