EGARCH — Exponential GARCH

EGARCH (Exponential GARCH) extends GARCH to capture the asymmetric / leverage effect — in financial markets, bad news (negative shocks) typically raises volatility more than good news of the same magnitude. Standard GARCH cannot distinguish the sign of shocks; EGARCH can.

When to use

Use EGARCH when you suspect volatility responds asymmetrically to positive/negative shocks (very common in stock returns). It models log variance, so no positivity constraints are needed.

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Model specification

$$\ln(\sigma_t^2) = \omega + \alpha \left( |z_{t-1}| - E|z_{t-1}| \right) + \gamma \, z_{t-1} + \beta \ln(\sigma_{t-1}^2)$$

where $z_{t-1} = \varepsilon_{t-1}/\sigma_{t-1}$. The parameter $\gamma \ne 0$ measures the leverage effect: $\gamma < 0$ ⇒ negative shocks raise volatility more.

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Running in EcoLab

1. Modeling module → Univariate time series family → EGARCH.
2. Choose the returns series; declare the order and mean equation.
3. Run; check the sign/significance of $\gamma$ (leverage); export the replication code.

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Replication code

Stata

* --- EGARCH(1,1) ---
arch ret, earch(1) egarch(1)

* Conditional variance
predict sigma2, variance

* Check leverage coefficient (gamma)
* A significant negative gamma confirms leverage effect

R

# --- EGARCH(1,1) ---
library(rugarch)

spec <- ugarchspec(
  variance.model = list(model = "eGARCH", garchOrder = c(1, 1)),
  mean.model     = list(armaOrder = c(0, 0), include.mean = TRUE),
  distribution.model = "std"
)

fit <- ugarchfit(spec = spec, data = ret)
show(fit)

# Check the leverage parameter (alpha1 in rugarch = gamma)
# Negative and significant => leverage effect confirmed
plot(sigma(fit), type = "l", main = "EGARCH Conditional Volatility")

Python

from arch import arch_model
import matplotlib.pyplot as plt

# Estimate EGARCH(1,1)
model = arch_model(ret, vol='EGARCH', p=1, q=1, dist='t')
result = model.fit(disp='off')
print(result.summary())

# The gamma parameter captures leverage
# Negative & significant => bad news increases volatility more
fig, ax = plt.subplots()
ax.plot(result.conditional_volatility)
ax.set_title('EGARCH(1,1) Conditional Volatility')
plt.show()

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Limitations

• Parameter interpretation is more complex than GARCH.
• Sensitive to the distributional assumption; needs a large enough sample.

Video tutorial

Video Tutorial: Running EGARCH in EcoLab

See also

• ARCH/GARCH · ARIMA · Catalog