Forecasting next value range based on variance and historical values

Question

If I know next variance from GARCH(1,1) model, can I get $X_{i+1}$?

For example:

$$ \sigma^2_n = \frac{1}{N} \sum_{i=1}^n(X_i-\mu_i)^2 $$

where

$$ \mu_i = \frac{1}{n} \sum_{i=1}^nX_i. $$

So

$$ \mu_{i+1} = \frac{1}{n} \left(\sum_{i=1}^nX_i + X_{n+1}\right). $$

I know the variance $\sigma^2_{n+1}$ and $X_n$, $\mu_n$, $n$, $n+1$.

Can I calculate $X_{i+1}$?

Maybe something like (not the correct formula I guess...):

$$ \sigma^2_{n+1} = \frac{1}{N+1} \sum_{i=1}^{n+1}(X_i-\mu_{n+1})^2 $$

to

$$ \sigma^2_{n+1} someVariable = X_{i+1} $$

Thanks!

Answer 1

If you need a point forecast of $X_{t+1}$ (which you indicate in the body of your question), you should build a conditional mean model for $X_t$.

If you need a range forecast of $X_{t+1}$ (which you indicate in the title of your question), you would need a predicted density or predicted quantiles of $X_{t+1}$.

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A GARCH model involves a conditional mean and a conditional variance specification, e.g.

$$ \begin{aligned} X_t &= \sigma_t \varepsilon_t \\ \sigma_t^2 &= \omega + \alpha_1 X_{t-1}^2 + \beta_1 \sigma_{t-1}^2 \end{aligned} $$

with a density specification for $\varepsilon_t$, e.g. $\varepsilon_t \sim i.i.d.(0,1,\theta)$ for some density function $d$ with zero mean, unit variance and possibly extra parameters $\theta$.

Given the estimated model, you can obtain the predicted density of $X_{t+1}$, $d(0,\hat\sigma^2_{t+1},\hat\theta)$, where hats denote fitted values and

$$ \hat\sigma_{t+1}^2 = \hat\omega + \hat\alpha_1 X_{t}^2 + \hat\beta_1 \hat\sigma_{t}^2. $$

You can obtain an $x\%$ confidence interval for $X_{t+1}$ using the quantiles of the predicted density of $X_{t+1}$. This will be the likely range of $X_{t+1}$, where "likely" is quantified as $x\%$ confidence.