Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates

<!DOCTYPE html> <html lang="en" xmlns:mml="http://www.w3.org/1998/Math/MathML"> <head> <meta charset="utf-8"> <meta http-equiv="x-ua-compatible" content="ie=edge"> <title>Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates</title>  <link href="https://www.degruyter.com/document/doi/10.1515/jtse-2018-0024/html" rel="canonical"></link>  <link rel="preload" href="/assets/fonts/montserrat-v15-latin-ext_cyrillic-ext-regular.woff2" as="font" type="font/woff2" crossorigin="anonymous"> <link rel="preload" href="/assets/fonts/montserrat-v25-latin-ext_cyrillic-ext-500.woff2" as="font" type="font/woff2" crossorigin="anonymous"> <link rel="preload" href="/assets/fonts/montserrat-v15-latin-ext_cyrillic-ext-700.woff2" as="font" type="font/woff2" crossorigin="anonymous">  <link rel="preload" href="/assets/fonts/merriweather-v30-latin-ext_cyrillic-ext-300.woff2" as="font" type="font/woff2" crossorigin="anonymous"> <link rel="preload" href="/assets/fonts/merriweather-v22-latin-ext_cyrillic-ext-regular.woff2" as="font" type="font/woff2" crossorigin="anonymous"> <link rel="preload" href="/assets/fonts/merriweather-v22-latin-ext_cyrillic-ext-700.woff2" as="font" type="font/woff2" crossorigin="anonymous">  <link rel="preload" href="/assets/stylesheets/fontawesome.woff2" as="font" type="font/woff2" crossorigin="anonymous"> <link rel="stylesheet" media="all" href='/assets/stylesheets/02f295d1f6a8e9c282e7831cd493e8ff-bootstrap.purged.min.css' /> <link rel="stylesheet" media="all" href='/assets/stylesheets/028959ba60d23cf0bdb77660595d35e1-main.min.css' /> <link rel="stylesheet" media="all" href='/assets/stylesheets/13761c60ba5201f465803f3005ccd7ef-fontawesome-codes.css' /> <link rel="shortcut icon" type="image/x-icon" href='/assets/images/ec7d7606b4e2f3f921b5e1700948efb6-favicon.ico' /> <link rel="alternate" hreflang="en" href="https://www.degruyter.com/document/doi/10.1515/jtse-2018-0024/html?lang=en" /> <link rel="alternate" hreflang="de" href="https://www.degruyter.com/document/doi/10.1515/jtse-2018-0024/html?lang=de" /> <link rel="alternate" hreflang="x-default" href="https://www.degruyter.com/document/doi/10.1515/jtse-2018-0024/html" />  <link rel="schema.dcterms" href="http://purl.org/dc/terms/"> <meta name="dcterms.rightsHolder" content="Walter de Gruyter GmbH"> <meta name="dcterms.rights" content="De Gruyter expressly reserves the right to use all content for commercial text and data mining within the meaning of Section 44b of the German Copyright Act."> <link rel="dns-prefetch" href="https://www.google-analytics.com" /> <meta name="google" content="notranslate" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="description" content="Recent developments in econometric methods enable estimation and testing of general long memory processes, which include the general Gegenbauer process. This paper considers the error correction model for a vector general long memory process, which encompasses the vector autoregressive fractionally integrated moving average and general Gegenbauer processes. We modify the tests for unit roots and cointegration, based on the concept of heterogeneous autoregression. The Monte Carlo simulations show that the finite sample properties of the modified tests for unit roots are satisfactory, while the conventional tests suffer from size distortion. The experiments also indicate that the modified tests for cointegration improve the problem of finding too many cointegration relationships which arises for fractionally integrated series. Empirical results for interest rates series for the USA and Australia indicate that: (1) the modified unit root test detected unit roots for all series; (2) after differencing, all series favour the general Gegenbauer (GG) process; (3) the modified test for cointegration found only two cointegrating vectors; and (4) the zero interest rate policy in the USA had no effect on the cointegrating vectors for the two countries."