time_series_finance.tex

% Financial Time Series Analysis Template
% Topics: Stylized facts, ARIMA modeling, GARCH volatility, cointegration, regime switching
% Style: Econometric research report with financial data analysis

\documentclass[a4paper, 11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath, amssymb}
\usepackage{graphicx}
\usepackage{siunitx}
\usepackage{booktabs}
\usepackage{subcaption}
\usepackage[makestderr]{pythontex}

% Theorem environments
\newtheorem{definition}{Definition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}{Example}[section]
\newtheorem{remark}{Remark}[section]

\title{Financial Time Series Analysis: ARIMA, GARCH, and Cointegration}
\author{Quantitative Finance Research Group}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
This report presents a comprehensive econometric analysis of financial time series using modern
time series techniques. We examine the stylized facts of asset returns including fat tails,
volatility clustering, and leverage effects. We develop ARIMA models for mean dynamics, GARCH
models for conditional heteroskedasticity, test for cointegration between asset pairs, and
implement Markov regime-switching models to capture bull and bear market dynamics. Computational
analysis demonstrates parameter estimation, diagnostic testing, and out-of-sample forecasting
performance across multiple financial time series.
\end{abstract}

\section{Introduction}

Financial time series exhibit distinct empirical regularities that distinguish them from
idealized Gaussian white noise. Understanding these stylized facts and developing appropriate
models is crucial for risk management, derivative pricing, and portfolio optimization.

\begin{definition}[Stylized Facts of Returns]
Financial asset returns typically display:
\begin{itemize}
\item \textbf{Heavy tails}: Return distributions have excess kurtosis relative to the normal
\item \textbf{Volatility clustering}: Large changes tend to be followed by large changes
\item \textbf{Leverage effect}: Negative returns increase volatility more than positive returns
\item \textbf{Absence of autocorrelation}: Returns are approximately uncorrelated
\item \textbf{Long memory in volatility}: Autocorrelation in squared/absolute returns decays slowly
\end{itemize}
\end{definition}

\section{Theoretical Framework}

\subsection{ARIMA Models}

\begin{definition}[ARIMA Process]
An ARIMA$(p,d,q)$ model for a time series $\{y_t\}$ is:
\begin{equation}
\phi(L)(1-L)^d y_t = \theta(L)\varepsilon_t
\end{equation}
where $\phi(L) = 1 - \phi_1 L - \cdots - \phi_p L^p$ is the autoregressive polynomial,
$\theta(L) = 1 + \theta_1 L + \cdots + \theta_q L^q$ is the moving average polynomial,
$L$ is the lag operator, and $\varepsilon_t \sim \text{WN}(0, \sigma^2)$.
\end{definition}

The parameters $p$, $d$, $q$ control the autoregressive order, degree of differencing, and
moving average order respectively. Stationarity requires the roots of $\phi(z)=0$ to lie
outside the unit circle. The autocorrelation function (ACF) and partial autocorrelation
function (PACF) provide diagnostic tools for model identification.

\begin{theorem}[Box-Jenkins Identification]
For ARIMA model selection:
\begin{itemize}
\item AR$(p)$: ACF decays exponentially, PACF cuts off after lag $p$
\item MA$(q)$: ACF cuts off after lag $q$, PACF decays exponentially
\item ARMA$(p,q)$: Both ACF and PACF decay exponentially
\end{itemize}
\end{theorem}

\subsection{GARCH Models}

\begin{definition}[GARCH Process]
A GARCH$(p,q)$ model specifies conditional mean and variance:
\begin{align}
r_t &= \mu + \varepsilon_t \\
\varepsilon_t &= \sigma_t z_t, \quad z_t \sim \text{i.i.d.}(0,1) \\
\sigma_t^2 &= \omega + \sum_{i=1}^{p} \alpha_i \varepsilon_{t-i}^2 + \sum_{j=1}^{q} \beta_j \sigma_{t-j}^2
\end{align}
where $\omega > 0$, $\alpha_i \geq 0$, $\beta_j \geq 0$, and $\sum \alpha_i + \sum \beta_j < 1$.
\end{definition}

The GARCH(1,1) model is widely used in practice due to its parsimony and ability to capture
volatility clustering. The parameters $\alpha$ and $\beta$ control the reaction to past
shocks and persistence of volatility respectively.

\begin{definition}[GJR-GARCH]
To capture asymmetric volatility (leverage effect):
\begin{equation}
\sigma_t^2 = \omega + (\alpha + \gamma I_{t-1})\varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2
\end{equation}
where $I_{t-1} = 1$ if $\varepsilon_{t-1} < 0$ and zero otherwise. Positive $\gamma$
implies negative shocks increase volatility more than positive shocks.
\end{definition}

\subsection{Cointegration}

\begin{definition}[Cointegration]
Two I$(1)$ series $y_t$ and $x_t$ are cointegrated if there exists $\beta$ such that
$z_t = y_t - \beta x_t$ is I$(0)$. The vector $[1, -\beta]$ is the cointegrating vector.
\end{definition}

\begin{theorem}[Engle-Granger Two-Step]
Test for cointegration:
\begin{enumerate}
\item Estimate $\hat{\beta}$ by regressing $y_t$ on $x_t$
\item Test residuals $\hat{z}_t = y_t - \hat{\beta}x_t$ for unit root
\item Reject unit root implies cointegration exists
\end{enumerate}
\end{theorem}

Cointegrated series share common stochastic trends and cannot drift apart indefinitely. This
forms the basis for pairs trading strategies and error correction models.

\subsection{Regime Switching}

\begin{definition}[Markov Regime-Switching Model]
A two-state model with state-dependent dynamics:
\begin{equation}
r_t = \mu_{s_t} + \sigma_{s_t} \varepsilon_t, \quad s_t \in \{1, 2\}
\end{equation}
where $s_t$ follows a Markov chain with transition matrix:
\begin{equation}
P = \begin{pmatrix} p_{11} & 1-p_{11} \\ 1-p_{22} & p_{22} \end{pmatrix}
\end{equation}
\end{definition}

\section{Computational Analysis}

\subsection{Data Generation and Stylized Facts}

We begin by simulating financial returns with realistic properties and examining their
empirical characteristics. The data generating process incorporates time-varying volatility
and regime switches to mimic observed market behavior.

\begin{pycode}
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
from scipy.optimize import minimize
import warnings
warnings.filterwarnings('ignore')

np.random.seed(42)

# Simulate GARCH(1,1) returns
def simulate_garch11(n, omega, alpha, beta, mu=0.0):
    """Simulate GARCH(1,1) process"""
    returns = np.zeros(n)
    variance = np.zeros(n)
    variance[0] = omega / (1 - alpha - beta)

    for t in range(1, n):
        variance[t] = omega + alpha * returns[t-1]**2 + beta * variance[t-1]
        returns[t] = mu + np.sqrt(variance[t]) * np.random.standard_t(df=6)

    return returns, variance

# GARCH parameters (typical equity values)
omega = 0.00001
alpha = 0.08
beta = 0.90
persistence = alpha + beta
unconditional_vol = np.sqrt(omega / (1 - persistence))

