ARCH models capture time-varying volatility in financial time series, revealing patterns of high and low volatility periods. They use conditional mean and variance equations to model the behavior of returns, with parameters estimated through maximum likelihood estimation.
These models help interpret the impact of past volatility on current volatility, but have limitations. They may require high orders, assume symmetric responses to shocks, and don't account for the leverage effect, leading to potential improvements in more advanced models.
ARCH Models
Structure of ARCH models
- Capture time-varying volatility in financial time series (stock returns, exchange rates)
- Volatility clustering: periods of high volatility followed by periods of low volatility (calm periods)
- Basic structure of ARCH(q) model:
- Conditional mean equation: $y_t = \mu_t + \varepsilon_t$
- $\mu_t$: conditional mean of $y_t$ given past information (moving average, regression model)
- $\varepsilon_t$: error term with conditional variance $\sigma_t^2$
- Conditional variance equation: $\sigma_t^2 = \alpha_0 + \alpha_1 \varepsilon_{t-1}^2 + \ldots + \alpha_q \varepsilon_{t-q}^2$
- $\alpha_0 > 0$ and $\alpha_i \geq 0$ for $i = 1, \ldots, q$ to ensure positive variance
- $q$: order of the ARCH model, determines the number of lagged squared errors included (ARCH(1), ARCH(2))
- Assume error term follows a conditional normal distribution:
- $\varepsilon_t | \Omega_{t-1} \sim N(0, \sigma_t^2)$, where $\Omega_{t-1}$ is the information set up to time $t-1$ (past returns, volatility)
Parameter estimation for ARCH
- Maximum likelihood estimation (MLE) estimates ARCH model parameters
- Log-likelihood function for an ARCH(q) model:
- $\ln L(\theta) = -\frac{1}{2} \sum_{t=1}^T \left[ \ln(2\pi) + \ln(\sigma_t^2) + \frac{\varepsilon_t^2}{\sigma_t^2} \right]$
- $\theta$: vector of parameters $(\mu, \alpha_0, \alpha_1, \ldots, \alpha_q)$
- $T$: number of observations (daily, weekly, monthly returns)
- MLE finds parameter values that maximize the log-likelihood function
- Numerical optimization methods (BFGS, Nelder-Mead) find optimal parameters
- Standard errors of estimated parameters obtained from inverse of Hessian matrix evaluated at maximum likelihood estimates
Interpretation of ARCH coefficients
- $\alpha_0$: unconditional variance of the error term (baseline volatility)
- $\alpha_i$: impact of squared error at lag $i$ on current conditional variance
- Higher $\alpha_i$ values indicate stronger influence of past volatility on current volatility (persistence)
- Goodness of fit measures for ARCH models:
- Likelihood ratio test: compares fit of ARCH model to constant variance model
- Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC): trade-off between model fit and complexity
- Lower AIC and BIC values indicate better-fitting model
- Ljung-Box test on standardized residuals: checks for remaining autocorrelation in squared standardized residuals
- Significant test suggests ARCH model may not have captured all volatility clustering
Limitations of ARCH models
- Non-negativity constraint on parameters:
- ARCH models require $\alpha_0 > 0$ and $\alpha_i \geq 0$ for $i = 1, \ldots, q$ to ensure positive conditional variance
- Restrictive constraint may limit model's ability to capture certain volatility patterns (negative coefficients)
- Often require high order (large $q$) to adequately capture volatility dynamics
- High-order models can lead to overparameterization and difficulties in estimation (convergence issues)
- Assume symmetric response to positive and negative shocks
- Financial time series may exhibit asymmetric volatility (negative shocks have larger impact than positive shocks)
- Do not account for leverage effect
- Leverage effect: negative correlation between past returns and future volatility (Black, 1976)
- Models like exponential GARCH (EGARCH) and GJR-GARCH designed to capture leverage effect