/> <meta property="og:url" content="https://www.degruyter.com/document/doi/10.1515/jtse-2018-0024/html"/><meta property="og:site_name" content="De Gruyter"/><meta property="og:title" content="Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates"/><meta property="og:type" content="article"/><meta property="og:locale" content="en"/><meta property="og:image" content="https://www.degruyter.com/document/cover/journal_key/JTSE/product"/><meta property="og:image:type" content="image/jpeg"/><meta property="og:description" content="Recent developments in econometric methods enable estimation and testing of general long memory processes, which include the general Gegenbauer process. This paper considers the error correction model for a vector general long memory process, which encompasses the vector autoregressive fractionally integrated moving average and general Gegenbauer processes. We modify the tests for unit roots and cointegration, based on the concept of heterogeneous autoregression. The Monte Carlo simulations show that the finite sample properties of the modified tests for unit roots are satisfactory, while the conventional tests suffer from size distortion. The experiments also indicate that the modified tests for cointegration improve the problem of finding too many cointegration relationships which arises for fractionally integrated series. Empirical results for interest rates series for the USA and Australia indicate that: (1) the modified unit root test detected unit roots for all series; (2) after differencing, all series favour the general Gegenbauer (GG) process; (3) the modified test for cointegration found only two cointegrating vectors; and (4) the zero interest rate policy in the USA had no effect on the cointegrating vectors for the two countries."/><meta property="og:locale:alternate" content="de"/><meta property="article:author" content="Manabu Asai"/><meta property="article:author" content="Shelton Peiris"/><meta property="article:author" content="Michael McAleer"/><meta property="article:author" content="David E. Allen"/><meta property="article:tag" content="long memory processes"/><meta property="article:tag" content="Gegenbauer process"/><meta property="article:tag" content="Dickey–Fuller Tests"/><meta property="article:tag" content="cointegration"/><meta property="article:tag" content="differencing"/><meta property="article:tag" content="interest rates"/><meta property="article:published_time" content="2020-01-02"/><meta property="article:section" content="Journal of Time Series Econometrics"/> <meta name="citation_issue" content="1" /> <meta name="citation_issn" content="1941-1928" /> <meta name="citation_language" content='en' /> <meta name="citation_volume" content="12" /> <meta name="citation_publisher" content='De Gruyter' /> <meta name="citation_keywords" content='long memory processes; Gegenbauer process; Dickey–Fuller Tests; cointegration; differencing; interest rates' /> <meta name="citation_author" content="Manabu Asai" /> <meta name="citation_author" content="Shelton Peiris" /> <meta name="citation_author" content="Michael McAleer" /> <meta name="citation_author" content="David E. Allen" /> <meta name="citation_title" content='Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates' /> <meta name="citation_xml_url" content="https://www.degruyter.com/document/doi/10.1515/jtse-2018-0024/xml" /> <meta name="citation_journal_title" content="Journal of Time Series Econometrics" /> <meta name="citation_publication_date" content='2020/01/02' /> <meta name="citation_doi" content="10.1515/jtse-2018-0024" /> <script type="application/ld+json">{"author":[{"@type":"Person","name":"Manabu Asai"},{"@type":"Person","name":"Shelton Peiris"},{"@type":"Person","name":"Michael McAleer"},{"@type":"Person","name":"David E. 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Allen</span> <contributor-popdown name="David E. Allen" position="4" email="" affiliations="Department of Finance, School of Business and Law, Edith Cowan University, Perth, Australia; Department of Finance, Asia University, Taichung, Taiwan; School of Mathematics and Statistics, University of Sydney, Sydney, New South Wales, Australia" orcid="https://orcid.org/0000-0001-7782-0865" > </contributor-popdown> <a href="https://orcid.org/0000-0001-7782-0865" target="_blank"> <img alt="ORCID logo" src='/assets/images/661a4eb80b7527d6467b6e5742778cc1-orcid.svg' width="16" height="16" /> </a> </span></span> </li> </ul> <div class="subTitleInfoProductPage">From the journal <a class="ga_parent ga_parent_journal" href="/journal/key/jtse/html">Journal