# Simulate 2000 daily returns
n_obs = 2000
returns, volatility = simulate_garch11(n_obs, omega, alpha, beta, mu=0.0005)

# Calculate cumulative returns (log prices)
log_prices = np.cumsum(returns)
prices = 100 * np.exp(log_prices)

# Stylized facts analysis
returns_centered = returns - np.mean(returns)
skewness = stats.skew(returns)
kurtosis = stats.kurtosis(returns, fisher=True)  # Excess kurtosis
jb_stat, jb_pvalue = stats.jarque_bera(returns)

# Autocorrelation function for returns and squared returns
max_lag = 30
acf_returns = np.array([np.corrcoef(returns[:-i], returns[i:])[0,1]
                        if i > 0 else 1.0 for i in range(max_lag)])
acf_squared = np.array([np.corrcoef(returns[:-i]**2, returns[i:]**2)[0,1]
                        if i > 0 else 1.0 for i in range(max_lag)])

# Ljung-Box test statistic
def ljung_box(acf, n_obs, max_lag):
    """Ljung-Box Q statistic"""
    lags = np.arange(1, max_lag + 1)
    Q = n_obs * (n_obs + 2) * np.sum(acf[1:]**2 / (n_obs - lags))
    return Q

Q_returns = ljung_box(acf_returns, n_obs, max_lag)
Q_squared = ljung_box(acf_squared, n_obs, max_lag)

# Plot 1: Price and returns evolution
fig = plt.figure(figsize=(14, 12))

ax1 = fig.add_subplot(3, 3, 1)
time = np.arange(n_obs)
ax1.plot(time, prices, 'b-', linewidth=1.2)
ax1.set_xlabel('Time (days)')
ax1.set_ylabel('Price')
ax1.set_title('Simulated Asset Price Path')
ax1.grid(True, alpha=0.3)

ax2 = fig.add_subplot(3, 3, 2)
ax2.plot(time, returns * 100, 'b-', linewidth=0.8, alpha=0.7)
ax2.set_xlabel('Time (days)')
ax2.set_ylabel('Returns (\%)')
ax2.set_title('Daily Returns (Volatility Clustering)')
ax2.grid(True, alpha=0.3)
ax2.axhline(y=0, color='black', linewidth=0.8)

ax3 = fig.add_subplot(3, 3, 3)
ax3.plot(time, np.sqrt(volatility) * 100, 'r-', linewidth=1.2)
ax3.set_xlabel('Time (days)')
ax3.set_ylabel('Volatility (\%)')
ax3.set_title('Conditional Volatility (GARCH)')
ax3.grid(True, alpha=0.3)

# Plot 2: Return distribution (fat tails)
ax4 = fig.add_subplot(3, 3, 4)
ax4.hist(returns * 100, bins=50, density=True, alpha=0.7,
         color='steelblue', edgecolor='black', label='Empirical')
x_norm = np.linspace(returns.min() * 100, returns.max() * 100, 200)
ax4.plot(x_norm, stats.norm.pdf(x_norm, np.mean(returns) * 100, np.std(returns) * 100),
         'r-', linewidth=2, label='Normal')
ax4.set_xlabel('Returns (\%)')
ax4.set_ylabel('Density')
ax4.set_title(f'Return Distribution (Kurt={kurtosis:.2f})')
ax4.legend(fontsize=8)
ax4.set_yscale('log')
ax4.set_ylim(1e-4, 1e2)

# Plot 3: Q-Q plot
ax5 = fig.add_subplot(3, 3, 5)
stats.probplot(returns, dist="norm", plot=ax5)
ax5.set_title('Normal Q-Q Plot (Fat Tails)')
ax5.grid(True, alpha=0.3)

# Plot 4: ACF of returns
ax6 = fig.add_subplot(3, 3, 6)
lags = np.arange(max_lag)
ax6.bar(lags, acf_returns, width=0.6, color='steelblue', edgecolor='black')
conf_int = 1.96 / np.sqrt(n_obs)
ax6.axhline(y=conf_int, color='red', linestyle='--', linewidth=1.5)
ax6.axhline(y=-conf_int, color='red', linestyle='--', linewidth=1.5)
ax6.axhline(y=0, color='black', linewidth=0.8)
ax6.set_xlabel('Lag')
ax6.set_ylabel('ACF')
ax6.set_title(f'ACF of Returns (Q={Q_returns:.1f})')
ax6.set_ylim(-0.2, 0.2)

# Plot 5: ACF of squared returns (volatility clustering)
ax7 = fig.add_subplot(3, 3, 7)
ax7.bar(lags, acf_squared, width=0.6, color='coral', edgecolor='black')
ax7.axhline(y=conf_int, color='red', linestyle='--', linewidth=1.5)
ax7.axhline(y=-conf_int, color='red', linestyle='--', linewidth=1.5)
ax7.axhline(y=0, color='black', linewidth=0.8)
ax7.set_xlabel('Lag')
ax7.set_ylabel('ACF')
ax7.set_title(f'ACF of Squared Returns (Q={Q_squared:.1f})')

# Save figure state
plt.tight_layout()
plt.savefig('time_series_finance_stylized_facts.pdf', dpi=150, bbox_inches='tight')
plt.close()

# Store results for table
stylized_facts = {
    'mean': np.mean(returns) * 100,
    'std': np.std(returns) * 100,
    'skew': skewness,
    'kurt': kurtosis,
    'jb_stat': jb_stat,
    'jb_pvalue': jb_pvalue,
    'Q_returns': Q_returns,
    'Q_squared': Q_squared
}
\end{pycode}

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{time_series_finance_stylized_facts.pdf}
\caption{Stylized facts of financial returns: (a) Simulated price path exhibiting realistic
dynamics; (b) Returns showing pronounced volatility clustering where periods of high volatility
follow periods of high volatility; (c) Time-varying conditional volatility estimated from the
GARCH process; (d) Return distribution displaying excess kurtosis and fat tails relative to
the normal distribution (log scale emphasizes tail behavior); (e) Q-Q plot showing systematic
departure from normality in both tails; (f) Autocorrelation function of returns showing no
significant serial correlation; (g) ACF of squared returns displaying strong positive
autocorrelation indicating volatility persistence and long memory.}
\label{fig:stylized}
\end{figure}

The simulated returns exhibit the canonical stylized facts observed in equity markets. The
volatility clustering is evident in panel (b), where tranquil and turbulent periods alternate.
The return distribution shows excess kurtosis of \py{f"{kurtosis:.2f}"}, substantially higher
than the normal distribution's value of zero, confirming the presence of fat tails. The
Ljung-Box statistic for squared returns (\py{f"{Q_squared:.1f}"}) far exceeds critical
values, providing strong evidence of conditional heteroskedasticity.

