Session 3: Asymmetric GARCH Models

Course: Advanced Volatility Modeling¶

Learning Objectives¶

By the end of this session, students will be able to:

Understand the leverage effect and its economic interpretation
Derive and estimate EGARCH, GJR-GARCH, and TGARCH models
Compare news impact curves across models
Test for asymmetry in volatility response

1. The Leverage Effect¶

1.1 Empirical Evidence¶

Black (1976) documented that stock returns and volatility are negatively correlated:

Negative returns → Increased future volatility
Positive returns → Decreased (or unchanged) future volatility

This asymmetry is called the leverage effect.

1.2 Economic Explanations¶

Leverage hypothesis (Black, 1976; Christie, 1982):

When stock price falls, debt-to-equity ratio rises
Higher leverage → higher risk → higher volatility

Volatility feedback hypothesis (French et al., 1987; Campbell & Hentschel, 1992):

News of higher future volatility raises required return
Stock price must fall immediately to deliver higher expected return

1.3 News Impact Curve¶

The News Impact Curve (NIC) shows how today’s shock affects tomorrow’s variance:

NIC(\epsilon_{t-1}) = E[\sigma_t^2 | \epsilon_{t-1}, \sigma_{t-1}^2 = \bar{\sigma}^2]

(1)

For symmetric GARCH(1,1), the NIC is a parabola centered at zero.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import scipy.stats as stats
from scipy.optimize import minimize
import yfinance as yf
from arch import arch_model
import warnings
warnings.filterwarnings('ignore')

plt.style.use('seaborn-v0_8-whitegrid')
plt.rcParams['figure.figsize'] = (12, 6)
plt.rcParams['font.size'] = 11

np.random.seed(42)

# Download S&P 500 data
spy = yf.download('SPY', start='2010-01-01', end='2024-12-31', progress=False)
if isinstance(spy.columns, pd.MultiIndex):
    spy.columns = spy.columns.get_level_values(0)

spy['returns'] = np.log(spy['Close'] / spy['Close'].shift(1))
returns = spy['returns'].dropna() * 100  # Percentage returns

print(f"Sample: {returns.index[0].date()} to {returns.index[-1].date()}")
print(f"Observations: {len(returns)}")

YF.download() has changed argument auto_adjust default to True
Sample: 2010-01-05 to 2024-12-30
Observations: 3772

# Empirical evidence of leverage effect
def empirical_leverage_effect(returns, lags=20):
    """
    Compute cross-correlation between returns and future squared returns.
    Leverage effect implies negative correlation at positive lags.
    """
    ret = returns - returns.mean()
    sq_ret = ret ** 2
    
    correlations = []
    for lag in range(-lags, lags + 1):
        if lag < 0:
            corr = ret.iloc[-lag:].corr(sq_ret.iloc[:lag])
        elif lag > 0:
            corr = ret.iloc[:-lag].corr(sq_ret.iloc[lag:])
        else:
            corr = ret.corr(sq_ret)
        correlations.append(corr)
    
    return np.arange(-lags, lags + 1), np.array(correlations)

lags, corrs = empirical_leverage_effect(returns)

fig, ax = plt.subplots(figsize=(12, 5))
colors = ['green' if c > 0 else 'red' for c in corrs]
ax.bar(lags, corrs, color=colors, alpha=0.7, edgecolor='white')
ax.axhline(0, color='black', linewidth=0.5)
ax.axvline(0, color='black', linewidth=0.5, linestyle='--')
ax.set_xlabel('Lag k')
ax.set_ylabel('Corr(r_t, r²_{t+k})')
ax.set_title('S&P 500: Cross-Correlation of Returns with Future Squared Returns', fontsize=13)

plt.tight_layout()
plt.show()

print(f"\nCorr(r_t, r²_{{t+1}}): {corrs[lags==1][0]:.4f}")
print(f"Corr(r_t, r²_{{t+5}}): {corrs[lags==5][0]:.4f}")
print("\nNegative values at positive lags confirm the leverage effect.")


Corr(r_t, r²_{t+1}): -0.1962
Corr(r_t, r²_{t+5}): -0.1962

Negative values at positive lags confirm the leverage effect.

