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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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  • 1
    $\begingroup$ 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