\subsection{ARIMA Model Estimation}

We now construct an ARIMA model for a synthetic time series with both autoregressive and moving
average components. Model identification proceeds via ACF and PACF analysis, followed by
maximum likelihood estimation.

\begin{pycode}
# Simulate ARMA(2,1) process
def simulate_arma(n, phi, theta, sigma):
    """Simulate ARMA process"""
    p = len(phi)
    q = len(theta)
    y = np.zeros(n + max(p, q))
    eps = sigma * np.random.randn(n + max(p, q))

    for t in range(max(p, q), n + max(p, q)):
        ar_part = sum(phi[i] * y[t-i-1] for i in range(p))
        ma_part = sum(theta[i] * eps[t-i-1] for i in range(q))
        y[t] = ar_part + ma_part + eps[t]

    return y[max(p, q):]

# True ARMA(2,1) parameters
phi_true = [0.5, -0.3]
theta_true = [0.4]
sigma_arma = 1.0

# Simulate 500 observations
n_arma = 500
y_arma = simulate_arma(n_arma, phi_true, theta_true, sigma_arma)

# Calculate ACF and PACF
def acf_pacf(y, max_lag):
    """Compute ACF and PACF"""
    n = len(y)
    y_centered = y - np.mean(y)

    # ACF
    acf = np.correlate(y_centered, y_centered, mode='full')
    acf = acf[n-1:] / acf[n-1]
    acf = acf[:max_lag]

    # PACF using Durbin-Levinson recursion
    pacf = np.zeros(max_lag)
    pacf[0] = acf[0]

    for k in range(1, max_lag):
        num = acf[k] - sum(pacf[j] * acf[k-j-1] for j in range(k))
        den = 1 - sum(pacf[j] * acf[j] for j in range(k))
        pacf_k = num / den if abs(den) > 1e-10 else 0

        # Update previous PACF values
        pacf_old = pacf[:k].copy()
        for j in range(k):
            pacf[j] = pacf_old[j] - pacf_k * pacf_old[k-j-1]
        pacf[k] = pacf_k

    return acf, pacf

acf_y, pacf_y = acf_pacf(y_arma, 25)

# Estimate ARMA(2,1) using conditional maximum likelihood
def arma_loglik(params, y):
    """ARMA log-likelihood (simplified)"""
    phi1, phi2, theta1, sigma = params
    n = len(y)

    # Initialize
    residuals = np.zeros(n)
    y_mean = np.mean(y)
    y_centered = y - y_mean

    # Compute residuals recursively
    for t in range(2, n):
        pred = phi1 * y_centered[t-1] + phi2 * y_centered[t-2]
        if t > 2:
            pred += theta1 * residuals[t-1]
        residuals[t] = y_centered[t] - pred

    # Log-likelihood (excluding first two obs)
    resid_used = residuals[2:]
    loglik = -0.5 * n * np.log(2 * np.pi) - 0.5 * n * np.log(sigma**2) \
             - 0.5 * np.sum(resid_used**2) / sigma**2

    return -loglik  # Minimize negative log-likelihood

# Initial guess
init_params = [0.3, 0.0, 0.0, 1.0]

# Optimize with bounds
bounds = [(-0.99, 0.99), (-0.99, 0.99), (-0.99, 0.99), (0.01, 10)]
result_arma = minimize(arma_loglik, init_params, args=(y_arma,),
                       bounds=bounds, method='L-BFGS-B')
phi1_est, phi2_est, theta1_est, sigma_est = result_arma.x

# One-step ahead forecasts
n_forecast = 50
y_forecast = np.zeros(n_forecast)
y_history = list(y_arma[-10:])  # Use last 10 observations

for h in range(n_forecast):
    pred = phi1_est * (y_history[-1] - np.mean(y_arma)) + \
           phi2_est * (y_history[-2] - np.mean(y_arma)) + np.mean(y_arma)
    y_forecast[h] = pred
    y_history.append(pred)

# Forecast standard errors (approximate)
forecast_se = sigma_est * np.sqrt(1 + np.arange(1, n_forecast + 1) * 0.1)

# Plot ARIMA analysis
fig2 = plt.figure(figsize=(14, 10))

ax1 = fig2.add_subplot(2, 3, 1)
ax1.plot(y_arma, 'b-', linewidth=1.2)
ax1.set_xlabel('Time')
ax1.set_ylabel('Value')
ax1.set_title('ARMA(2,1) Time Series')
ax1.grid(True, alpha=0.3)

ax2 = fig2.add_subplot(2, 3, 2)
lags_acf = np.arange(len(acf_y))
ax2.bar(lags_acf, acf_y, width=0.6, color='steelblue', edgecolor='black')
conf_int_acf = 1.96 / np.sqrt(n_arma)
ax2.axhline(y=conf_int_acf, color='red', linestyle='--', linewidth=1.5)
ax2.axhline(y=-conf_int_acf, color='red', linestyle='--', linewidth=1.5)
ax2.axhline(y=0, color='black', linewidth=0.8)
ax2.set_xlabel('Lag')
ax2.set_ylabel('ACF')
ax2.set_title('Autocorrelation Function')

ax3 = fig2.add_subplot(2, 3, 3)
ax3.bar(lags_acf, pacf_y, width=0.6, color='coral', edgecolor='black')
ax3.axhline(y=conf_int_acf, color='red', linestyle='--', linewidth=1.5)
ax3.axhline(y=-conf_int_acf, color='red', linestyle='--', linewidth=1.5)
ax3.axhline(y=0, color='black', linewidth=0.8)
ax3.set_xlabel('Lag')
ax3.set_ylabel('PACF')
ax3.set_title('Partial Autocorrelation Function')

ax4 = fig2.add_subplot(2, 3, 4)
time_forecast = np.arange(n_arma - 50, n_arma + n_forecast)
ax4.plot(np.arange(n_arma - 50, n_arma), y_arma[-50:], 'b-',
         linewidth=1.2, label='Observed')
ax4.plot(np.arange(n_arma, n_arma + n_forecast), y_forecast, 'r-',
         linewidth=1.5, label='Forecast')
ax4.fill_between(np.arange(n_arma, n_arma + n_forecast),
                  y_forecast - 1.96 * forecast_se,
                  y_forecast + 1.96 * forecast_se,
                  alpha=0.3, color='red')
ax4.axvline(x=n_arma, color='black', linestyle='--', linewidth=1)
ax4.set_xlabel('Time')
ax4.set_ylabel('Value')
ax4.set_title('ARMA Forecast (95\% CI)')
ax4.legend(fontsize=9)
ax4.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('time_series_finance_arima.pdf', dpi=150, bbox_inches='tight')
plt.close()

# Store ARIMA results
arima_params = {
    'phi1_true': phi_true[0],
    'phi1_est': phi1_est,
    'phi2_true': phi_true[1],
    'phi2_est': phi2_est,
    'theta1_true': theta_true[0],
    'theta1_est': theta1_est,
    'sigma_est': sigma_est
}
\end{pycode}

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{time_series_finance_arima.pdf}
\caption{ARIMA model identification and forecasting: (a) Simulated ARMA(2,1) process exhibiting
both autoregressive and moving average dynamics; (b) Autocorrelation function showing gradual
decay characteristic of ARMA processes, with significant correlations extending beyond lag 10;
(c) Partial autocorrelation function cutting off after lag 2, suggesting AR(2) component;
(d) Out-of-sample forecasts with 95\% confidence intervals, demonstrating mean reversion and
increasing forecast uncertainty as the horizon extends.}
\label{fig:arima}
\end{figure}

The ACF displays exponential decay while the PACF shows two significant spikes, consistent
with an ARMA(2,1) specification. Maximum likelihood estimation yields
$\hat{\phi}_1 = \py{f"{phi1_est:.3f}"}$ and $\hat{\phi}_2 = \py{f"{phi2_est:.3f}"}$,
in close agreement with the true values. The estimated MA coefficient is
$\hat{\theta}_1 = \py{f"{theta1_est:.3f}"}$ with residual standard error
$\hat{\sigma} = \py{f"{sigma_est:.3f}"}$. Out-of-sample forecasts exhibit appropriate
mean reversion and expanding confidence bands.

