GARCH Models

Overview

Definition

GARCH (Generalized AutoRegressive Conditional Heteroskedasticity) models are used to estimate the volatility of a time series. While ARIMA models the conditional mean, GARCH models the conditional variance ( $σ_{t}^{2}$ ).

1. The Volatility Problem

In financial time series (stock returns), we often observe Volatility Clustering: Large changes tend to be followed by large changes, and small by small.

This violates the constant variance (homoscedasticity) assumption of standard regression.
GARCH treats this heteroscedasticity not as a bug, but as a feature to be modeled.

2. The Model Structure

GARCH(p, q):

σ_{t}^{2} = ω + \sum_{i = 1}^{q} α_{i} ϵ_{t - i}^{2} + \sum_{j = 1}^{p} β_{j} σ_{t - j}^{2}

Where:

$ω$ : Constant baseline variance.
$ϵ_{t - i}^{2}$ : Past squared residuals (ARCH term) — "News" or "Shock".
$σ_{t - j}^{2}$ : Past variance (GARCH term) — "Persistence" (Memory).

GARCH(1,1): The most common specification.

σ_{t}^{2} = ω + α_{1} ϵ_{t - 1}^{2} + β_{1} σ_{t - 1}^{2}

Worked Example: Risk Management

Value at Risk (VaR)

You manage a portfolio.

Today: Return $r_{t} = - 2 %$ .
Current Volatility: $σ_{t}^{2} = 1.5$ .
Model: GARCH(1,1) with $ω = 0.1, α = 0.2, β = 0.7$ .

Question: Forecast tomorrow's volatility ( $σ_{t + 1}^{2}$ ).

Solution:

σ_{t + 1}^{2} = ω + α ϵ_{t}^{2} + β σ_{t}^{2}

Identifying Terms:
- $ϵ_{t}^{2}$ is the squared shock from today. If we assume mean return is 0, $ϵ_{t} \approx - 2$ . So $ϵ_{t}^{2} = 4$ .
- $σ_{t}^{2} = 1.5$ .
Calculation:
$σ_{t + 1}^{2} = 0.1 + (0.2 \times 4) + (0.7 \times 1.5)$ $σ_{t + 1}^{2} = 0.1 + 0.8 + 1.05 = 1.95$

Conclusion:
Volatility is predicted to increase from 1.5 to 1.95. The market is getting riskier because of today's large drop (shock).

Assumptions

Stationarity: The underlying series (returns) must be stationary (no trend).
Mean Reversion: Volatility is expected to revert to a long-run average ( $α + β < 1$ ).
Clustering: Volatility is not constant but clusters in periods of calm and stress.

Limitations

Pitfalls

Explosive Models: If $α + β \geq 1$ , the variance explodes to infinity (Integrated GARCH). Real financial data usually sums to ~0.99.
Ignores Direction: Standard GARCH assumes positive shocks and negative shocks affect volatility equally. In reality, market crash (bad news) spikes volatility more than a rally (good news). Use EGARCH or GJR-GARCH for asymmetric effects.
High Frequency: Does not work well on low-frequency data (e.g., yearly). Best for daily/intraday.

4. Interpretation Guide

Coefficient	Meaning
$α$ (ARCH)	Reaction to new "shocks". High $α$ = spiky volatility.
$β$ (GARCH)	Persistence of old variance. High $β$ = volatility dies out slowly.
$ω > 0$	Baseline variance. Essential for stability.
$α + β \approx 1$	High persistence (common in finance).

5. Python Implementation (arch package)

Note: The standard library statsmodels has limited GARCH support. arch is the industry standard.

from arch import arch_model
import numpy as np

# Sample Returns
returns = np.random.normal(0, 1, 1000)

# Fit GARCH(1,1)
model = arch_model(returns, vol='Garch', p=1, q=1)
results = model.fit(disp='off')

print(results.summary())

# Forecasting Volatility
forecasts = results.forecast(horizon=5)
print(forecasts.variance[-1:])

ARIMA Models - Often combined (ARIMA-GARCH).
Heteroscedasticity - The phenomenon being modeled.
Stationarity (ADF & KPSS)

GARCH Models

Overview

1. The Volatility Problem

2. The Model Structure

Worked Example: Risk Management

Assumptions

Limitations

4. Interpretation Guide

5. Python Implementation (arch package)

6. Related Concepts