Heteroskedasticity

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue" stylesrc="../_private/template.htm" bgcolor="#FFFFFF" link="#669933" vlink="#000000" alink="#000000"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table1"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table2"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table3"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Heteroskedasticity</td> <td width="45" align="right" valign="bottom"> <map name="FPMap6"> <area href="http://www.contingencyanalysis.com" shape="rect" coords="1, 0, 44, 11" target="_top"> </map> <img border="0" src="../images/home.gif" usemap="#FPMap6" width="45" height="12"></td> </tr> </table> </td> </tr> <tr> <td width="100%" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="0%" valign="top" bgcolor="#FFFFCC"> <table border="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" cellspacing="1" id="table4"> <tr> <td width="0%" align="left" valign="top"> <p class="header2_black">Explained:<br> <img border="0" src="../images/blank_1x100.gif" width="100" height="1"></td> <td width="100%" align="left" valign="top"><p class="word_list"> <a href="#conditionally heteroskedastic">conditional heteroskedasticity</a><p class="word_list"> <a href="#conditionally homoskedastic">conditional homoskedasticity</a><p class="word_list"> <a href="#heteroskedastic">heteroskedasticity</a><p class="word_list"> <a href="#homoskedastic">homoskedasticity</a><p class="word_list"> <a href="#regime-switching models">regime switching model</a><p class="word_list"> <a href="#unconditionally heteroskedastic">unconditional heteroskedasticity</a><p class="word_list"> <a href="#unconditionally homoskedastic">unconditional homoskedasticity</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <p class="body">A <a href="../articles/time_series_stochastic_process.htm"> univariate stochastic process</a> <i><b>X</b></i> is said to be <b> <a name="homoskedastic">homoskedastic</a></b> if <a href="../articles/standard_deviation.htm">standard deviations</a> of terms <span style="font-family: Times New Roman; letter-spacing: -0.2pt"> <img border="0" src="../formulas/time_series_univariate_term.gif" align="absbottom" width="19" height="18">are constant for all times <i>t</i>. Otherwise, it is said to be <b> <a name="heteroskedastic">heteroskedastic</a></b>. This is illustrated with realizations of two stochastic processes in Exhibit 1.</span></p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="325" cellpadding="0" id="table5"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Homoskedastic vs. Heteroskedastic</span><br> <span class="exhibit_subheader">Exhibit 1</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td> <img border="0" src="../images/ex1_heteroskedasticity.gif" width="260" height="195"></td> </tr> <tr> <td> <p class="exhibit_legend">Realizations of two processes are indicated. The first exhibits homoskedasticity. The second exhibits heteroskedasticity.</td> </tr> </table> </center> </div> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table6"> <tr> <td width="336" height="14"> <p style="margin-top: 0; margin-bottom: 0"><font size="2"> </font><font size="4"> </font></td> <td width="9" height="14"> <p style="margin-top: 0; margin-bottom: 0"></td> </tr> <tr> <td width="336"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script> <p style="margin-top: 0; margin-bottom: 0"> </td> <td width="9"> <p style="margin-top: 0; margin-bottom: 0"> </td> </tr> </table> <p class="body">Heteroskedasticity is an important concept in finance because asset <a href="../articles/return.htm">returns</a> in the capital, commodity and energy markets often exhibit heteroskedasticity.</p> <p class="body">Heteroskedasticity can take two forms. A process is <a name="unconditionally heteroskedastic">unconditionally heteroskedastic</a> if unconditional standard deviations<img border="0" src="../symbols/t_sigma.gif" align="absbottom" width="13" height="19"> are not constant. It is <a name="conditionally heteroskedastic">conditionally heteroskedastic</a> if conditional standard deviations<img border="0" src="../symbols/t_t-1_sigma.gif" align="absbottom" width="36" height="20"> are not constant (see the <i><a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation conventions</a></i> documentation).</p> <p class="body">For example, <a href="../articles/common_stock.htm">stock</a> or <a href="../articles/coupon_bond.htm">bond</a> returns tend to be conditionally heteroskedastic. The prices exhibit non-constant <a href="../articles/volatility.htm">volatility</a>, but periods of low or high volatility are generally not known in advance. New England electricity prices, on the other hand, exhibit unconditional heteroskedasticity. The prices tend to have higher volatilities during the Summer than during other seasons. This is predictable, so the electricity prices exhibit unconditional heteroskedasticity.</p> <p class="body">If a process is unconditionally heteroskedastic, then it is necessarily conditionally heteroskedastic. The converse is not true. If a process is not unconditionally heteroskedastic or not conditionally heteroskedastic, it is said to be <b><a name="unconditionally homoskedastic">unconditionally homoskedastic</a></b> or <b><a name="conditionally homoskedastic"> conditionally homoskedastic</a></b>, respectively.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table7"> <tr> <td align="right" width="9"> </td> <td width="336"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> </table> <p class="body"> All these notions extend to higher dimensions, A <a href="../articles/time_series_stochastic_process.htm">multivariate stochastic process</a> <i><b>X</b></i> is said to be homoskedastic if its <a href="../articles/correlation.htm">covariance matrix</a> is constant for all times <i>t</i>, etc.</p> <p class="body"> In finance, a variety of models are used for conditionally heteroskedastic processes. These include</p> <p class="body" align="left" style="margin-top: 16; margin-bottom: 16"> <img border="0" src="../images/bullet.gif" width="20" height="10"><a href="../articles/ARCH_GARCH.htm">autoregressive conditional heteroskedastic</a> (ARCH) models;</p> <p class="body" align="left" style="margin-top: 16; margin-bottom: 16"> <img border="0" src="../images/bullet.gif" width="20" height="10"><a href="../articles/ARCH_GARCH.htm">generalized ARCH</a> (GARCH) models</p> <p class="body" align="left" style="margin-top: 16; margin-bottom: 16"> <img border="0" src="../images/bullet.gif" width="20" height="10"><b><a name="regime-switching models">regime-switching models</a></b>; and</p> <p class="body" align="left" style="margin-bottom: 30; margin-top:16"> <img border="0" src="../images/bullet.gif" width="20" height="10"><a href="../articles/stochastic_volatility_model.htm">stochastic volatility models</a>.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table8"> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Internal Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table9"> <tr> <td width="100%"> <span style="font-family: Times New Roman; 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