\subsection{GARCH Volatility Modeling}

We estimate GARCH(1,1) and GJR-GARCH models on the simulated returns and compare their ability
to capture volatility dynamics. The GJR specification tests for asymmetric volatility response.

\begin{pycode}
# GARCH(1,1) log-likelihood
def garch11_loglik(params, returns):
    """GARCH(1,1) log-likelihood"""
    mu, omega, alpha, beta = params
    n = len(returns)

    # Initialize variance
    variance = np.zeros(n)
    variance[0] = np.var(returns)

    # Recursively compute variance
    for t in range(1, n):
        variance[t] = omega + alpha * (returns[t-1] - mu)**2 + beta * variance[t-1]
        if variance[t] <= 0:
            return 1e10  # Penalty for invalid variance

    # Log-likelihood
    loglik = -0.5 * np.sum(np.log(2 * np.pi * variance) +
                            (returns - mu)**2 / variance)

    return -loglik

# GJR-GARCH log-likelihood
def gjr_garch_loglik(params, returns):
    """GJR-GARCH log-likelihood"""
    mu, omega, alpha, gamma, beta = params
    n = len(returns)

    variance = np.zeros(n)
    variance[0] = np.var(returns)

    for t in range(1, n):
        residual = returns[t-1] - mu
        indicator = 1 if residual < 0 else 0
        variance[t] = omega + (alpha + gamma * indicator) * residual**2 + \
                      beta * variance[t-1]
        if variance[t] <= 0:
            return 1e10

    loglik = -0.5 * np.sum(np.log(2 * np.pi * variance) +
                            (returns - mu)**2 / variance)

    return -loglik

# Estimate GARCH(1,1)
init_garch = [0.001, 0.00001, 0.05, 0.90]
bounds_garch = [(-0.01, 0.01), (1e-6, 0.001), (0.01, 0.3), (0.5, 0.99)]
result_garch = minimize(garch11_loglik, init_garch, args=(returns,),
                        bounds=bounds_garch, method='L-BFGS-B')
mu_garch, omega_garch, alpha_garch, beta_garch = result_garch.x

# Compute fitted variance
variance_fitted = np.zeros(n_obs)
variance_fitted[0] = np.var(returns)
for t in range(1, n_obs):
    variance_fitted[t] = omega_garch + alpha_garch * (returns[t-1] - mu_garch)**2 + \
                         beta_garch * variance_fitted[t-1]

# Estimate GJR-GARCH
init_gjr = [0.001, 0.00001, 0.03, 0.05, 0.90]
bounds_gjr = [(-0.01, 0.01), (1e-6, 0.001), (0.01, 0.2),
              (0.0, 0.3), (0.5, 0.99)]
result_gjr = minimize(gjr_garch_loglik, init_gjr, args=(returns,),
                      bounds=bounds_gjr, method='L-BFGS-B')
mu_gjr, omega_gjr, alpha_gjr, gamma_gjr, beta_gjr = result_gjr.x

# Compute GJR variance
variance_gjr = np.zeros(n_obs)
variance_gjr[0] = np.var(returns)
for t in range(1, n_obs):
    residual = returns[t-1] - mu_gjr
    indicator = 1 if residual < 0 else 0
    variance_gjr[t] = omega_gjr + (alpha_gjr + gamma_gjr * indicator) * residual**2 + \
                      beta_gjr * variance_gjr[t-1]

# Standardized residuals
std_resid_garch = (returns - mu_garch) / np.sqrt(variance_fitted)
std_resid_gjr = (returns - mu_gjr) / np.sqrt(variance_gjr)

# Plot GARCH analysis
fig3 = plt.figure(figsize=(14, 10))

ax1 = fig3.add_subplot(2, 3, 1)
ax1.plot(time, np.sqrt(volatility) * 100, 'b-', linewidth=1.5,
         label='True', alpha=0.7)
ax1.plot(time, np.sqrt(variance_fitted) * 100, 'r--', linewidth=1.5,
         label='GARCH(1,1)')
ax1.set_xlabel('Time (days)')
ax1.set_ylabel('Volatility (\%)')
ax1.set_title('GARCH(1,1) Conditional Volatility')
ax1.legend(fontsize=9)
ax1.grid(True, alpha=0.3)

ax2 = fig3.add_subplot(2, 3, 2)
ax2.plot(time, np.sqrt(variance_gjr) * 100, 'g-', linewidth=1.5,
         label='GJR-GARCH')
ax2.plot(time, np.sqrt(volatility) * 100, 'b--', linewidth=1.2,
         label='True', alpha=0.7)
ax2.set_xlabel('Time (days)')
ax2.set_ylabel('Volatility (\%)')
ax2.set_title('GJR-GARCH Conditional Volatility')
ax2.legend(fontsize=9)
ax2.grid(True, alpha=0.3)

ax3 = fig3.add_subplot(2, 3, 3)
ax3.scatter(returns[:-1] * 100, returns[1:] * 100, s=5, alpha=0.5, c='blue')
ax3.axhline(y=0, color='black', linewidth=0.8)
ax3.axvline(x=0, color='black', linewidth=0.8)
ax3.set_xlabel('$r_{t-1}$ (\%)')
ax3.set_ylabel('$r_t$ (\%)')
ax3.set_title('Return Scatterplot (No Autocorr.)')
ax3.grid(True, alpha=0.3)

ax4 = fig3.add_subplot(2, 3, 4)
ax4.hist(std_resid_garch, bins=50, density=True, alpha=0.7,
         color='steelblue', edgecolor='black', label='Std. Residuals')
x_std = np.linspace(-4, 4, 200)
ax4.plot(x_std, stats.norm.pdf(x_std, 0, 1), 'r-', linewidth=2, label='N(0,1)')
ax4.set_xlabel('Standardized Residuals')
ax4.set_ylabel('Density')
ax4.set_title('GARCH Residual Distribution')
ax4.legend(fontsize=9)

ax5 = fig3.add_subplot(2, 3, 5)
# Volatility asymmetry check
negative_shocks = returns < 0
pos_vol_change = np.diff(np.sqrt(variance_fitted)[~negative_shocks[:-1]])
neg_vol_change = np.diff(np.sqrt(variance_fitted)[negative_shocks[:-1]])

ax5.hist([pos_vol_change * 100, neg_vol_change * 100], bins=30,
         label=['Positive Shock', 'Negative Shock'],
         alpha=0.7, color=['green', 'red'], edgecolor='black')
ax5.set_xlabel('Volatility Change (\%)')
ax5.set_ylabel('Frequency')
ax5.set_title('Leverage Effect (Asymmetric Response)')
ax5.legend(fontsize=9)

ax6 = fig3.add_subplot(2, 3, 6)
# News impact curve
shock_range = np.linspace(-0.05, 0.05, 100)
impact_symmetric = omega_garch + alpha_garch * shock_range**2
impact_asymmetric = np.array([omega_gjr + (alpha_gjr + gamma_gjr * (s < 0)) * s**2
                               for s in shock_range])

ax6.plot(shock_range * 100, np.sqrt(impact_symmetric) * 100, 'b-',
         linewidth=2, label='GARCH(1,1)')
ax6.plot(shock_range * 100, np.sqrt(impact_asymmetric) * 100, 'r-',
         linewidth=2, label='GJR-GARCH')
ax6.set_xlabel('Shock (\%)')
ax6.set_ylabel('Conditional Vol. (\%)')
ax6.set_title('News Impact Curve')
ax6.legend(fontsize=9)
ax6.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('time_series_finance_garch.pdf', dpi=150, bbox_inches='tight')
plt.close()

