Forecasting using GARCH in R

Question

I am using the predict and ugarchforecast functions in R. When I fit my models and try to forecast, I get either only increasing or decreasing values for sigma, does anyone know why?

Thank you

Example:

eGARCHfit2 = ugarchspec(variance.model=list(model="eGARCH", garchOrder=c(1,1)), mean.model=list(armaOrder=c(0,0), include.mean=TRUE), distribution.model="norm") eGARCH2 <- ugarchfit(brentlog2, spec=eGARCHfit2)

ugarchforecast(eGARCH2, data =brentlog2, n.ahead = 21)
* GARCH Model Forecast * ------------------------------------
* Model: eGARCH
* Horizon: 21 Roll Steps: 0 Out of Sample: 0 0-roll

forecast [T0=1976-06-26 01:00:00]:
Series Sigma
T+1 0.0002619 0.008350
T+2 0.0002619 0.008387
T+3 0.0002619 0.008423
T+4 0.0002619 0.008459
T+5 0.0002619 0.008496
T+6 0.0002619 0.008532
T+7 0.0002619 0.008569
T+8 0.0002619 0.008605
T+9 0.0002619 0.008642
T+10 0.0002619 0.008678

The first value is the mean which is always constant and the second one is sigma which is always increasing as you can see.

Answer 1

This should follow from the properties of the forecast - for example the GARCH(1,1) forecast for $h$ steps is computing the conditional expectation of $\sigma^2_{t+h}$ based on the information set-up in $t$. This can be computed recursively by

$$ V(\varepsilon_{t+h}|F_t)=\omega+\alpha\varepsilon_{t+h-1|F_t}+\beta\sigma^2_{t+h-1|F_t}\\ =\omega\sum\limits_{i=0}^{h-2}(\alpha+\beta)^i+(\alpha+\beta)^{h-1}\sigma^2_{t+1} $$ something similar but more complicated should hold for the EGARCH model. Stationarity for $\varepsilon_t^2$ hold if $|\alpha+\beta|<1$. If we assume stationarity the second term of the formula above should decrease with $h$. The first term is an increasing function in $h$ for $\omega>0$ which is a standard assumption to ensure the positiveness of the conditional variance. Therefore it seems to me like there is no monotonicity in $h$ but one saddle point at which the forecast starts to increase.