2. GJR-GARCH (Glosten-Jagannathan-Runkle)¶

2.1 Model Specification¶

GJR-GARCH(1,1) adds an asymmetric term for negative shocks:

\boxed{\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \gamma \epsilon_{t-1}^2 I_{t-1} + \beta \sigma_{t-1}^2}

(2)

where $I_{t-1} = \mathbf{1}_{\{\epsilon_{t-1} < 0\}}$ is an indicator for negative shocks.

Interpretation:

Positive shock ( $\epsilon_{t-1} > 0$ ): Impact is $\alpha$
Negative shock ( $\epsilon_{t-1} < 0$ ): Impact is $\alpha + \gamma$
$\gamma > 0$ indicates leverage effect

2.2 Stationarity Condition¶

\alpha + \frac{\gamma}{2} + \beta < 1

(3)

2.3 News Impact Curve¶

NIC(\epsilon) = \omega + \beta\bar{\sigma}^2 + \begin{cases} \alpha \epsilon^2 & \text{if } \epsilon \geq 0 \\ (\alpha + \gamma) \epsilon^2 & \text{if } \epsilon < 0 \end{cases}

(4)

def simulate_gjr_garch(n, omega, alpha, gamma, beta, dist='normal', df=5):
    """
    Simulate GJR-GARCH(1,1) process.
    """
    eps = np.zeros(n)
    sigma2 = np.zeros(n)
    
    if dist == 'normal':
        z = np.random.standard_normal(n)
    else:
        z = np.random.standard_t(df, n) / np.sqrt(df / (df - 2))
    
    uncond_var = omega / (1 - alpha - gamma/2 - beta)
    sigma2[0] = uncond_var
    eps[0] = np.sqrt(sigma2[0]) * z[0]
    
    for t in range(1, n):
        indicator = 1 if eps[t-1] < 0 else 0
        sigma2[t] = omega + alpha * eps[t-1]**2 + gamma * eps[t-1]**2 * indicator + beta * sigma2[t-1]
        eps[t] = np.sqrt(sigma2[t]) * z[t]
    
    return eps, sigma2

# Simulate GJR-GARCH
n = 2000
omega, alpha, gamma, beta = 0.05, 0.05, 0.10, 0.85

np.random.seed(42)
eps_gjr, sigma2_gjr = simulate_gjr_garch(n, omega, alpha, gamma, beta)

fig, axes = plt.subplots(2, 1, figsize=(14, 8))

axes[0].plot(eps_gjr, color='steelblue', linewidth=0.5)
axes[0].set_title('Simulated GJR-GARCH(1,1) Returns', fontsize=12)
axes[0].set_ylabel('Returns')

axes[1].plot(np.sqrt(sigma2_gjr), color='red', linewidth=0.8)
axes[1].set_title('Conditional Volatility', fontsize=12)
axes[1].set_ylabel('σ_t')

plt.tight_layout()
plt.show()

# Fit GJR-GARCH to S&P 500
model_gjr = arch_model(returns, vol='Garch', p=1, o=1, q=1, dist='t')
fit_gjr = model_gjr.fit(disp='off')

print("GJR-GARCH(1,1) Estimation Results")
print("="*60)
print(fit_gjr.summary())

alpha_hat = fit_gjr.params['alpha[1]']
gamma_hat = fit_gjr.params['gamma[1]']

print(f"\n\nInterpretation:")
print(f"Impact of positive shock: α = {alpha_hat:.4f}")
print(f"Impact of negative shock: α + γ = {alpha_hat + gamma_hat:.4f}")
print(f"Asymmetry ratio: (α + γ)/α = {(alpha_hat + gamma_hat)/max(alpha_hat, 0.001):.2f}")