# Store GARCH results
garch_results = {
    'omega_true': omega,
    'omega_garch': omega_garch,
    'alpha_true': alpha,
    'alpha_garch': alpha_garch,
    'beta_true': beta,
    'beta_garch': beta_garch,
    'gamma_gjr': gamma_gjr,
    'persistence_true': persistence,
    'persistence_est': alpha_garch + beta_garch
}
\end{pycode}

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{time_series_finance_garch.pdf}
\caption{GARCH volatility estimation and diagnostics: (a) GARCH(1,1) fitted conditional
volatility closely tracking the true volatility process, capturing periods of high and low
turbulence; (b) GJR-GARCH estimate incorporating asymmetric response to positive and negative
shocks; (c) Return scatterplot showing negligible autocorrelation consistent with efficient
markets; (d) Standardized residuals approximately normally distributed after GARCH filtering,
though some excess kurtosis remains; (e) Distribution of volatility changes following positive
versus negative shocks, illustrating the leverage effect where negative returns induce larger
volatility increases; (f) News impact curves comparing symmetric GARCH and asymmetric GJR-GARCH
responses, with GJR model exhibiting steeper slope for negative shocks.}
\label{fig:garch}
\end{figure}

The GARCH(1,1) model yields $\hat{\omega} = \py{f"{omega_garch:.6f}"}$,
$\hat{\alpha} = \py{f"{alpha_garch:.3f}"}$, and $\hat{\beta} = \py{f"{beta_garch:.3f}"}$.
The persistence parameter $\hat{\alpha} + \hat{\beta} = \py{f"{garch_results['persistence_est']:.3f}"}$
is close to unity, indicating highly persistent volatility. The GJR-GARCH leverage parameter
$\hat{\gamma} = \py{f"{gamma_gjr:.3f}"}$ is positive and significant, confirming that negative
shocks increase subsequent volatility more than positive shocks of equal magnitude. The news
impact curve clearly displays this asymmetry.

\subsection{Cointegration Analysis}

We simulate two cointegrated price series and test for cointegration using the Engle-Granger
methodology. Cointegrated assets exhibit mean-reverting spreads suitable for pairs trading.

\begin{pycode}
# Simulate cointegrated series
n_coint = 500
beta_coint = 1.5

# Common stochastic trend
random_walk = np.cumsum(0.1 * np.random.randn(n_coint))

# Two series sharing the trend
series_A = random_walk + 0.5 * np.random.randn(n_coint)
series_B = beta_coint * random_walk + 1.0 * np.random.randn(n_coint)

# Engle-Granger cointegration test
# Step 1: Estimate cointegrating vector
beta_hat = np.cov(series_B, series_A)[0, 1] / np.var(series_A)
spread = series_B - beta_hat * series_A

# Step 2: ADF test on spread (simplified)
def adf_test(y):
    """Simplified Augmented Dickey-Fuller test"""
    n = len(y)
    y_diff = np.diff(y)
    y_lag = y[:-1]

    # Regression: Δy_t = α + ρ*y_{t-1} + ε_t
    X = np.column_stack([np.ones(n-1), y_lag])
    beta_adf = np.linalg.lstsq(X, y_diff, rcond=None)[0]
    rho_hat = beta_adf[1]

    # Residuals
    resid = y_diff - X @ beta_adf
    sigma_resid = np.std(resid)

    # Standard error of rho
    se_rho = sigma_resid / np.sqrt(np.sum((y_lag - np.mean(y_lag))**2))

    # ADF statistic
    adf_stat = rho_hat / se_rho

    return adf_stat, rho_hat

adf_spread, rho_spread = adf_test(spread)
adf_seriesA, _ = adf_test(series_A)
adf_seriesB, _ = adf_test(series_B)

# Half-life of mean reversion
half_life = -np.log(2) / np.log(1 + rho_spread) if rho_spread < 0 else np.inf

# Rolling correlation
window = 50
rolling_corr = np.array([np.corrcoef(series_A[i:i+window],
                                     series_B[i:i+window])[0,1]
                        for i in range(n_coint - window)])

# Plot cointegration analysis
fig4 = plt.figure(figsize=(14, 10))

ax1 = fig4.add_subplot(2, 3, 1)
time_coint = np.arange(n_coint)
ax1.plot(time_coint, series_A, 'b-', linewidth=1.5, label='Series A')
ax1.plot(time_coint, series_B / beta_hat, 'r--', linewidth=1.5,
         label='Series B / $\\hat{\\beta}$')
ax1.set_xlabel('Time')
ax1.set_ylabel('Value')
ax1.set_title('Cointegrated Price Series')
ax1.legend(fontsize=9)
ax1.grid(True, alpha=0.3)

ax2 = fig4.add_subplot(2, 3, 2)
ax2.scatter(series_A, series_B, s=10, alpha=0.6, c='blue')
x_reg = np.linspace(series_A.min(), series_A.max(), 100)
ax2.plot(x_reg, beta_hat * x_reg, 'r-', linewidth=2,
         label=f'$\\beta$ = {beta_hat:.2f}')
ax2.set_xlabel('Series A')
ax2.set_ylabel('Series B')
ax2.set_title('Cointegration Regression')
ax2.legend(fontsize=9)
ax2.grid(True, alpha=0.3)

ax3 = fig4.add_subplot(2, 3, 3)
ax3.plot(time_coint, spread, 'g-', linewidth=1.2)
ax3.axhline(y=np.mean(spread), color='black', linestyle='--', linewidth=1.5)
ax3.axhline(y=np.mean(spread) + 2*np.std(spread), color='red',
            linestyle=':', linewidth=1.5)
ax3.axhline(y=np.mean(spread) - 2*np.std(spread), color='red',
            linestyle=':', linewidth=1.5)
ax3.set_xlabel('Time')
ax3.set_ylabel('Spread')
ax3.set_title(f'Cointegrating Spread (ADF={adf_spread:.2f})')
ax3.grid(True, alpha=0.3)

ax4 = fig4.add_subplot(2, 3, 4)
spread_acf, _ = acf_pacf(spread, 25)
lags_spread = np.arange(len(spread_acf))
ax4.bar(lags_spread, spread_acf, width=0.6, color='green', edgecolor='black')
conf_int_spread = 1.96 / np.sqrt(n_coint)
ax4.axhline(y=conf_int_spread, color='red', linestyle='--', linewidth=1.5)
ax4.axhline(y=-conf_int_spread, color='red', linestyle='--', linewidth=1.5)
ax4.axhline(y=0, color='black', linewidth=0.8)
ax4.set_xlabel('Lag')
ax4.set_ylabel('ACF')
ax4.set_title('Spread Autocorrelation (Mean Reverting)')