of Time Series Econometrics</a></div> <div class="doi"><a href="https://doi.org/10.1515/jtse-2018-0024" class="linkWithoutStyle subTitleInfoProductPage ga_doi" target="_blank">https://doi.org/10.1515/jtse-2018-0024</a></div> </div> </div> </div> <div class="d-flex align-items-center flex-wrap px-4 px-lg-2 "> <button id='citationsModalButton' type="button" class="btn btn-main-content ga_cite_this me-2" href='#citationsModal' data-bs-toggle="modal" data-bs-target="#citationsModal" data-doi="10.1515/jtse-2018-0024" aria-controls='citationsModal'> Cite this </button> <button id="socialModalButton" type="button" class="btn btn-main-content ga_share_this me-2" href="#socialModal" data-bs-toggle="modal" data-bs-target="#socialModal" aria-controls="socialModal"> Share this </button> <div class="dimensions __dimensions_badge_embed__ ga_dimensions_citation me-2" data-doi="10.1515/jtse-2018-0024" data-hide-zero-citations="true" data-legend="never" data-style="large_rectangle"></div> </div> <div id="div-document-progress-bar" class="sticky-top d-none"> <div class="row"> <div class="col-12"> <div class="progress progress-bar-toolbar"> <div id="document-progress-bar" class="progress-bar bg-primary" role="progressbar" aria-label="Document progress bar" aria-valuenow="0" aria-valuemin="0" aria-valuemax="100"></div> </div> </div> </div> </div> <div id="document-main-content" class='row'> <div class="container-fluid px-3 px-lg-0 py-2"> <div class="col"> <div class="d-none analyticsHolder" data-subjects='EC|EC-03|EC-03-02|MT|MT-09' data-publisherCode='DEG' data-license='restricted' data-publisher='De Gruyter' data-contentName='Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates' data-doi='10.1515/jtse-2018-0024' data-parentIdentifier='JTSE' data-parentName='Journal of Time Series Econometrics' data-languages='en' ></div> <div id="documentContent" class="content py-2" data-doi='10.1515/jtse-2018-0024' data-accessrestricted="true" data-countertype="document"> <div class="px-2"> <div class="alert alert-info-box" role="alert"> <div id="noAccessReasonSpinner"> <div class="spinner-border text-primary" role="status"></div> </div> <div id="noAccessReasonContent"> <small class="fa fas fa-block"></small> <small id="noAccessReason"></small> <div class="previewAlertContainer"> <small class="previewAlert">Showing a limited preview of this publication:</small> </div> </div> </div> <div id="text-container"> <div xmlns:dgdoi="http://degruyter.com/resources/doi-from-crossref" xmlns:dgpm="http://degruyter.com/resources/fetched-pubmed-id" class="contentWrapper"><div class="article" lang="en"><div class="abstract"> <h2 class="subheading">Abstract</h2> <p>Recent developments in econometric methods enable estimation and testing of general long memory processes, which include the general Gegenbauer process. This paper considers the error correction model for a vector general long memory process, which encompasses the vector autoregressive fractionally integrated moving average and general Gegenbauer processes. We modify the tests for unit roots and cointegration, based on the concept of heterogeneous autoregression. The Monte Carlo simulations show that the finite sample properties of the modified tests for unit roots are satisfactory, while the conventional tests suffer from size distortion. The experiments also indicate that the modified tests for cointegration improve the problem of finding too many cointegration relationships which arises for fractionally integrated series. Empirical results for interest rates series for the USA and Australia indicate that: (1) the modified unit root test detected unit roots for all series; (2) after differencing, all series favour the general Gegenbauer (GG) process; (3) the modified test for cointegration found only two cointegrating vectors; and (4) the zero interest rate policy in the USA had no effect on the cointegrating vectors for the two countries.