ARCH

#Statistics

Autoregressive Conditional Heteroskedasticity (ARCH) is a statistical model used in time series analysis to describe the variance of the errors in a regression model, where the variance at a given time is dependent on the magnitudes of the previous time periods' error terms.

ARCH is a statistical model utilized primarily in time series analysis to assess and forecast volatility, particularly in financial markets. Developed by economist Robert F.Engle in the 1980s, the ARCH model has become fundamental in economics for understanding how volatility clusters over time.

Key Concepts

Definition and Functionality

ARCH models describe the variance of the current error term as a function of past error terms, capturing the phenomenon where high volatility follows high volatility and low follows low. This characteristic is essential in financial contexts, where understanding risk and volatility is crucial for asset management and investment strategies.

Volatility Clustering

The concept of volatility clustering implies that periods of high volatility are likely to be followed by more high volatility, while periods of low volatility tend to be followed by low volatility. This behavior is particularly relevant in financial data, such as stock prices or exchange rates, where market conditions can lead to sudden changes in variance.

Model Structure

ARCH model

σ_{t}^{2} = α_{0} + \sum_{i = 1}^{q} α_{i} ϵ_{t - i}^{2}

where:

$σ^{2}$ is the conditional variance,
$α_{0}$ is the constant,
$α_{i}$ are parameters to be estimated,
$ϵ_{t - i}$ are past error terms.