Bayesian GARCH(1,1) MCMC Simulation

This project implements a Bayesian GARCH(1,1) model using a Random Walk Metropolis (RWM) sampler. It allows for estimating the parameters ($\omega$, $\alpha$, $\beta$) of a GARCH model from both simulated data and real-world financial returns (e.g., EUR/USD).

Project Overview

The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model is a standard tool in financial econometrics for modeling time-varying volatility. This implementation uses Bayesian inference to estimate parameter posteriors, providing a robust way to quantify uncertainty.

Key Features

• Simulation: Generate synthetic GARCH(1,1) processes with known parameters.
• MCMC Sampler: A custom Random Walk Metropolis-Hastings implementation.
• Diagnostics: Trace plots, posterior density histograms, and autocorrelation plots.
• Real Data: Integration with yfinance to fetch and analyze real financial time series.
• Volatility Estimation: Calculation of posterior mean conditional variance.

Workflow Explanation

The following diagram illustrates the project's logic and execution flow:

graph TD
    A[Start] --> B{Data Source}
    B -->|Simulated| C[Generate GARCH 1,1 Data]
    B -->|Real| D[Fetch EUR/USD via yfinance]
    
    C --> E[Initialize Parameters]
    D --> E
    
    E --> F[Run RWM MCMC Sampler]
    
    subgraph MCMC Loop
        F --> G[Propose New Parameters θ*]
        G --> H[Calculate Log-Likelihood & Log-Prior]
        H --> I[Accept/Reject θ* based on Metropolis Ratio]
        I --> J{More Iterations?}
        J -->|Yes| G
    end
    
    J -->|No| K[Post-processing]
    K --> L[Discard Burn-in Samples]
    L --> M[Generate Diagnostic Plots]
    M --> N[Output Posterior Summary]
    N --> O[End]

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Mathematical Model

GARCH(1,1) Model

$y_t = \sqrt{h_t} \epsilon_t, \quad \epsilon_t \sim N(0,1)$ $h_t = \omega + \alpha y_{t-1}^2 + \beta h_{t-1}$

Constraints

• $\omega > 0$
• $\alpha \ge 0, \beta \ge 0$
• $\alpha + \beta < 1$ (Stationarity)

Bayesian Setup

• Likelihood: Gaussian likelihood for the returns $y_t$.
• Priors:
  • $p(\omega) \propto 1/\omega$ (Jeffreys-type prior)
  • $p(\alpha) \sim \text{Uniform}(0, 1)$
  • $p(\beta) \sim \text{Uniform}(0, 1)$
  • Subject to $\alpha + \beta < 1$.

Installation

1. Clone the repository:

   git clone <repository-url>
   cd monte_carlo

2. Install dependencies:

   pip install -r requirements.txt

Usage

You can run the main simulation and estimation script directly:

python garch_mcmc.py

Or explore the provided Jupyter notebooks in the notebooks/ directory for step-by-step analysis:

• notebooks/Q1.ipynb: Initial exploration and simulation.
• notebooks/Q2.ipynb: Implementation of the sampler.
• notebooks/Q3 and Q4.ipynb: Advanced diagnostics and real data application.

Results

The script generates several diagnostic plots saved in the outputs/ directory:

• outputs/garch_sim_diagnostics.png: MCMC trace and posterior plots for simulated data.
• outputs/garch_sim_volatility.png: Simulated returns vs. estimated volatility.
• outputs/garch_real_diagnostics.png: Diagnostics for real financial data.
• outputs/garch_persistence.png: Posterior distribution of the persistence parameter ($\alpha + \beta$).

References

• Mira, Solgi & Imparato (2013) "Zero variance Markov chain Monte Carlo for Bayesian estimators", Statistics and Computing.
• Ardia (2008) / Nakatsuma (2000) for GARCH prior setups.