r"""
Nelson's EGARCH
===============

GARCH reacts to the size of yesterday's shock but not its sign, and its
coefficients must be kept positive. Daniel Nelson (1991) modelled the
logarithm of the variance instead,

.. math::

   \ln\sigma_t^2 = \omega + \beta\ln\sigma_{t-1}^2
   + \alpha\bigl(|z_{t-1}| - E|z|\bigr) + \gamma z_{t-1},

which is positive by construction and asymmetric when :math:`\gamma \ne 0`.
For stock returns :math:`\gamma < 0`: a fall raises volatility more than a
rise of the same size, Black's (1976) "leverage effect".
"""

# %%
import matplotlib.pyplot as plt
import numpy as np

from financialkit.time_series import EGARCH, GARCH
from financialkit.time_series.visualizers import plot_news_impact

# %%
# The news impact curve
# ---------------------
# Tomorrow's variance as a function of today's standardized shock, starting
# from the same variance. GARCH's curve is a symmetric parabola; EGARCH's is
# exponential and tilted towards bad news.

egarch = EGARCH(omega=0.0, alpha=0.15, gamma=-0.08, beta=0.97)
garch = GARCH(omega=0.03, alpha=0.08, beta=0.89)
ax = plot_news_impact({"GARCH(1,1)": garch, "EGARCH(1,1)": egarch}, var_prev=1.0)
ax.set_title("News impact: bad news raises EGARCH volatility more")

# %%
# Estimating the asymmetry
# ------------------------
# Fitting both models to EGARCH returns: the EGARCH fit recovers a
# significantly negative :math:`\gamma`, and its log-likelihood beats
# GARCH's with one extra parameter.

r = egarch.simulate(3000, rng=1).returns
eg = EGARCH.fit(r)
ga = GARCH.fit(r)
for k in ("alpha", "gamma", "beta"):
    print(f"{k:5s} true {egarch.params[k]:+.3f}  fitted {eg.params[k]:+.3f} +/- {eg.std_errors[k]:.3f}")
print(f"log-likelihood: EGARCH {eg.loglik:.1f}, GARCH {ga.loglik:.1f}; AIC {eg.aic:.1f} vs {ga.aic:.1f}")

fig, ax = plt.subplots(figsize=(8, 3.5))
ax.plot(np.sqrt(eg.variance[:600]), label="EGARCH")
ax.plot(np.sqrt(ga.variance[:600]), label="GARCH", alpha=0.8)
ax.set_xlabel("day")
ax.set_ylabel(r"fitted $\sigma_t$")
ax.legend()
plt.show()