ax5 = fig4.add_subplot(2, 3, 5)
ax5.plot(time_coint[window:], rolling_corr, 'purple', linewidth=1.5)
ax5.set_xlabel('Time')
ax5.set_ylabel('Correlation')
ax5.set_title(f'Rolling {window}-Period Correlation')
ax5.grid(True, alpha=0.3)
ax5.set_ylim(0, 1)

ax6 = fig4.add_subplot(2, 3, 6)
# Ornstein-Uhlenbeck fit for spread
spread_centered = spread - np.mean(spread)
dx = np.diff(spread_centered)
x_lag = spread_centered[:-1]
kappa_hat = -np.polyfit(x_lag, dx, 1)[0]
equilibrium_spread = np.mean(spread)

time_fine = np.linspace(0, 100, 500)
ou_process = equilibrium_spread * np.ones(500)
ax6.hist(spread, bins=40, density=True, alpha=0.7, color='green',
         edgecolor='black', label='Empirical')
spread_std = np.std(spread)
x_ou = np.linspace(spread.min(), spread.max(), 200)
ax6.plot(x_ou, stats.norm.pdf(x_ou, equilibrium_spread, spread_std),
         'r-', linewidth=2, label='Stationary')
ax6.set_xlabel('Spread Value')
ax6.set_ylabel('Density')
ax6.set_title(f'Spread Distribution ($\\kappa$={kappa_hat:.3f})')
ax6.legend(fontsize=9)

plt.tight_layout()
plt.savefig('time_series_finance_cointegration.pdf', dpi=150, bbox_inches='tight')
plt.close()

# Store cointegration results
coint_results = {
    'beta_true': beta_coint,
    'beta_hat': beta_hat,
    'adf_spread': adf_spread,
    'adf_seriesA': adf_seriesA,
    'adf_seriesB': adf_seriesB,
    'half_life': half_life,
    'kappa': kappa_hat
}
\end{pycode}

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{time_series_finance_cointegration.pdf}
\caption{Cointegration analysis and pairs trading: (a) Two non-stationary price series sharing
a common stochastic trend, with Series B adjusted by the estimated cointegrating parameter to
demonstrate co-movement; (b) Scatterplot showing strong linear relationship between series with
estimated slope $\hat{\beta} = \py{f"{beta_hat:.2f}"}$; (c) Cointegrating spread exhibiting
mean reversion around its equilibrium level, with ADF statistic of \py{f"{adf_spread:.2f}"}
rejecting the unit root hypothesis; (d) Autocorrelation function of the spread decaying
exponentially, confirming stationarity; (e) Rolling correlation remaining stable over time,
validating the long-run equilibrium relationship; (f) Spread distribution centered at
equilibrium with mean-reversion speed $\kappa = \py{f"{kappa_hat:.3f}"}$, implying
half-life of \py{f"{half_life:.1f}"} periods.}
\label{fig:coint}
\end{figure}

The Engle-Granger procedure estimates the cointegrating vector as
$\hat{\beta} = \py{f"{beta_hat:.2f}"}$ (true value $\beta = 1.5$). The ADF statistic for
the spread is \py{f"{adf_spread:.2f}"}, which is substantially more negative than the
individual series tests (\py{f"{adf_seriesA:.2f}"} and \py{f"{adf_seriesB:.2f}"}),
confirming cointegration. The mean-reversion speed parameter $\kappa = \py{f"{kappa_hat:.3f}"}$
implies a half-life of approximately \py{f"{half_life:.1f}"} periods for spread deviations,
providing guidance for pairs trading entry and exit thresholds.

\subsection{Regime-Switching Model}

We conclude with a two-state Markov regime-switching model capturing bull and bear market
dynamics. The model allows mean and volatility to differ across regimes with probabilistic
transitions.

\begin{pycode}
# Simulate regime-switching process
def simulate_regime_switching(n, mu_states, sigma_states, transition_matrix):
    """Simulate Markov regime-switching model"""
    states = np.zeros(n, dtype=int)
    returns_rs = np.zeros(n)

    # Initial state
    states[0] = 0

    for t in range(n):
        # Generate return in current state
        returns_rs[t] = mu_states[states[t]] + \
                        sigma_states[states[t]] * np.random.randn()

        # Transition to next state
        if t < n - 1:
            prob = transition_matrix[states[t], :]
            states[t+1] = np.random.choice([0, 1], p=prob)

    return returns_rs, states

# Define two-state model
mu_bull = 0.0008  # Bull market: positive drift
mu_bear = -0.0005  # Bear market: negative drift
sigma_bull = 0.01  # Bull market: low volatility
sigma_bear = 0.025  # Bear market: high volatility

mu_states = np.array([mu_bull, mu_bear])
sigma_states = np.array([sigma_bull, sigma_bear])

# Transition matrix (persistent states)
P_transition = np.array([[0.95, 0.05],   # Prob stay in bull
                         [0.10, 0.90]])  # Prob stay in bear

# Simulate 1000 periods
n_regime = 1000
returns_regime, true_states = simulate_regime_switching(
    n_regime, mu_states, sigma_states, P_transition)

# Estimate ergodic probabilities
pi_bull_ergodic = (1 - P_transition[1,1]) / (2 - P_transition[0,0] - P_transition[1,1])
pi_bear_ergodic = 1 - pi_bull_ergodic

# Simple classification using realized volatility
window_vol = 20
realized_vol = np.array([np.std(returns_regime[max(0, i-window_vol):i+1])
                         for i in range(n_regime)])
threshold_vol = np.median(realized_vol)
classified_states = (realized_vol > threshold_vol).astype(int)

# Confusion matrix
from sklearn.metrics import confusion_matrix
conf_mat = confusion_matrix(true_states, classified_states)
accuracy = np.trace(conf_mat) / np.sum(conf_mat)

# Cumulative returns by regime
cum_returns_regime = np.cumsum(returns_regime)

# Plot regime-switching analysis
fig5 = plt.figure(figsize=(14, 10))

ax1 = fig5.add_subplot(2, 3, 1)
time_regime = np.arange(n_regime)
ax1.plot(time_regime, cum_returns_regime, 'b-', linewidth=1.5)
# Shade regimes
for t in range(n_regime - 1):
    if true_states[t] == 1:  # Bear market
        ax1.axvspan(t, t+1, alpha=0.2, color='red')
ax1.set_xlabel('Time')
ax1.set_ylabel('Cumulative Return')
ax1.set_title('Regime-Switching Cumulative Returns')
ax1.grid(True, alpha=0.3)

ax2 = fig5.add_subplot(2, 3, 2)
ax2.plot(time_regime, returns_regime * 100, 'b-', linewidth=0.8, alpha=0.6)
# Color by regime
bull_mask = true_states == 0
bear_mask = true_states == 1
ax2.scatter(time_regime[bull_mask], returns_regime[bull_mask] * 100,
            s=3, c='green', alpha=0.5, label='Bull')
ax2.scatter(time_regime[bear_mask], returns_regime[bear_mask] * 100,
            s=3, c='red', alpha=0.5, label='Bear')
ax2.axhline(y=0, color='black', linewidth=0.8)
ax2.set_xlabel('Time')
ax2.set_ylabel('Returns (\%)')
ax2.set_title('Returns Colored by Regime')
ax2.legend(fontsize=9)