</p> </div><div class="keywords mb-3">Keywords: <a href="/search?query=keywordValues%3A%28%22long%20memory%20processes%22%29%20AND%20journalKey%3A%28%22JTSE%22%29&documentVisibility=all&documentTypeFacet=article" class="ga_keyword">long memory processes</a>; <a href="/search?query=keywordValues%3A%28%22Gegenbauer%20process%22%29%20AND%20journalKey%3A%28%22JTSE%22%29&documentVisibility=all&documentTypeFacet=article" class="ga_keyword">Gegenbauer process</a>; <a href="/search?query=keywordValues%3A%28%22Dickey%E2%80%93Fuller%20Tests%22%29%20AND%20journalKey%3A%28%22JTSE%22%29&documentVisibility=all&documentTypeFacet=article" class="ga_keyword">Dickey–Fuller Tests</a>; <a href="/search?query=keywordValues%3A%28%22cointegration%22%29%20AND%20journalKey%3A%28%22JTSE%22%29&documentVisibility=all&documentTypeFacet=article" class="ga_keyword">cointegration</a>; <a href="/search?query=keywordValues%3A%28%22differencing%22%29%20AND%20journalKey%3A%28%22JTSE%22%29&documentVisibility=all&documentTypeFacet=article" class="ga_keyword">differencing</a>; <a href="/search?query=keywordValues%3A%28%22interest%20rates%22%29%20AND%20journalKey%3A%28%22JTSE%22%29&documentVisibility=all&documentTypeFacet=article" class="ga_keyword">interest rates</a></div><div class="keywords mb-3">JEL Classification: <a href="/search?query=keywordValues%3A%28%22C22%22%29%20AND%20journalKey%3A%28%22JTSE%22%29&documentVisibility=all&documentTypeFacet=article" class="ga_keyword">C22</a>; <a href="/search?query=keywordValues%3A%28%22C32%22%29%20AND%20journalKey%3A%28%22JTSE%22%29&documentVisibility=all&documentTypeFacet=article" class="ga_keyword">C32</a>; <a href="/search?query=keywordValues%3A%28%22C51%22%29%20AND%20journalKey%3A%28%22JTSE%22%29&documentVisibility=all&documentTypeFacet=article" class="ga_keyword">C51</a></div><div class="contrib-group"></div><div class="back"> <div class="acknowledgements" id="j_jtse-2018-0024_ack_001"> <h2 class="subheading">Acknowledgements</h2> <p>Initial draft of this article was written while the first author was a Visiting Scholar at the University of Sydney. The authors are most grateful to Yoshi Baba, Karen Lewis, the Editor and two anonymous reviewers for very helpful comments and suggestions. The first author acknowledges the financial support of the Japan Ministry of Education, Culture, Sports, Science and Technology, the Japan Society for the Promotion of Science (Grant Number: JSPS KAKENHI JP16K03603), the School of Mathematics and Statistics at The University of Sydney, the Zengin Foundation for Studies on Economics and Finance, and the Australian Academy of Science. The second author acknowledges the support from the Faculty of Economics at Soka University. The third author is most grateful for the financial support of the Australian Research Council, Ministry of Science and Technology (MOST), Taiwan, and the Japan Society for the Promotion of Science. The fourth author acknowledges the Australian Research Council.</p> </div> <span class="app-group"> <div class="app" id="j_jtse-2018-0024_app_001"> <h2 class="subheading">A Appendix</h2> <section id="j_jtse-2018-0024_s_014"> <h3 class="subheading">A.1 Identification and Estimation of Gegenbauer Frequencies</h3> <p>We discuss the identification of <em>k</em> and give a short explanation of semiparametric estimation of the location frequency parameters. First, we explain the semiparametric technique of <a href="#j_jtse-2018-0024_ref_020" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_020" data-bs-toggle="tooltip" title="Hidalgo, J., and P. Soulier. 2004. “Estimation of the Location and Exponent of the Spectral Singularity of a Long Memory Process.” Journal of Time Series Analysis 25: 55–81.10.1111/j.1467-9892.2004.00337.xSearch in Google Scholar">Hidalgo and Soulier (2004)</a> for estimating the location frequency parameters of <span class="inline-formula" id="j_jtse-2018-0024_ineq_126"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math></span></span> for the univariate GG process. We assume that <em>k</em> is known until we discuss the identification of parameters.