GJR-GARCH(1,1) Estimation Results
============================================================
                      Constant Mean - GJR-GARCH Model Results                       
====================================================================================
Dep. Variable:                      returns   R-squared:                       0.000
Mean Model:                   Constant Mean   Adj. R-squared:                  0.000
Vol Model:                        GJR-GARCH   Log-Likelihood:               -4643.32
Distribution:      Standardized Student's t   AIC:                           9298.65
Method:                  Maximum Likelihood   BIC:                           9336.06
                                              No. Observations:                 3772
Date:                      Thu, Feb 05 2026   Df Residuals:                     3771
Time:                              09:32:08   Df Model:                            1
                                 Mean Model                                 
============================================================================
                 coef    std err          t      P>|t|      95.0% Conf. Int.
----------------------------------------------------------------------------
mu             0.0747  1.054e-02      7.091  1.334e-12 [5.406e-02,9.536e-02]
                               Volatility Model                              
=============================================================================
                 coef    std err          t      P>|t|       95.0% Conf. Int.
-----------------------------------------------------------------------------
omega          0.0288  4.634e-03      6.223  4.885e-10  [1.976e-02,3.792e-02]
alpha[1]   2.0125e-12  1.317e-02  1.529e-10      1.000 [-2.580e-02,2.580e-02]
gamma[1]       0.2917  3.587e-02      8.131  4.244e-16      [  0.221,  0.362]
beta[1]        0.8303  1.690e-02     49.133      0.000      [  0.797,  0.863]
                              Distribution                              
========================================================================
                 coef    std err          t      P>|t|  95.0% Conf. Int.
------------------------------------------------------------------------
nu             5.8145      0.566     10.264  1.024e-24 [  4.704,  6.925]
========================================================================

Covariance estimator: robust


Interpretation:
Impact of positive shock: α = 0.0000
Impact of negative shock: α + γ = 0.2917
Asymmetry ratio: (α + γ)/α = 291.69

3. EGARCH (Exponential GARCH)¶

3.1 Model Specification¶

Nelson (1991) proposed EGARCH, which models log-variance:

\boxed{\ln(\sigma_t^2) = \omega + \alpha g(z_{t-1}) + \beta \ln(\sigma_{t-1}^2)}

(5)

where $z_t = \epsilon_t / \sigma_t$ and:

g(z_t) = \theta z_t + \gamma(|z_t| - E[|z_t|])

(6)

3.2 Key Properties¶

Advantages over GJR-GARCH:

No parameter restrictions for positivity (since we model log-variance)
Continuous news impact curve at zero
Log-linear structure facilitates interpretation

Asymmetry: $\theta < 0$ implies leverage effect

# Fit EGARCH to S&P 500
model_egarch = arch_model(returns, vol='EGARCH', p=1, o=1, q=1, dist='t')
fit_egarch = model_egarch.fit(disp='off')

print("EGARCH(1,1) Estimation Results")
print("="*60)
print(fit_egarch.summary())

print(f"\n\nNote: In EGARCH, alpha[1] < 0 indicates leverage effect.")
print(f"alpha[1] = {fit_egarch.params['alpha[1]']:.4f}")

EGARCH(1,1) Estimation Results
============================================================
                        Constant Mean - EGARCH Model Results                        
====================================================================================
Dep. Variable:                      returns   R-squared:                       0.000
Mean Model:                   Constant Mean   Adj. R-squared:                  0.000
Vol Model:                           EGARCH   Log-Likelihood:               -4630.50
Distribution:      Standardized Student's t   AIC:                           9273.01
Method:                  Maximum Likelihood   BIC:                           9310.42
                                              No. Observations:                 3772
Date:                      Thu, Feb 05 2026   Df Residuals:                     3771
Time:                              09:32:20   Df Model:                            1
                                 Mean Model                                 
============================================================================
                 coef    std err          t      P>|t|      95.0% Conf. Int.
----------------------------------------------------------------------------
mu             0.0694  1.062e-02      6.532  6.508e-11 [4.855e-02,9.018e-02]
                               Volatility Model                               
==============================================================================
                  coef    std err          t      P>|t|       95.0% Conf. Int.
------------------------------------------------------------------------------
omega      -9.4307e-03  5.012e-03     -1.882  5.987e-02 [-1.925e-02,3.919e-04]
alpha[1]        0.1876  2.126e-02      8.823  1.115e-18      [  0.146,  0.229]
gamma[1]       -0.1973  1.610e-02    -12.257  1.534e-34      [ -0.229, -0.166]
beta[1]         0.9642  6.178e-03    156.062      0.000      [  0.952,  0.976]
                              Distribution                              
========================================================================
                 coef    std err          t      P>|t|  95.0% Conf. Int.
------------------------------------------------------------------------
nu             6.0058      0.601      9.989  1.697e-23 [  4.827,  7.184]
========================================================================