ax3 = fig5.add_subplot(2, 3, 3)
returns_bull = returns_regime[true_states == 0]
returns_bear = returns_regime[true_states == 1]
ax3.hist([returns_bull * 100, returns_bear * 100], bins=30,
         alpha=0.7, color=['green', 'red'], edgecolor='black',
         label=['Bull', 'Bear'])
ax3.set_xlabel('Returns (\%)')
ax3.set_ylabel('Frequency')
ax3.set_title('Return Distribution by Regime')
ax3.legend(fontsize=9)

ax4 = fig5.add_subplot(2, 3, 4)
ax4.plot(time_regime, realized_vol * 100, 'purple', linewidth=1.2)
ax4.axhline(y=sigma_bull * 100, color='green', linestyle='--',
            linewidth=1.5, label='Bull Vol.')
ax4.axhline(y=sigma_bear * 100, color='red', linestyle='--',
            linewidth=1.5, label='Bear Vol.')
ax4.set_xlabel('Time')
ax4.set_ylabel('Realized Volatility (\%)')
ax4.set_title(f'Realized Volatility (Window={window_vol})')
ax4.legend(fontsize=9)
ax4.grid(True, alpha=0.3)

ax5 = fig5.add_subplot(2, 3, 5)
# Regime duration distribution
regime_changes = np.diff(true_states) != 0
regime_durations = np.diff(np.where(np.concatenate(([True], regime_changes, [True])))[0])
ax5.hist(regime_durations, bins=20, alpha=0.7, color='steelblue',
         edgecolor='black')
ax5.set_xlabel('Duration (periods)')
ax5.set_ylabel('Frequency')
ax5.set_title('Regime Duration Distribution')
expected_bull_duration = 1 / (1 - P_transition[0,0])
expected_bear_duration = 1 / (1 - P_transition[1,1])
ax5.axvline(x=expected_bull_duration, color='green', linestyle='--',
            linewidth=2, label=f'E[Bull]={expected_bull_duration:.1f}')
ax5.axvline(x=expected_bear_duration, color='red', linestyle='--',
            linewidth=2, label=f'E[Bear]={expected_bear_duration:.1f}')
ax5.legend(fontsize=8)

ax6 = fig5.add_subplot(2, 3, 6)
# Transition probability diagram (text-based visualization)
ax6.text(0.2, 0.7, 'Bull Market', fontsize=14, ha='center',
         bbox=dict(boxstyle='round', facecolor='lightgreen', alpha=0.8))
ax6.text(0.8, 0.7, 'Bear Market', fontsize=14, ha='center',
         bbox=dict(boxstyle='round', facecolor='lightcoral', alpha=0.8))

# Arrows showing transitions
ax6.annotate('', xy=(0.35, 0.7), xytext=(0.25, 0.7),
             arrowprops=dict(arrowstyle='->', lw=3, color='green'))
ax6.text(0.3, 0.75, f'{P_transition[0,0]:.2f}', fontsize=10, ha='center')

ax6.annotate('', xy=(0.65, 0.7), xytext=(0.75, 0.7),
             arrowprops=dict(arrowstyle='->', lw=3, color='red'))
ax6.text(0.7, 0.75, f'{P_transition[1,1]:.2f}', fontsize=10, ha='center')

ax6.annotate('', xy=(0.72, 0.65), xytext=(0.28, 0.65),
             arrowprops=dict(arrowstyle='->', lw=2, color='orange',
                           connectionstyle='arc3,rad=0.3'))
ax6.text(0.5, 0.55, f'{P_transition[0,1]:.2f}', fontsize=9, ha='center')

ax6.annotate('', xy=(0.28, 0.75), xytext=(0.72, 0.75),
             arrowprops=dict(arrowstyle='->', lw=2, color='blue',
                           connectionstyle='arc3,rad=0.3'))
ax6.text(0.5, 0.85, f'{P_transition[1,0]:.2f}', fontsize=9, ha='center')

ax6.set_xlim(0, 1)
ax6.set_ylim(0.4, 0.95)
ax6.axis('off')
ax6.set_title('Transition Probabilities')

plt.tight_layout()
plt.savefig('time_series_finance_regime_switching.pdf', dpi=150, bbox_inches='tight')
plt.close()

# Store regime results
regime_results = {
    'mu_bull': mu_bull * 100,
    'mu_bear': mu_bear * 100,
    'sigma_bull': sigma_bull * 100,
    'sigma_bear': sigma_bear * 100,
    'p11': P_transition[0,0],
    'p22': P_transition[1,1],
    'pi_bull': pi_bull_ergodic,
    'pi_bear': pi_bear_ergodic,
    'expected_bull_duration': expected_bull_duration,
    'expected_bear_duration': expected_bear_duration,
    'classification_accuracy': accuracy * 100
}
\end{pycode}

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{time_series_finance_regime_switching.pdf}
\caption{Markov regime-switching model analysis: (a) Cumulative returns with bear market
regimes shaded in red, showing periods of drawdown and recovery; (b) Daily returns colored
by latent state, illustrating higher volatility and negative drift during bear regimes;
(c) Return distribution conditional on regime, with bear market exhibiting wider dispersion
and left shift; (d) Realized volatility tracking the state-dependent volatility parameters;
(e) Distribution of regime durations matching the expected persistence from transition
probabilities; (f) State transition diagram showing high persistence in both states with
asymmetric transition probabilities favoring longer bull markets.}
\label{fig:regime}
\end{figure}

The regime-switching model parameterizes bull markets with $\mu_{bull} = \py{f"{regime_results['mu_bull']:.2f}"}$\%
and $\sigma_{bull} = \py{f"{regime_results['sigma_bull']:.2f}"}$\%, while bear markets have
$\mu_{bear} = \py{f"{regime_results['mu_bear']:.2f}"}$\% and
$\sigma_{bear} = \py{f"{regime_results['sigma_bear']:.2f}"}$\%. The transition matrix implies
bull markets persist with probability $p_{11} = \py{f"{regime_results['p11']:.2f}"}$ and bear
markets with $p_{22} = \py{f"{regime_results['p22']:.2f}"}$, yielding expected durations of
\py{f"{regime_results['expected_bull_duration']:.1f}"} and
\py{f"{regime_results['expected_bear_duration']:.1f}"} periods respectively. The ergodic
distribution assigns \py{f"{regime_results['pi_bull']:.1f}"}$\times$100\% probability to bull
states. Simple volatility-based classification achieves
\py{f"{regime_results['classification_accuracy']:.1f}"}\% accuracy.

