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1

let' s define a ARMA-GARCH model: $y_{t} = \mu_{t} + \epsilon_{t}$ where $\mu_{t} $ is the conditional mean process (ARMA(p,q) part, $\mu_{t} = E(y_{t}|\mathcal{F}_{t-1})$) . The errors (or mean residuals) re defined by: $\epsilon_{t} = \sigma_{t} \eta_{t}$ where $\eta_{t}$ is a white noise (0,1) Then : $Var[\epsilon_{t}]= \sigma_{t}^{2}$. next see ...


0

I have found the mistake. The ugarchfit function sets automatically non negativity constraints for all coefficients- This makes sense since the alpha in our case shouldn't be negative. However, when releasing the constraint to negative values you get the right results. The only explanation I can think of is that in the course of optimisation, temporarily ...


2

If your question is: "Given all the information available up to time $t$, if I compute the 1 period ahead forecast $r_{t+1}$, is the conditional volatility over $[t,t+1[$ given by $\sqrt{r_{t+1}}$?", the answer is NO. To compute the 1 period ahead conditional variance, you should use your model equations (see this post which might help you better understand ...


4

Assume that your stationary time series (here a daily close-to-close log-returns' series) is modelled as follows $\forall t \in \mathcal{T}=\{1,...,N\}$ \begin{align} r_t &= E_{t-1}[r_t] + \epsilon_t \\ &= E_{t-1}[r_t] + \sigma_t z_t \end{align} with $z_t \sim N(0,1)$ and $\{z_t\}_{t \in \mathcal{T}}$ are IID. The above equations suggest that, ...


0

A limitation of both papers is they focus on point estimates, i.e they compare $\sigma_{t}$ with $h_{t}$ in the loss functions of the SPA Tests. A possible suggestion to overcome it, is to use a loss function based on density forecast, in order to capture the whole forecast density distribution and not only a single point. This may have important ...


1

When calculating the simple arithmetic mean, each observation has an equal weight: $$ \hat \mu^{simple} = \frac{1}{T}\sum_{t=1}^T x_t.$$ If the observations are $i.i.d.$, $\hat \mu^{simple}$ is an efficient estimator of the population mean. When estimating the mean of a GARCH process, $\hat \mu^{simple}$ is no longer efficient. It makes sense to ...


1

Try the mgarch package, it's available at CRAN. In this link you will find an example from Prof. Zivot.


6

Consider the GARCH(1,1) process \begin{align} r_{t+1} &= \sigma_{t+1} z_{t+1} \\ \sigma^2_{t+1} &= \omega+\alpha r^2_t +\beta \sigma^2_{t} \end{align} for the returns $r_t$, with ${z_t} \sim N (0,1)$ IID. In what follows, let us distinguish the conditional return variance $$ V [ r_{t+1} \vert \mathcal{F}_t ] = \sigma^2_{t+1} $$ from the ...



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