r"""
Christoffersen's independence test
==================================

Kupiec's test counts exceptions but ignores when they happen. A VaR
that is right on average but slow to react produces exceptions in
bunches, exactly when a bank can least afford them. Christoffersen
(1998) fitted a two-state Markov chain to the hit sequence and tested
whether a hit is more likely the day after a hit,
:math:`\pi_{11} > \pi_{01}`; adding Kupiec's statistic gives the
conditional-coverage test, :math:`LR_{cc} = LR_{uc} + LR_{ind} \sim \chi^2_2`.
"""

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

from financialkit.risk import christoffersen_independence, christoffersen_test, kupiec_test, parametric_var, var_exceptions
from financialkit.risk.visualizers import plot_backtest
from financialkit.time_series import GARCH

# %%
# A constant VaR on a GARCH market
# --------------------------------
#
# Returns follow a GARCH(1,1) process, simulated with
# :class:`financialkit.time_series.GARCH`. One VaR uses the unconditional
# volatility, the other the GARCH conditional volatility. Both are
# exceeded about as often as they should be, so Kupiec's test is
# satisfied, but the constant VaR's exceptions arrive in the volatile
# spells.

n = 2000
model = GARCH(omega=2e-6, alpha=0.20, beta=0.78)
path = model.simulate(n, rng=0)
r, sig2 = path.returns, path.variance
static = parametric_var(0.0, np.sqrt(model.unconditional_variance), 0.99)
dynamic = parametric_var(0.0, np.sqrt(sig2), 0.99)
for name, var in [("constant VaR", static), ("GARCH VaR", dynamic)]:
    hits = var_exceptions(r, var)
    uc, ind, cc = kupiec_test(hits), christoffersen_independence(hits), christoffersen_test(hits)
    print(f"{name:12s}: {uc.n_exceptions} hits; p(uc) = {uc.p_value:.3f}, p(ind) = {ind.p_value:.4f}, p(cc) = {cc.p_value:.4f}")

fig, axes = plt.subplots(2, 1, figsize=(8, 6), sharex=True)
plot_backtest(r, np.full(n, static), ax=axes[0])
axes[0].set_title("constant VaR: exceptions cluster")
plot_backtest(r, dynamic, ax=axes[1])
axes[1].set_title("conditional VaR: exceptions scattered")
fig.tight_layout()

# %%
# How often each test notices
# ---------------------------
#
# On one path the evidence can go either way, so repeat the experiment.
# The independence test rejects the constant VaR on most paths, and the
# conditional VaR at about its 5% size. Kupiec also rejects the constant
# VaR fairly often: GARCH returns are fat-tailed, so a normal VaR with
# the right variance is exceeded too often on average.

paths = [model.simulate(n, rng=seed) for seed in range(1, 301)]
R = np.array([p.returns for p in paths])
S2 = np.array([p.variance for p in paths])
for name, V in [("constant VaR", np.full_like(R, static)), ("GARCH VaR", parametric_var(0.0, np.sqrt(S2), 0.99))]:
    H = var_exceptions(R, V)
    uc = np.mean([kupiec_test(h).reject() for h in H])
    ind = np.mean([christoffersen_independence(h).reject() for h in H])
    print(f"{name:12s}: Kupiec rejects {uc:.0%}, independence rejects {ind:.0%} of 300 paths")

plt.show()