# Solved – Estimating the confidence interval for the volatility of a GARCH model

This question is a followup of my previous question
Forecasting with ARIMA and GARCH: does my plan look alright?

I have a times series \$r_t\$ and I am trying to estimate its volatility with a GARCH model as in the referred question.

You can for example consider the following time series:

``  -0.008230499 -0.025105921 -0.025752496  0.025752496   -0.008510690  0.041847110 -0.033336420  0.041499731   -0.008163311  0.024292693 -0.032523192 -0.008298803   0.080042708  0.000000000  0.088292607  0.041385216  -0.013605652  0.046831300  0.006514681 -0.019672766   0.013158085 -0.006557401  0.019544596  0.092373320   0.116474991  0.020726131  0.169418152  0.167054085  -0.128832872  0.056695344 -0.032002731 -0.016393810  -0.151399646  0.104879631  0.159701110 -0.029964789  -0.003809528 -0.034955015 -0.011928571 -0.016129382   0.012121361 -0.004024150 -0.004040410 -0.008130126  -0.033198069 -0.065382759  0.017857617  0.034786116  -0.017241806 -0.026433257  0.013303966  0.000000000  -0.045052664  0.013730193 -0.009132484 -0.013857035  -0.023530497 -0.019231362 -0.070380797 -0.005221944   0.015584731 -0.010362787 -0.048009219 -0.005479466 >  ``

the elements of which are not autocorrelated.

The volatility is then estimated using a GARCH(1,1) model and predicted from it as follows:

``G.A = garchFit(formula = A~garch(1, 1), data = diff_log_close_price, trace = F) G.A.est = predict(G.A, 30, plot=T, nx=nrow(closing_price)) volatility.A = c(volatility(G.A, type = "sigma"), G.A.est\$standardDeviation) ``

Questions:

1. How can I estimate the confidence interval for the volatility from a theoretical point of view?
2. Is there an R function that does it?

I need the intervals for both predicted and historical volatility.

Contents

Due to the fact that I am not an avid R user, I will only contribute to question 1).

The GARCH process is defined by the mean equation begin{equation} r_t = mu + sigma_t z_t end{equation} and the GARCH equation begin{equation} sigma_{t+1}^2 = omega + alpha varepsilon_t^2 + betasigma_t^2 end{equation} combined with an assumption about \$z_t\$, e.g. \$z_tsim N(0,1)\$.

I have never seen people showcase confidence intervals for the fitted volatility process – in the GARCH process, the volatility at time t+1 is not stohastic, but know at time t.

It will however make sense, to showcase confidence intervals for \$varepsilon_{t+1vert t} sim N(0,sigma_{t+1}^2)\$ or \$varepsilon_{t+1vert t}^2\$.

To construct confidence intervals for GARCH forecasts multiple periods ahead (\$sigma_{t+kvert t}^2\$), where no closed form distribution is available, the standard procedure is to simulate a large number of paths and take the desired quantile based on either the assumed distribution of \$z_t\$ applied in the estimation or the bootstrapped distribution.

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