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:

` [1] -0.008230499 -0.025105921 -0.025752496 0.025752496 [5] -0.008510690 0.041847110 -0.033336420 0.041499731 [9] -0.008163311 0.024292693 -0.032523192 -0.008298803 [13] 0.080042708 0.000000000 0.088292607 0.041385216 [17] -0.013605652 0.046831300 0.006514681 -0.019672766 [21] 0.013158085 -0.006557401 0.019544596 0.092373320 [25] 0.116474991 0.020726131 0.169418152 0.167054085 [29] -0.128832872 0.056695344 -0.032002731 -0.016393810 [33] -0.151399646 0.104879631 0.159701110 -0.029964789 [37] -0.003809528 -0.034955015 -0.011928571 -0.016129382 [41] 0.012121361 -0.004024150 -0.004040410 -0.008130126 [45] -0.033198069 -0.065382759 0.017857617 0.034786116 [49] -0.017241806 -0.026433257 0.013303966 0.000000000 [53] -0.045052664 0.013730193 -0.009132484 -0.013857035 [57] -0.023530497 -0.019231362 -0.070380797 -0.005221944 [61] 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:**

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

I need the intervals for both predicted and historical volatility.

**Contents**hide

#### Best Answer

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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