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Generalized Heteroskedasticity ACF for Moving Average Models in Explicit Forms

By Samir Khaled Safi

Abstract

<!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:PunctuationKerning /> <w:ValidateAgainstSchemas /> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables /> <w:SnapToGridInCell /> <w:WrapTextWithPunct /> <w:UseAsianBreakRules /> <w:DontGrowAutofit /> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--> <p class="MsoNormal" style="text-align: justify; direction: ltr; unicode-bidi: embed;">The autocorrelation function (ACF) measures the correlation between observations at different<span style="mso-spacerun: yes;">&nbsp;&nbsp; </span>distances apart. We derive explicit equations for generalized heteroskedasticity ACF for moving average of order q, MA(q). We consider two cases: Firstly: when the disturbance term follow the general covariance matrix structure Cov(w<sub>i</sub>, w<sub>j</sub>)=<span style="font-family: Symbol; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">S</span></span> with <span style="font-family: Symbol; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">s</span></span><sub>i,j</sub> <span style="font-family: Symbol; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">&sup1;</span></span> 0 <span style="font-family: Symbol; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">"</span></span> i<span style="font-family: Symbol; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">&sup1;</span></span>j<span style="position: relative; top: 7.0pt; mso-text-raise: -7.0pt;"> </span>. Secondly: when the diagonal elements of <span style="font-family: Symbol; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">S</span></span> are not all identical but <span style="font-family: Symbol; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">s</span></span><sub>ij</sub> = 0 <span style="font-family: Symbol; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">"</span></span> i<span style="font-family: Symbol; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">&sup1;</span></span>j, i.e. <span style="font-family: Symbol; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">S</span></span>=diag(<span style="font-family: Symbol; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">s</span></span><sub>11</sub>, <span style="font-family: Symbol; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">s</span></span><sub>22,&hellip;,</sub><span style="font-family: Symbol; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">s</span></span><sub>tt</sub>). The forms of the explicit equations depend essentially on the moving average coefficients and covariance structure of the disturbance terms.</p> <p class="MsoNormal"><span style="mso-spacerun: yes;">&nbsp;</span></p> <!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles> </xml><![endif]--><!--[if gte mso 10]> <style>\ud /* Style Definitions */\ud table.MsoNormalTable\ud {mso-style-name:"جدول عادي";\ud mso-tstyle-rowband-size:0;\ud mso-tstyle-colband-size:0;\ud mso-style-noshow:yes;\ud mso-style-parent:"";\ud mso-padding-alt:0cm 5.4pt 0cm 5.4pt;\ud mso-para-margin:0cm;\ud mso-para-margin-bottom:.0001pt;\ud mso-pagination:widow-orphan;\ud font-size:10.0pt;\ud font-family:"Times New Roman";\ud mso-ansi-language:#0400;\ud mso-fareast-language:#0400;\ud mso-bidi-language:#0400;}\ud </style> <![endif]--

Topics: Heteroscedasticity, Homoscedasticity, Autocorrelation, Moving Average, Covariance., Mathematics, QA1-939, Statistics, HA1-4737
Publisher: University of the Punjab
Year: 2014
OAI identifier: oai:doaj.org/article:e5bd013d612849e69333f13a98d87410
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