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<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial","sans-serif"">Hi there,
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial","sans-serif""><o:p> </o:p></span></p>
<p class="MsoListParagraph" style="text-indent:-18.0pt;mso-list:l2 level1 lfo1"><![if !supportLists]><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif""><span style="mso-list:Ignore">1)<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">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<o:p></o:p></span></p>
<p class="MsoListParagraph"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif""><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:9.0pt;font-family:"Courier New"">Data <- read.csv("WTI_logreturnsUS.csv", header = TRUE, sep = ";", dec=".")<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="IT" style="font-size:9.0pt;font-family:"Courier New"">renditen <- Data$LogReturnsWTI<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="IT" style="font-size:9.0pt;font-family:"Courier New"">Data_WTI <- renditen<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:9.0pt;font-family:"Courier New"">nrenditen = renditen - mean(renditen)<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:9.0pt;font-family:"Courier New"">external <- Data$LogReturnsStocks<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:9.0pt;font-family:"Courier New"">dim(external) <- c(length(external),1)<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:9.0pt;font-family:"Courier New"">mean_WTI <- mean(renditen)<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:9.0pt;font-family:"Courier New""><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:9.0pt;font-family:"Courier New"">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")<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:9.0pt;font-family:"Courier New"">fit <- ugarchfit(spec,nrenditen)<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:9.0pt;font-family:"Courier New""><o:p> </o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif""><o:p> </o:p></span></p>
<p class="MsoPlainText" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">likelihood 2326.425 2319.141<o:p></o:p></span></p>
<p class="MsoPlainText" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">
<o:p></o:p></span></p>
<p class="MsoPlainText" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">mxreg1 -0.3644 -0.3124<o:p></o:p></span></p>
<p class="MsoPlainText" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">omega -0.3081</span><span lang="EN-US">
</span><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">-0.0474<o:p></o:p></span></p>
<p class="MsoPlainText" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">alpha1 -0.1288</span><span lang="EN-US">
</span><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">-0.1090<o:p></o:p></span></p>
<p class="MsoPlainText" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">alpha2 -0.1308
</span><span lang="EN-US"> </span><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">0.0644<o:p></o:p></span></p>
<p class="MsoPlainText" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">gamma1 0.2575</span><span lang="EN-US">
</span><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">0.2189<o:p></o:p></span></p>
<p class="MsoPlainText" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">gamm</span><span style="font-size:10.0pt;font-family:"Arial","sans-serif"">a2 0.2340</span>
<span style="font-size:10.0pt;font-family:"Arial","sans-serif"">-0.1441<o:p></o:p></span></p>
<p class="MsoPlainText" style="margin-left:35.4pt"><span style="font-size:10.0pt;font-family:"Arial","sans-serif"">beta1 -0.6917 0.9999<o:p></o:p></span></p>
<p class="MsoPlainText" style="margin-left:35.4pt"><span style="font-size:10.0pt;font-family:"Arial","sans-serif"">beta2 0.9764 0.4135<o:p></o:p></span></p>
<p class="MsoPlainText" style="margin-left:35.4pt"><span style="font-size:10.0pt;font-family:"Arial","sans-serif"">beta3 0.6738 -0.4199<o:p></o:p></span></p>
<p class="MsoPlainText" style="margin-left:35.4pt"><span style="font-size:10.0pt;font-family:"Arial","sans-serif"">skew 0.9659 0.9708<o:p></o:p></span></p>
<p class="MsoPlainText" style="margin-left:35.4pt"><span style="font-size:10.0pt;font-family:"Arial","sans-serif"">shape 1.9107 1.9373<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span style="font-size:9.0pt;font-family:"Courier New""><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">Why do I see these differences?<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif""><o:p> </o:p></span></p>
<p class="MsoListParagraph" style="margin-left:35.4pt;text-indent:-18.0pt;mso-list:l2 level1 lfo1">
