Simulation with Copula-GARCH Model Clause Samples
Simulation with Copula-GARCH Model. The joint distribution of generated random samples is governed by the mixture Gaussian- ▇▇▇▇▇▇▇ copula model. I still need to specify some model to best mimic the marginal distributions of asset returns. GARCH(1,1) process is a well-known parsimonious model for stock returns. Following ▇▇▇▇, Tu, and ▇▇▇▇ (2007), I model the marginal distributions of the return series with a GARCH(1,1) with no ARMA components. Basically, the return series is modeled to be equal to an expected return component plus a random error term that follows a GARCH(1,1) process. I first fit the copula-GARCH model to the data to estimate the related parameters using Maximum Likelihood (ML) approach. I then plug the ML estimates back in the model and use it as the data generating process (DGP) in simulation. To be conservative, instead of using portfolios that show clear asymmetric dependence, such as the smallest stock portfolio or momentum portfolios, etc., I use the 5th smallest value- weighted size portfolio and the market return to estimate copula and GARCH parameters. Empirically, I do not find any evidence for asymmetric dependence for size 5 portfolio, hence using it to calibrate the parameters impose a harder challenge for the tests. It is interesting to see whether the tests have reasonable power under such parameter settings. ▇▇▇▇, ▇▇, and ▇▇▇▇ (2007) has done a similar practice in their simulation exercises. [Insert Table 1.1 about here] Table 1.1 gives the ML estimates from fitting the data to the GARCH(1,1) process using the full sample period. Panel A lists the parameter estimates for return series of value-weighted size 5 portfolio and Panel B lists the estimates for the market return series. All the estimated parameters are statistically significant at 5% level. Taking those estimates as the population parameters, I are able to simulate the data with the copula-GARCH model using the following detailed steps.
1. For a given κ, draw a bivariate uniform random sample of size T from the mixture Gaussian-▇▇▇▇▇▇▇ copula model;
2. Apply inverse standard normal CDF transformation to get a bivariate standard normal random sample with pre-specified dependence structure;
3. Feed each series of the joint normal random sample into the univariate GARCH(1,1) process as the innovation terms to generate simulated joint return series;
4. The simulated data vectors will each follow a GARCH(1,1) process and the perceived dependence structure governed by the mixture copula model.
5. Repea...