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 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.

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.