GARCH(1,1) Deep Dive

Volatility clusters, squared returns exhibit long memory, and the news impact curve shows symmetric response to shocks.

Persistence α+β

0.980

Uncond. Vol (ann.)

1587.5%

Vol half-life (days)

34.3

Excess kurtosis

4.43

Returns with ±2σₜ Conditional Volatility Envelope

Conditional Vol — annualised (%)

ACF of rₜ² — volatility clustering

News Impact Curve — symmetric response to shocks

μ (mean return)0.020

ω (constant)0.020

α (ARCH effect)0.08

β (GARCH persist)0.90

Observations1000

Seed42

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

σ_{\infty}^{2} = \frac{ω}{1 - α - β}, Excess kurtosis = 3 \frac{1 - ( α + β ) ^{2}}{1 - 2 α ^{2} - ( α + β ) ^{2}}

Push α+β close to 1 to see extremely persistent volatility. At α+β ≥ 1 the process is IGARCH — integrated, unconditional variance is infinite.