Covariance estimator: robust


Note: In EGARCH, alpha[1] < 0 indicates leverage effect.
alpha[1] = 0.1876

4. News Impact Curves Comparison¶

# Fit symmetric GARCH for comparison
model_garch = arch_model(returns, vol='Garch', p=1, q=1, dist='t')
fit_garch = model_garch.fit(disp='off')

def compute_news_impact_curve(fit, model_type, shock_range, uncond_var):
    """Compute news impact curve for a given model."""
    params = fit.params
    sample_std = np.sqrt(uncond_var)
    shocks = shock_range * sample_std
    
    if model_type == 'GARCH':
        omega = params['omega']
        alpha = params['alpha[1]']
        beta = params['beta[1]']
        nic = omega + beta * uncond_var + alpha * shocks**2
        
    elif model_type == 'GJR':
        omega = params['omega']
        alpha = params['alpha[1]']
        gamma = params['gamma[1]']
        beta = params['beta[1]']
        nic = omega + beta * uncond_var + alpha * shocks**2
        nic[shocks < 0] += gamma * shocks[shocks < 0]**2
        
    elif model_type == 'EGARCH':
        omega = params['omega']
        alpha = params['alpha[1]']
        gamma = params['gamma[1]']
        beta = params['beta[1]']
        z = shocks / sample_std
        log_var = omega + beta * np.log(uncond_var) + alpha * z + gamma * (np.abs(z) - np.sqrt(2/np.pi))
        nic = np.exp(log_var)
    
    return shocks, nic

# Compute NICs
shock_range = np.linspace(-4, 4, 200)
uncond_var = returns.var()

shocks_g, nic_garch = compute_news_impact_curve(fit_garch, 'GARCH', shock_range, uncond_var)
shocks_j, nic_gjr = compute_news_impact_curve(fit_gjr, 'GJR', shock_range, uncond_var)
shocks_e, nic_egarch = compute_news_impact_curve(fit_egarch, 'EGARCH', shock_range, uncond_var)

# Plot
fig, ax = plt.subplots(figsize=(12, 7))

ax.plot(shocks_g, np.sqrt(nic_garch), 'b-', linewidth=2, label='GARCH(1,1)')
ax.plot(shocks_j, np.sqrt(nic_gjr), 'r--', linewidth=2, label='GJR-GARCH(1,1)')
ax.plot(shocks_e, np.sqrt(nic_egarch), 'g-.', linewidth=2, label='EGARCH(1,1)')

ax.axvline(0, color='gray', linestyle=':', alpha=0.5)
ax.set_xlabel('Shock ε_{t-1}', fontsize=12)
ax.set_ylabel('Conditional Volatility σ_t', fontsize=12)
ax.set_title('News Impact Curves: Symmetric vs Asymmetric Models', fontsize=14)
ax.legend(loc='upper center', fontsize=11)

plt.tight_layout()
plt.show()

print("\nKey observations:")
print("1. GARCH is symmetric (parabola centered at zero)")
print("2. GJR-GARCH has a kink at zero - steeper on the left (negative shocks)")
print("3. EGARCH is smooth and asymmetric")


Key observations:
1. GARCH is symmetric (parabola centered at zero)
2. GJR-GARCH has a kink at zero - steeper on the left (negative shocks)
3. EGARCH is smooth and asymmetric

5. Testing for Asymmetry¶

5.1 Sign Bias Test (Engle & Ng, 1993)¶

Regress squared standardized residuals on sign indicators:

\hat{z}_t^2 = c_0 + c_1 S_{t-1}^- + c_2 S_{t-1}^- \hat{\epsilon}_{t-1} + c_3 S_{t-1}^+ \hat{\epsilon}_{t-1} + u_t

(7)

where $S_t^- = \mathbf{1}_{\{\epsilon_t < 0\}}$ and $S_t^+ = 1 - S_t^-$ .

