https://github.com/beliavsky/garch

Simulation and estimation of ARCH and GARCH processes, used to model the time-varying standard deviation (volatility) of asset returns, with conditional distributions such as the normal, Laplace, and Student t.
https://github.com/beliavsky/garch

finance gjr-garch laplace-distribution monte-carlo nelder-mead normal-distribution probability-distribution quantitative-finance simulation t-distribution volatility volatility-modeling

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Simulation and estimation of ARCH and GARCH processes, used to model the time-varying standard deviation (volatility) of asset returns, with conditional distributions such as the normal, Laplace, and Student t.

• Host: GitHub
• URL: https://github.com/beliavsky/garch
• Owner: Beliavsky
• Created: 2025-02-06T17:51:24.000Z (11 months ago)
• Default Branch: main
• Last Pushed: 2025-03-14T02:42:27.000Z (10 months ago)
• Last Synced: 2025-03-14T03:28:53.472Z (10 months ago)
• Topics: finance, gjr-garch, laplace-distribution, monte-carlo, nelder-mead, normal-distribution, probability-distribution, quantitative-finance, simulation, t-distribution, volatility, volatility-modeling
• Language: Fortran
• Homepage:
• Size: 221 KB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 3
• Metadata Files:
  • Readme: README.md

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README

          
## GJR-GARCH Simulation and Estimation

The program `xgjr_garch.f90` simulates 10000 observations of a GJR-GARCH(1,1) process and then fits a GJR-GARCH model to the simulated returns. The [GJR-GARCH](https://frds.io/algorithms/gjr-garch/) model "extends the basic GARCH(1,1) by accounting for leverage effects, where bad news (negative returns) has a greater impact on volatility than good news." Other programs fit ARCH and GARCH models to financial returns, computed from daily ETF closing prices in `spy_efa_eem_tlt.csv`, with some results [here](https://github.com/Beliavsky/GARCH/blob/main/etf_garch_results.md).

### Code Overview

- **Simulation:**  

  The program generates 10,000 observations from a GJR-GARCH process using the true parameters.

- **Estimation:**  

  The model is then fitted to the simulated returns using the Nelder-Mead algorithm. The estimated parameters, negative log-likelihood, and other diagnostic statistics are output.

- **Diagnostics:**  

  The output includes basic statistics (mean, standard deviation, skewness, kurtosis, min, and max) for both the true conditional standard deviations (`sigma`) and the estimated standard deviations (`sigma_est`).  

  Additional diagnostics include:

  - Kurtosis of returns and standardized returns.

  - Autocorrelation function (ACF) of squared returns and of the squared standardized returns.

  - Correlation between the true and estimated volatilities.

## Explanation of Sample Output in `results.txt`

### 1. Model Specification

- **GARCH Model:**  

  The selected model is printed as **gjr_garch**.

  

- **Conditional Distribution:**  

  The returns are modeled using a **normal** distribution.

### 2. True and Estimated Parameters

The following table compares the true parameters with the estimated parameters obtained from fitting the model:

| Parameter | True Value | Estimated Value |

|-----------|------------|-----------------|

| **mu**    | 0.000000   | -0.017736       |

| **omega** | 0.100000   | 0.117329        |

| **alpha** | 0.100000   | 0.087711        |

| **gamma** | 0.100000   | 0.101340        |

| **beta**  | 0.800000   | 0.800561        |

This close agreement indicates that the estimation procedure is performing well.

### 3. Log-Likelihood

- The log-likelihood is reported as **-16677.550339**.  

- The value computed by the negative log-likelihood function matches this value, confirming the consistency of the likelihood calculation.

### 4. Volatility Statistics

- **True Volatility (`sigma`):**  

  Basic statistics (mean, standard deviation, skew, kurtosis, min, and max) are computed for the simulated conditional standard deviation. For example, the mean is approximately 1.321965 with a standard deviation of 0.363676.

- **Estimated Volatility (`sigma_est`):**  

  The corresponding statistics for the estimated volatilities are nearly identical to the true values (e.g., mean ≈ 1.319146), indicating a good model fit.

### 5. Additional Diagnostics

- **Kurtosis:**  

  The kurtosis of the raw returns is near unity, while the kurtosis of the standardized returns (both ret/sigma and ret/sigma_est) is close to zero. This suggests that the model has successfully normalized the returns.

- **Correlation:**  

  The correlation between `sigma` and `sigma_est` is extremely high (≈ 0.999808), demonstrating that the estimated volatility closely tracks the true volatility.

- **Autocorrelation Function (ACF):**  

  - The ACF of squared returns shows moderate autocorrelation at low lags, a common sign of volatility clustering.  

  - In contrast, the ACF of the squared standardized returns (ret/sigma_est)^2 is near zero, confirming that the model has effectively removed the time-dependence in the volatility.