GARCH MCMC

Time Series Volatility#

Description#

Fitting GARCH models via maximum likelihood estimation can sometimes be a bit tricky. Here we offer an alternative using markov chain monte carlo for GARCH(1, 1) and GARCH(2, 1) models.

Model: GARCH(1, 1)#

$$ y_{t} = \mu + \epsilon_{t}, \epsilon_{t} \sim N(0, \sigma_{t}^{2}) $$ $$ \sigma_{t}^{2} = \alpha_{0} + \alpha_{1} \epsilon_{t-1}^{2} + g_{1} \sigma_{t-1}^{2} $$

Model: GARCH(2, 1)#

$$ y_{t} = \mu + \epsilon_{t}, \epsilon_{t} \sim N(0, \sigma_{t}^{2}) $$ $$ \sigma_{t}^{2} = \alpha_{0} + \alpha_{1} \epsilon_{t-1}^{2} + g_{1} \sigma_{t-1}^{2} + g_{2} \sigma_{t-2}^{2} $$

Input#

• draws: Run the sample this many times.
• burn-in: Discard these many samples from the beginning draws. Burn-in must be less than draws.
• timeout: Time limit in minutes.

Returns#

• coef: Coefficient estimates for $g$ for GARCH parameters and $a$ for ARCH parameters
• serr: Standard Errors
• $ \sigma_{t} $: volatility process