Document Zbl 0898.62141

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Econom.</a> <a href="/?q=in%3A41848" title="Articles in this Issue">79, No. 1, 97-127 (1997)</a>. </div> <div class="abstract">Summary: A family of parametric GARCH models, defined in terms of an auxiliary process and referred to as the augmented GARCH process, is proposed. The strict stationarity of the augmented GARCH process is characterized and this process is shown to contain many existing parametric GARCH models. The augmented GARCH process can serve as a general alternative for the Lagrange multiplier test of many existing GARCH specifications. The diffusion limit of the augmented GARCH process is shown to contain many bivariate diffusion processes that are commonly used for modeling stochastic volatility in the finance literature. <br class="zbmathjax-paragraph">This convergence result generalizes that of <span class="zbmathjax-textit">D. Nelson</span> [ibid. 45, No. 1/2, 7-38 (1990; <a href="/0719.60089">Zbl 0719.60089</a>)] to cover a substantially larger class of $\text{GARCH} (1,1)$ models and also extends to the $\text{GARCH} (p,q)$ specification. The augmented GARCH process can be used as a direct approximation to the stochastic volatility models, or as the score generator in the efficient method of moments estimation of these models.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A1042462">Cited in <strong>4</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A1042462">Cited in <strong>68</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62P20" title="MSC2020">62P20</a> </td> <td class="space"> Applications of statistics to economics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60J70" title="MSC2020">60J70</a> </td> <td class="space"> Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62M10" title="MSC2020">62M10</a> </td> <td class="space"> Time series, auto-correlation, regression, etc. in statistics (GARCH) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ALyapunov+exponent">Lyapunov exponent</a>; <a href="/?q=ut%3Alocal+time">local time</a>; <a href="/?q=ut%3Astationarity">stationarity</a>; <a href="/?q=ut%3Aaugmented+GARCH+process">augmented GARCH process</a>; <a href="/?q=ut%3ALagrange+multiplier+test">Lagrange multiplier test</a>; <a href="/?q=ut%3Adiffusion+processes">diffusion processes</a>; <a href="/?q=ut%3Astochastic+volatility">stochastic volatility</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0719.60089">Zbl 0719.60089</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0898.62141" data-ciurl="/ci/01042462" data-biburl="/bibtex/01042462.bib" data-amsurl="/amsrefs/01042462.bib" data-xmlurl="/xml/01042462.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01042462.pdf" title="Zbl 0898.62141 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/S0304-4076(97)00009-2" aria-label="DOI for “Augmented GARCH$(p,q)$ process and its diffusion limit”" title="10.1016/S0304-4076(97)00009-2">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Billingsley, P.: Convergence of probability measures. 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