Sign bias test: $H_0: c_1 = 0$
Negative size bias test: $H_0: c_2 = 0$
Positive size bias test: $H_0: c_3 = 0$
Joint test: $H_0: c_1 = c_2 = c_3 = 0$

import statsmodels.api as sm

def sign_bias_test(fitted_model):
    """
    Perform Engle-Ng sign bias tests for asymmetry.
    """
    std_resid = fitted_model.std_resid
    resid = fitted_model.resid
    
    # Create variables
    z2 = std_resid ** 2
    S_neg = (resid < 0).astype(float)
    S_pos = 1 - S_neg
    
    # Lag the indicators
    df = pd.DataFrame({
        'z2': z2,
        'S_neg_lag': S_neg.shift(1),
        'neg_size': S_neg.shift(1) * resid.shift(1),
        'pos_size': S_pos.shift(1) * resid.shift(1)
    }).dropna()
    
    # Regression
    X = sm.add_constant(df[['S_neg_lag', 'neg_size', 'pos_size']])
    y = df['z2']
    
    model = sm.OLS(y, X).fit()
    
    # Individual tests
    print("Engle-Ng Sign Bias Tests")
    print("="*60)
    print(f"\n{'Test':<25} {'Coef':>10} {'t-stat':>10} {'p-value':>10}")
    print("-"*60)
    
    tests = [
        ('Sign Bias', 'S_neg_lag'),
        ('Negative Size Bias', 'neg_size'),
        ('Positive Size Bias', 'pos_size')
    ]
    
    for test_name, var in tests:
        coef = model.params[var]
        tstat = model.tvalues[var]
        pval = model.pvalues[var]
        sig = '***' if pval < 0.01 else ('**' if pval < 0.05 else ('*' if pval < 0.1 else ''))
        print(f"{test_name:<25} {coef:>10.4f} {tstat:>10.2f} {pval:>10.4f} {sig}")
    
    # Joint test (F-test)
    r_matrix = np.eye(4)[1:]  # Test all except constant
    f_test = model.f_test(r_matrix)
    print(f"\n{'Joint Test':<25} {'F-stat':>10} {'':>10} {'p-value':>10}")
    print("-"*60)
    print(f"{'H0: c1=c2=c3=0':<25} {f_test.fvalue:>10.2f} {'':<10} {f_test.pvalue:>10.4f}")
    
    return model

# Test on symmetric GARCH residuals
print("Testing GARCH(1,1) residuals for asymmetry:")
sb_garch = sign_bias_test(fit_garch)

print("\n" + "="*60)
print("\nTesting GJR-GARCH(1,1) residuals for remaining asymmetry:")
sb_gjr = sign_bias_test(fit_gjr)

Testing GARCH(1,1) residuals for asymmetry:
Engle-Ng Sign Bias Tests
============================================================

Test                            Coef     t-stat    p-value
------------------------------------------------------------
Sign Bias                     0.3460       3.88     0.0001 ***
Negative Size Bias            0.0629       1.20     0.2287 
Positive Size Bias           -0.0678      -1.03     0.3045 

Joint Test                    F-stat               p-value
------------------------------------------------------------
H0: c1=c2=c3=0                  9.98                0.0000

============================================================

Testing GJR-GARCH(1,1) residuals for remaining asymmetry:
Engle-Ng Sign Bias Tests
============================================================

Test                            Coef     t-stat    p-value
------------------------------------------------------------
Sign Bias                     0.3186       3.13     0.0018 ***
Negative Size Bias            0.1583       2.62     0.0089 ***
Positive Size Bias           -0.0147      -0.20     0.8436 

Joint Test                    F-stat               p-value
------------------------------------------------------------
H0: c1=c2=c3=0                  4.94                0.0020

