[R-SIG-Finance] Update of rugarch package yields different results / questions on stationarity conditions

[Stefan.Jaeschke at rwe.com]

Hi there,


1)     I have recently updated the rugarch package to version 1.3-3 (I do not remember the previous number) and I am surprised to see different results when fitting a dataset, the loglikelihood is lower than before and the beta parameters have changed significantly. Below I put the code from the fit


Data <- read.csv("WTI_logreturnsUS.csv", header = TRUE, sep = ";", dec=".")
renditen <- Data$LogReturnsWTI
Data_WTI <- renditen
nrenditen = renditen - mean(renditen)
external <- Data$LogReturnsStocks
dim(external) <- c(length(external),1)
mean_WTI <- mean(renditen)

spec = ugarchspec(variance.model = list(model = "eGARCH", garchOrder = c(2,3), submodel = NULL, external.regressors = NULL, variance.targeting = FALSE), mean.model =list(armaOrder = c(0, 0), include.mean = FALSE, external.regressors = external), distribution.model = "sged")
fit <- ugarchfit(spec,nrenditen)




likelihood         2326.425          2319.141



mxreg1             -0.3644             -0.3124

omega              -0.3081                 -0.0474

alpha1              -0.1288                 -0.1090

alpha2              -0.1308                0.0644

gamma1           0.2575                   0.2189

gamma2           0.2340                  -0.1441

beta1                -0.6917             0.9999

beta2                0.9764              0.4135

beta3                0.6738              -0.4199

skew                0.9659             0.9708

shape               1.9107              1.9373

Why do I see these differences?


2)     Why do we need the following two conditions for strict stationarity of an EGARCH(q,p) model? I do refer to the ARMA representation in Nelson (1991), Equation (2.3)



a)     min(Mod(polyroot(c(1, -betas)))) > 1



b)    |beta_i| < 1, i = 1,...,p



Whereas condition a) is clear to me (stationarity of AR processes), I don't see we should restrict the parameter |beta_i| < 1. Could somewhen help on that? Why are the parameters regarding q not involved in the conditions at all?


3)     In general, I am aware of conditions for stationarity for conditional mean processes (e.g. ARMA-models) or conditional variance processes (e.g. GARCH-models). I am struggling a bit to find sufficient conditions for (strikt) stationarity in case of combinations. For instance, an ARMA(1,0)-GARCH(1,1) or ARMA(0,1)-EGARCH(2,3) model. Can I take the conditions for mean/variance separately and join them in the end? They should interact somehow, shouldn't they? If anybody could help me on that, I would be very pleased.

Many thanks in advance!

Mit freundlichen Grüßen / Kind regards

Stefan Jäschke
RWE Supply & Trading GmbH
Performance Controlling CAO Gas & VAC (MFC-GV)
Altenessener Str. 27
45141 Essen
Germany
Phone                      +49 201 5179-1674
Email                      stefan.jaeschke at rwe.com<mailto:stefan.jaeschke at rwe.com>
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