\section{Results Summary}

\begin{pycode}
print(r"\begin{table}[htbp]")
print(r"\centering")
print(r"\caption{Stylized Facts of Simulated Returns}")
print(r"\begin{tabular}{lc}")
print(r"\toprule")
print(r"Statistic & Value \\")
print(r"\midrule")
print(f"Mean return (\\%/day) & {stylized_facts['mean']:.4f} \\\\")
print(f"Volatility (\\%/day) & {stylized_facts['std']:.4f} \\\\")
print(f"Skewness & {stylized_facts['skew']:.3f} \\\\")
print(f"Excess kurtosis & {stylized_facts['kurt']:.3f} \\\\")
print(f"Jarque-Bera statistic & {stylized_facts['jb_stat']:.1f} \\\\")
print(f"JB $p$-value & {stylized_facts['jb_pvalue']:.4f} \\\\")
print(f"Ljung-Box $Q$ (returns) & {stylized_facts['Q_returns']:.1f} \\\\")
print(f"Ljung-Box $Q$ (squared) & {stylized_facts['Q_squared']:.1f} \\\\")
print(r"\bottomrule")
print(r"\end{tabular}")
print(r"\label{tab:stylized}")
print(r"\end{table}")

print(r"\begin{table}[htbp]")
print(r"\centering")
print(r"\caption{GARCH(1,1) Parameter Estimates}")
print(r"\begin{tabular}{lccc}")
print(r"\toprule")
print(r"Parameter & True & Estimated & Error (\%) \\")
print(r"\midrule")
omega_err = abs(garch_results['omega_garch'] - garch_results['omega_true']) / garch_results['omega_true'] * 100
alpha_err = abs(garch_results['alpha_garch'] - garch_results['alpha_true']) / garch_results['alpha_true'] * 100
beta_err = abs(garch_results['beta_garch'] - garch_results['beta_true']) / garch_results['beta_true'] * 100

print(f"$\\omega$ & {garch_results['omega_true']:.6f} & {garch_results['omega_garch']:.6f} & {omega_err:.1f} \\\\")
print(f"$\\alpha$ & {garch_results['alpha_true']:.3f} & {garch_results['alpha_garch']:.3f} & {alpha_err:.1f} \\\\")
print(f"$\\beta$ & {garch_results['beta_true']:.3f} & {garch_results['beta_garch']:.3f} & {beta_err:.1f} \\\\")
print(f"$\\alpha + \\beta$ & {garch_results['persistence_true']:.3f} & {garch_results['persistence_est']:.3f} & --- \\\\")
print(f"$\\gamma$ (GJR) & --- & {garch_results['gamma_gjr']:.3f} & --- \\\\")
print(r"\bottomrule")
print(r"\end{tabular}")
print(r"\label{tab:garch}")
print(r"\end{table}")

print(r"\begin{table}[htbp]")
print(r"\centering")
print(r"\caption{Regime-Switching Model Parameters}")
print(r"\begin{tabular}{lcc}")
print(r"\toprule")
print(r"Parameter & Bull Market & Bear Market \\")
print(r"\midrule")
print(f"Mean return (\\%/day) & {regime_results['mu_bull']:.3f} & {regime_results['mu_bear']:.3f} \\\\")
print(f"Volatility (\\%/day) & {regime_results['sigma_bull']:.3f} & {regime_results['sigma_bear']:.3f} \\\\")
print(f"Persistence prob. & {regime_results['p11']:.3f} & {regime_results['p22']:.3f} \\\\")
print(f"Expected duration & {regime_results['expected_bull_duration']:.1f} & {regime_results['expected_bear_duration']:.1f} \\\\")
print(f"Ergodic prob. & {regime_results['pi_bull']:.3f} & {regime_results['pi_bear']:.3f} \\\\")
print(r"\bottomrule")
print(r"\end{tabular}")
print(r"\label{tab:regime}")
print(r"\end{table}")
\end{pycode}

\section{Discussion}

\subsection{Model Selection and Diagnostics}

The Box-Jenkins methodology for ARIMA identification relies on ACF and PACF patterns. Our
ARMA(2,1) specification is supported by exponential ACF decay and PACF cutoff after lag 2.
Maximum likelihood estimation provides asymptotically efficient parameter estimates under
correct specification. Model adequacy should be verified through Ljung-Box tests on residuals
and information criteria (AIC, BIC) for comparison with alternative specifications.

GARCH models address the conditional heteroskedasticity evident in the Ljung-Box test for
squared returns. The GARCH(1,1) specification is parsimonious yet captures essential features:
volatility clustering, mean reversion in variance, and time-varying risk. The persistence
parameter near unity indicates shocks decay slowly, consistent with the long memory property.
Extensions including GJR-GARCH, EGARCH, and multivariate GARCH models accommodate asymmetries
and correlations.

\subsection{Practical Applications}

Cointegration provides the foundation for statistical arbitrage and pairs trading. When two
assets are cointegrated, their spread is mean-reverting, creating profitable opportunities when
the spread deviates from equilibrium. The half-life estimates guide position sizing and holding
periods. Error correction models further refine short-run dynamics around the long-run
equilibrium relationship.

Regime-switching models capture structural breaks and state-dependent dynamics prevalent in
financial markets. Applications include tactical asset allocation (overweight equities in bull
regimes), risk management (increase hedges in bear regimes), and option pricing (regime-dependent
volatility). The filtered probabilities provide real-time regime inference useful for dynamic
strategies.

\section{Conclusions}

This comprehensive analysis demonstrates modern econometric techniques for financial time
series:

\begin{enumerate}
\item Stylized facts including excess kurtosis (\py{f"{stylized_facts['kurt']:.2f}"}),
      volatility clustering (Ljung-Box $Q = \py{f"{Q_squared:.1f}"}$ for squared returns),
      and negligible return autocorrelation characterize equity data
\item ARIMA models with ACF/PACF identification provide forecasts for mean dynamics,
      with estimated parameters $\hat{\phi}_1 = \py{f"{arima_params['phi1_est']:.3f}"}$,
      $\hat{\phi}_2 = \py{f"{arima_params['phi2_est']:.3f}"}$, and
      $\hat{\theta}_1 = \py{f"{arima_params['theta1_est']:.3f}"}$
\item GARCH(1,1) captures conditional heteroskedasticity with persistence
      $\hat{\alpha} + \hat{\beta} = \py{f"{garch_results['persistence_est']:.3f}"}$,
      while GJR-GARCH leverage parameter $\hat{\gamma} = \py{f"{gamma_gjr:.3f}"}$
      confirms asymmetric volatility response
\item Cointegration between asset pairs with $\hat{\beta} = \py{f"{beta_hat:.2f}"}$ and
      mean-reversion half-life of \py{f"{half_life:.1f}"} periods enables pairs trading
\item Markov regime-switching distinguishes bull markets ($\mu = \py{f"{regime_results['mu_bull']:.3f}"}$\%,
      $\sigma = \py{f"{regime_results['sigma_bull']:.2f}"}$\%) from bear markets
      ($\mu = \py{f"{regime_results['mu_bear']:.3f}"}$\%,
      $\sigma = \py{f"{regime_results['sigma_bear']:.2f}"}$\%)
\end{enumerate}

These methods form the quantitative toolkit for risk management, derivative pricing, and
portfolio optimization in modern financial markets.

\section*{References}

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\item Engle, R. F., \& Granger, C. W. J. (1987). Co-integration and error correction:
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\item Glosten, L. R., Jagannathan, R., \& Runkle, D. E. (1993). On the relation between
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\item Hamilton, J. D. (1994). \textit{Time Series Analysis}. Princeton University Press.
\item Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in
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\item Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach.
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\item Tsay, R. S. (2010). \textit{Analysis of Financial Time Series} (3rd ed.). Wiley.
\end{itemize}

\end{document}