2
$\begingroup$

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!

$\endgroup$
2
  • 1
    $\begingroup$ What do you know about the sequence of $X$s? Are they independent? Do they have the same distribution? Do you know anything about their distribution? $\endgroup$ – shadowtalker May 18 '16 at 13:39
  • $\begingroup$ @ssdecontrol Yes I know all sequence of $X$. Actually variance I count by GARCH(1,1) model, but I need to calculate actual value($X_{n+1}$) from variance. sorry I'm tiro in statistics, if there any misunderstand please tell, thank you! $\endgroup$ – Nathaniel Chen May 18 '16 at 13:45
0
$\begingroup$

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 $X_{t+1}$.


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.

$\endgroup$
1
  • $\begingroup$ You are welcome, Nathaniel! $\endgroup$ – Richard Hardy May 21 '16 at 10:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.