In: Economics
show that GARCH(1,1) has time varying conditional variance
How to calculate the conditional variance of a time series?
Intuition
Firstly we model the conditional mean process (using a ARMA,ARFIMA...) and subtract it from the original returns series to obtain the "return residuals" : rt−μt=ϵt=σtzt where zt is an i.i.d process with Et(zt)=0 and Vart(zt)=1. Note that the conditional variance of ϵt is equal to σ2t.
However since we know that the variance is time varying we also know that σ2t has a time dependent structure and exhibits autocorrelations (so do the squares returns residuals). We can model it using GARCH class of models which can (very roughly) be seen as ARMA models for the conditional variance process.
Example of a Garch(1,1) : σ2t=a+αϵ2t−1+βσ2t−1
Once we fit our conditional variance models we will be left with the conditional variance process σ2t.At this point we know the conditional variance process σ2t and ϵ2t. This allow us to obtain the final standardized residuals series zt which is i.i.d and equal to ϵt/σt=zt.
Estimation
How do we estimate it ?
The simplest way is to rely on the Maximum Likelihood Estimation (MLE) method. We need to assume a distribution for the zt (the final residuals). Since we know that these residuals are i.i.d it is easy to compute the log-likelihood for a given zt serie. (to be more precise the typical arguments for the likelihood function are ϵ2t and σ2t)
Example : If we assume a normal distribution for zt the log likelihood (assuming no constant) is given by : LogLik=−12∑T1[log(2π)+log(σ2t)+z2t](=−12∑T1[log(2π)+log(σ2t)+ϵ2tσ2t])
But how can we practically obtain zt ? A solution is to use what we called "filters" takings as input the returns series and, based on a particular specifications (ex: arma(1,1)-garch(1,1)), returning σ2t. By "filtering" we mean that we applied the autoregressive framework (recursive algorithm on both the mean and variance) on the input serie (the return serie) to obtain as output the zt .