6. Model Comparison¶

# Compare all models
models = {
    'GARCH(1,1)': fit_garch,
    'GJR-GARCH(1,1)': fit_gjr,
    'EGARCH(1,1)': fit_egarch
}

comparison = []
for name, fit in models.items():
    comparison.append({
        'Model': name,
        'Log-Lik': fit.loglikelihood,
        'AIC': fit.aic,
        'BIC': fit.bic,
        'Params': fit.num_params
    })

comparison_df = pd.DataFrame(comparison).set_index('Model')
print("Model Comparison")
print("="*60)
print(comparison_df.round(2))
print(f"\nBest model by BIC: {comparison_df['BIC'].idxmin()}")

Model Comparison
============================================================
                Log-Lik      AIC      BIC  Params
Model                                            
GARCH(1,1)     -4711.96  9433.91  9465.09       5
GJR-GARCH(1,1) -4643.32  9298.65  9336.06       6
EGARCH(1,1)    -4630.50  9273.01  9310.42       6

Best model by BIC: EGARCH(1,1)

# Plot conditional volatility comparison
fig, axes = plt.subplots(3, 1, figsize=(14, 10), sharex=True)

for ax, (name, fit) in zip(axes, models.items()):
    vol = fit.conditional_volatility
    ax.plot(vol.index, vol.values, linewidth=0.8, label=name)
    ax.fill_between(vol.index, 0, vol.values, alpha=0.3)
    ax.set_ylabel('Volatility (%)')
    ax.set_title(name, fontsize=12)
    ax.set_ylim(0, vol.max() * 1.1)

plt.xlabel('Date')
plt.tight_layout()
plt.suptitle('Conditional Volatility: Model Comparison', y=1.01, fontsize=14, fontweight='bold')
plt.show()

7. Cryptocurrency Application¶

Does the leverage effect exist in cryptocurrency markets?

# Download Bitcoin data
btc = yf.download('BTC-USD', start='2018-01-01', end='2024-12-31', progress=False)
if isinstance(btc.columns, pd.MultiIndex):
    btc.columns = btc.columns.get_level_values(0)

btc_returns = (np.log(btc['Close'] / btc['Close'].shift(1)) * 100).dropna()

# Fit models
btc_garch = arch_model(btc_returns, vol='Garch', p=1, q=1, dist='t').fit(disp='off')
btc_gjr = arch_model(btc_returns, vol='Garch', p=1, o=1, q=1, dist='t').fit(disp='off')

print("Bitcoin: GARCH(1,1)")
print(btc_garch.summary())

print("\n\nBitcoin: GJR-GARCH(1,1)")
print(btc_gjr.summary())

btc_gamma = btc_gjr.params['gamma[1]']
print(f"\n\nBitcoin asymmetry parameter γ = {btc_gamma:.4f}")
if btc_gamma > 0:
    print("Bitcoin shows leverage effect (surprising for crypto!)")
else:
    print("Bitcoin shows inverse leverage effect (common in crypto)")

Bitcoin: GARCH(1,1)
                        Constant Mean - GARCH Model Results                         
====================================================================================
Dep. Variable:                        Close   R-squared:                       0.000
Mean Model:                   Constant Mean   Adj. R-squared:                  0.000
Vol Model:                            GARCH   Log-Likelihood:               -6402.25
Distribution:      Standardized Student's t   AIC:                           12814.5
Method:                  Maximum Likelihood   BIC:                           12843.7
                                              No. Observations:                 2555
Date:                      Thu, Feb 05 2026   Df Residuals:                     2554
Time:                              09:35:05   Df Model:                            1
                                 Mean Model                                
===========================================================================
                 coef    std err          t      P>|t|     95.0% Conf. Int.
---------------------------------------------------------------------------
mu             0.0740  4.189e-02      1.767  7.718e-02 [-8.071e-03,  0.156]
                              Volatility Model                              
============================================================================
                 coef    std err          t      P>|t|      95.0% Conf. Int.
----------------------------------------------------------------------------
omega          0.1355  8.336e-02      1.626      0.104  [-2.784e-02,  0.299]
alpha[1]       0.0723  1.091e-02      6.627  3.436e-11 [5.091e-02,9.368e-02]
beta[1]        0.9277  1.334e-02     69.522      0.000     [  0.902,  0.954]
                              Distribution                              
========================================================================
                 coef    std err          t      P>|t|  95.0% Conf. Int.
------------------------------------------------------------------------
nu             3.1152      0.155     20.057  1.755e-89 [  2.811,  3.420]
========================================================================