<![if !supportLists]><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif""><span style="mso-list:Ignore">2)<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">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)<o:p></o:p></span></p>
<p class="MsoListParagraph" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif""><o:p> </o:p></span></p>
<p class="MsoPlainText" style="margin-left:53.4pt;text-indent:-18.0pt;mso-list:l1 level1 lfo2">
<![if !supportLists]><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif""><span style="mso-list:Ignore">a)<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">min(Mod(polyroot(c(1, -betas)))) > 1
<o:p></o:p></span></p>
<p class="MsoPlainText"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif""><o:p> </o:p></span></p>
<p class="MsoPlainText" style="margin-left:53.4pt;text-indent:-18.0pt;mso-list:l1 level1 lfo2">
<![if !supportLists]><span style="font-size:10.0pt;font-family:"Arial","sans-serif""><span style="mso-list:Ignore">b)<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:10.0pt;font-family:"Arial","sans-serif"">|beta_i| < 1, i = 1,…,p
<o:p></o:p></span></p>
<p class="MsoListParagraph" style="margin-left:35.4pt"><span style="font-size:10.0pt;font-family:"Arial","sans-serif""><o:p> </o:p></span></p>
<p class="MsoListParagraph" style="margin-left:35.4pt"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">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?<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif""><o:p> </o:p></span></p>
<p class="MsoListParagraph" style="text-indent:-18.0pt;mso-list:l2 level1 lfo1"><![if !supportLists]><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif""><span style="mso-list:Ignore">3)<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif""><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif"">Many thanks in advance!<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US" style="font-size:10.0pt;font-family:"Arial","sans-serif""><o:p> </o:p></span></p>
<p class="MsoNormal" style="text-autospace:none"><span lang="EN-US" style="font-size:8.0pt;font-family:"Verdana","sans-serif";color:navy;mso-fareast-language:DE">Mit freundlichen Grüßen / Kind regards</span><span lang="EN-US" style="font-size:8.0pt;font-family:"Arial","sans-serif";mso-fareast-language:DE"><o:p></o:p></span></p>
<p class="MsoNormal" style="text-autospace:none"><span lang="EN-US" style="font-size:7.0pt;font-family:"Arial","sans-serif";mso-fareast-language:DE"><o:p> </o:p></span></p>
<p class="MsoNormal" style="text-autospace:none"><b><span lang="EN-US" style="font-size:7.0pt;font-family:"Verdana","sans-serif";color:navy;mso-fareast-language:DE">Stefan Jäschke</span></b><span lang="EN-US" style="font-size:7.0pt;font-family:"Verdana","sans-serif";color:navy;mso-fareast-language:DE"><o:p></o:p></span></p>
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<p class="MsoNormal" style="text-autospace:none"><span lang="EN-US" style="font-size:7.0pt;font-family:"Verdana","sans-serif";color:navy;mso-fareast-language:DE">Performance Controlling CAO Ga</span><span style="font-size:7.0pt;font-family:"Verdana","sans-serif";color:navy;mso-fareast-language:DE">s
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<a href="mailto:stefan.jaeschke@rwe.com"><span style="font-family:"Times New Roman","serif";color:blue">stefan.jaeschke@rwe.com</span></a><o:p></o:p></span></p>
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<p class="MsoNormal" style="text-autospace:none"><span lang="EN-GB" style="font-size:7.0pt;font-family:"Arial","sans-serif";color:navy;mso-fareast-language:DE">Supervisory Board: Peter Terium (Chairman)<o:p></o:p></span></p>
<p class="MsoNormal" style="text-autospace:none"><span lang="EN-GB" style="font-size:7.0pt;font-family:"Arial","sans-serif";color:navy;mso-fareast-language:DE">Management Board:<br>
Stefan Judisch (CEO), <br>
Dr Markus Krebber, Alan Robinson<br>
Headquarters: Essen<br>
Registered at Local District Court, Essen<br>
Commercial Registry No.: HRB 14327<br>
Sales Tax ID No.: DE 8130 22 070</span><span lang="EN-GB" style="font-size:7.0pt;font-family:"Arial","sans-serif";color:navy;mso-fareast-language:DE">
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<p class="MsoNormal" style="text-autospace:none"><span lang="EN-GB" style="font-size:7.0pt;font-family:"Arial","sans-serif";color:green;mso-fareast-language:DE">Please consider the environment before printing this e-mail</span><span lang="EN-GB" style="font-size:10.0pt;font-family:"Arial","sans-serif";mso-fareast-language:DE"><o:p></o:p></span></p>
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