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I want to forecast a differenced time series of an Index using the combined ARMA-GARCH model (because I want to forecast the mean and not the variance). My model is a ARMA(2,2)-GARCH(1,1) model. So the equations for the first forecast are:

Y(t+1)=Y(t)+Alpha(1)*(Y(t)-Y(t-1))+Alpha(2)*(Y(t-1)-Y(t-2)) - Beta(1)*e(t) - Beta(2)*e(t-1) + e(t+1)

with

e(t+1) = Sigma(t+1)*Z(t+1)  ,   Z(t+1)=N(0,1) 

and

Sigma^2 (t+1) = Omega + a(1)*u^2(t) + b(1)*Sigma^2(t) + b(2)*Sigma^2(t-1)

I tried it with the rugarch package and the ugarchforecast method:

GARCHspec <- ugarchspec( variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),mean.model = list(armaOrder = c(2, 2), include.mean = TRUE))

GARCHfit <- ugarchfit(GARCHspec, diffclosingkursu)

ugarchforecast(GARCHfit,n.ahead=250)

The forecast seems to be quite strange

*------------------------------------*
*       GARCH Model Forecast         *
*------------------------------------*
Model: sGARCH
Horizon: 30
Roll Steps: 0
Out of Sample: 0

0-roll forecast [T0=1976-11-23 01:00:00]:
     Series  Sigma
T+1   10.28 0.7802
T+2   10.30 0.8580
T+3   10.32 0.9264
T+4   10.34 0.9876
T+5   10.36 1.0429
T+6   10.38 1.0933
T+7   10.40 1.1395
T+8   10.43 1.1822
T+9   10.45 1.2217
T+10  10.47 1.2585
T+11  10.49 1.2927
T+12  10.51 1.3247
T+13  10.54 1.3547
T+14  10.56 1.3829
T+15  10.58 1.4093
T+16  10.60 1.4343
T+17  10.63 1.4578
T+18  10.65 1.4800
T+19  10.67 1.5010
T+20  10.69 1.5208
T+21  10.71 1.5396
T+22  10.74 1.5574
T+23  10.76 1.5743
T+24  10.78 1.5903
T+25  10.80 1.6056
T+26  10.82 1.6200
T+27  10.85 1.6338
T+28  10.87 1.6469
T+29  10.89 1.6593
T+30  10.91 1.6743

Also, how can it be, that every forecast for the same time series is the same but e(t+1) should be a random variable?

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closed as off-topic by Richard Hardy, Helin, LocalVolatility, Daneel Olivaw, amdopt Nov 3 '17 at 20:06

  • This question does not appear to be about quantitative finance within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Maybe better suited for Cross Validated SE? $\endgroup$ – LocalVolatility Oct 30 '17 at 13:10
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    $\begingroup$ I'm voting to close this question as off-topic because it is a statistical (rather than financial) question which has already been posted and answered on Cross Validated here. $\endgroup$ – Richard Hardy Nov 1 '17 at 20:29
  • $\begingroup$ yes please close it it was the wrong forum. I posted the same question in cross validated (although it has some reference to finance :) ) $\endgroup$ – user2968163 Nov 1 '17 at 21:31