Covariance estimator: robust


Bitcoin: GJR-GARCH(1,1)
                      Constant Mean - GJR-GARCH Model Results                       
====================================================================================
Dep. Variable:                        Close   R-squared:                       0.000
Mean Model:                   Constant Mean   Adj. R-squared:                  0.000
Vol Model:                        GJR-GARCH   Log-Likelihood:               -6402.20
Distribution:      Standardized Student's t   AIC:                           12816.4
Method:                  Maximum Likelihood   BIC:                           12851.5
                                              No. Observations:                 2555
Date:                      Thu, Feb 05 2026   Df Residuals:                     2554
Time:                              09:35:05   Df Model:                            1
                                 Mean Model                                
===========================================================================
                 coef    std err          t      P>|t|     95.0% Conf. Int.
---------------------------------------------------------------------------
mu             0.0749  4.176e-02      1.794  7.274e-02 [-6.910e-03,  0.157]
                               Volatility Model                               
==============================================================================
                  coef    std err          t      P>|t|       95.0% Conf. Int.
------------------------------------------------------------------------------
omega           0.1295      0.100      1.292      0.196   [-6.696e-02,  0.326]
alpha[1]        0.0739  1.130e-02      6.537  6.273e-11  [5.174e-02,9.605e-02]
gamma[1]   -5.6362e-03  2.032e-02     -0.277      0.782 [-4.547e-02,3.420e-02]
beta[1]         0.9289  1.652e-02     56.221      0.000      [  0.897,  0.961]
                              Distribution                              
========================================================================
                 coef    std err          t      P>|t|  95.0% Conf. Int.
------------------------------------------------------------------------
nu             3.1211      0.158     19.770  5.444e-87 [  2.812,  3.431]
========================================================================

Covariance estimator: robust


Bitcoin asymmetry parameter γ = -0.0056
Bitcoin shows inverse leverage effect (common in crypto)

8. Summary¶

Key Takeaways¶

Leverage effect: Negative shocks increase volatility more than positive shocks (in equities)
GJR-GARCH: Adds asymmetric term $\gamma \epsilon_{t-1}^2 I_{t-1}$ ; piecewise parabolic NIC
EGARCH: Models log-variance; smooth asymmetric NIC; no positivity constraints
Testing: Engle-Ng sign bias tests detect asymmetry in GARCH residuals
Crypto: May show inverse leverage effect (positive returns increase volatility)

Preview of Session 4¶

Session 4 covers advanced univariate models: FIGARCH (long memory), APARCH (power transformations), and Component GARCH (short-run vs long-run volatility).

Exercises¶

Exercise 1: Leverage Effect Across Assets¶

Compare the leverage effect (γ parameter) for: S&P 500, Gold, EUR/USD, and VIX. Which assets show the strongest asymmetry?

Exercise 2: TGARCH Implementation¶

Implement TGARCH(1,1) from scratch and compare its NIC to GJR-GARCH.

Exercise 3: Forecast Comparison¶

Compare 22-day ahead volatility forecasts from GARCH, GJR-GARCH, and EGARCH. Which model performs best during the COVID-19 crash?

Exercise 4: Time-Varying Asymmetry¶

Estimate GJR-GARCH on rolling 2-year windows for S&P 500. How does the γ parameter evolve over time?

Exercise 5: Inverse Leverage in Crypto¶

Test multiple cryptocurrencies (ETH, BNB, SOL) for the leverage effect. Is the inverse leverage effect consistent across crypto assets?

References¶

Black, F. (1976). Studies of stock price volatility changes. Proceedings of the 1976 Meetings of the American Statistical Association, 177-181.
Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59(2), 347-370.
Glosten, L. R., Jagannathan, R., & Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48(5), 1779-1801.
Engle, R. F., & Ng, V. K. (1993). Measuring and testing the impact of news on volatility. Journal of Finance, 48(5), 1749-1778.
Zakoian, J. M. (1994). Threshold heteroskedastic models. Journal of Economic Dynamics and Control, 18(5), 931-955.