ARCH LM Test & GARCH Models¶

While the ARCH model identifies that variance can be dependent on past shocks, we need a rigorous statistical way to detect these effects and a more parsimonious way to model them when they persist over long periods.

1. The ARCH Model: Testing and Selection¶

Testing for ARCH Effect: The Lagrange Multiplier (LM) Test¶

Also known as Engle's ARCH Test, this procedure determines if the residuals of a time series model exhibit conditional heteroskedasticity.

The Steps:
1. Estimate the Mean Equation: Fit a standard time series model (like an ARMA) to the data.
2. Square the Residuals: Use $\epsilon_t^2$ as a proxy for the variance at each time $t$.
3. Perform Auxiliary Regression: Regress the squared residuals on their own lags:
$$\epsilon_t^2 = \alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \dots + \alpha_q \epsilon_{t-q}^2 + v_t$$
4. Test for Significance:
* $H_{0}$: $\alpha_1 = \dots = \alpha_q = 0$ (No ARCH effects; variance is constant).
* $H_{1}$: At least one $\alpha_i \neq 0$ (Conditional heteroskedasticity exists).

Picking the Optimal ARCH Model¶

Order ($q$): The number of lagged squared residuals needed to capture the volatility structure.
Selection Strategy:
- PACF Analysis: Examine the Partial Autocorrelation Function of the squared residuals. Peaks in the PACF indicate the significant lags that should define the ARCH order.
- Information Criteria: Fit various $ARCH(q)$ models and select the one with the lowest AIC (Akaike Information Criterion).

2. The GARCH Model¶

Generalized Autoregressive Conditional Heteroskedasticity (GARCH) adds an "Autoregressive" component to the ARCH model. Instead of just looking at past shocks, it also looks at past variances.

Advantage

GARCH is much more efficient than ARCH. A high-order ARCH model can often be replaced by a very simple $GARCH(1,1)$ model, avoiding "parameter explosion."

$GARCH(p,q)$ Specification¶

The conditional variance $\sigma_t^2$ is defined as:
$$\sigma_{t}^{2} = \omega + \sum_{i=1}^{q} \alpha_{i} \epsilon_{t-1}^{2} + \sum_{j=1}^{p} \beta_{j} \sigma^{2}_{t-j}$$

$\omega > 0$: Constant term.
$\alpha_{i}$ (ARCH terms): Impact of past squared residuals (shocks).
$\beta_{j}$ (GARCH terms): Impact of past variances (persistence).

Key Features¶

Volatility Clustering: Successfully models the "bursts" of high and low volatility seen in financial markets.
Mean Reversion: If $\sum \alpha_{i} + \sum \beta_{j} < 1$, the volatility eventually reverts to a long-term stable level.
Leverage Effect Limitation: Standard GARCH assumes shocks have a symmetric effect. It cannot handle asymmetric "leverage effects" (where bad news spikes volatility more than good news). For this, we use EGARCH or GJR-GARCH.

3. The $GARCH(1,1)$ Model¶

This is the "workhorse" of volatility modelling in finance. It is highly effective because it captures long-term persistence with only three parameters in the variance equation.

Mean Equation: $y_{t} = \mu + \epsilon_{t}$
Variance Equation: $\sigma_{t}^{2} = \omega + \alpha \epsilon_{t-1}^{2} + \beta \sigma_{t-1}^{2}$

Stationarity Condition:
The model is stationary if $\alpha + \beta < 1$. If $\alpha + \beta$ is very close to 1, the volatility exhibits long-term memory or "persistence."

4. Implementation and Application¶

Steps to Fit a GARCH Model¶

Detection: Perform the ARCH LM test to confirm heteroskedasticity.
Selection: Use AIC/BIC to choose orders $p$ and $q$.
Estimation: Use Maximum Likelihood Estimation (MLE) to find the parameters.
Diagnostics: Check that the standardized residuals of the fitted model are now homoscedastic (no remaining ARCH effects).
Forecasting: Use the recursive formula to project future volatility.

Practical Use Cases¶

Asset Price Modelling: Predicting the "risk" of stock or commodity returns.
Option Pricing: Computing implied volatility, a critical input for the Black-Scholes model.
Risk Management (VaR): Calculating Value-at-Risk to estimate the maximum potential loss of a portfolio.
Stress Testing: Simulating how a portfolio behaves during hypothetical periods of extreme market volatility.