</p> <p>Let <span class="inline-formula" id="j_jtse-2018-0024_ineq_127"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:msub><mml:mi>I</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi mathvariant="bold-italic">z</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mi>λ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></span></span> be the periodogram defined by: </p><div class="formula" id="j_jtse-2018-0024_eq_025"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:msub><mml:mi>I</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi mathvariant="bold-italic">z</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mi>λ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>T</mml:mi><mml:mrow><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:msup><mml:mfenced open="|" close="|"><mml:mrow><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>T</mml:mi></mml:munderover></mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">z</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msub><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>t</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mspace width="1em"></mml:mspace><mml:mo>−</mml:mo><mml:mi>π</mml:mi><mml:mo><</mml:mo><mml:mi>λ</mml:mi><mml:mo>≤</mml:mo><mml:mi>π</mml:mi><mml:mo>,</mml:mo></mml:math></span></div> <p>for <span class="inline-formula" id="j_jtse-2018-0024_ineq_128"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>z</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:math></span></span>. Define <span class="inline-formula" id="j_jtse-2018-0024_ineq_129"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>n</mml:mi><mml:mo stretchy="false">˜</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mo fence="false" stretchy="false">⌊</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo fence="false" stretchy="false">⌋</mml:mo></mml:math></span></span>, and let <em>λ</em><sub><em>j</em></sub> = 2<em>j</em><em>π</em>/<em>T</em><span class="inline-formula" id="j_jtse-2018-0024_ineq_130"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:mo>−</mml:mo><mml:mover><mml:mi>n</mml:mi><mml:mo stretchy="false">˜</mml:mo></mml:mover><mml:mo>≤</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:mover><mml:mi>n</mml:mi><mml:mo stretchy="false">˜</mml:mo></mml:mover><mml:mo stretchy="false">)</mml:mo></mml:math></span></span> be the Fourier frequencies.</p> <p>For purposes of introducing the approach of <a href="#j_jtse-2018-0024_ref_020" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_020" data-bs-toggle="tooltip" title="Hidalgo, J., and P. Soulier. 2004. “Estimation of the Location and Exponent of the Spectral Singularity of a Long Memory Process.” Journal of Time Series Analysis 25: 55–81.10.1111/j.1467-9892.2004.00337.xSearch in Google Scholar">Hidalgo and Soulier (2004)</a>, we consider a simple case of a univariate process which produces <em>I</em><sub><em>T</em></sub>(<em>λ</em>), under the assumptions <em>d</em> = 0, <em>ω</em><sub>1</sub>≠ 0, <em>ω</em><sub>2</sub>≠ 0, <span class="inline-formula" id="j_jtse-2018-0024_ineq_131"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>d</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>≥</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></span></span>, and <em>k</em> = 2. Then we can estimate <em>ω</em><sub>1</sub> consistently as: </p><div class="formula" id="j_jtse-2018-0024_eq_026"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:msub><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy="false">ˆ</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mi>T</mml:mi></mml:mfrac><mml:mtext>arg</mml:mtext><mml:munder><mml:mrow><mml:mtext>max</mml:mtext></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mi>I</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width="1em"></mml:mspace></mml:math></span></div> <p>where <span class="inline-formula" id="j_jtse-2018-0024_ineq_132"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>z</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>T</mml:mi><mml:mo form="prefix">exp</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>−</mml:mo><mml:msqrt><mml:mo>ln</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:msqrt><mml:mo stretchy="false">)</mml:mo></mml:math></span></span>, and <em>n</em> is an integer between 1 and <span class="inline-formula" id="j_jtse-2018-0024_ineq_133"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">⌊</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo fence="false" stretchy="false">⌋</mml:mo></mml:math></span></span>, satisfying at least: </p><div class="formula" id="j_jtse-2018-0024_eq_027"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:mfrac><mml:mn>1</mml:mn><mml:mi>n</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mi>n</mml:mi><mml:mi>T</mml:mi></mml:mfrac><mml:mo stretchy="false">→</mml:mo><mml:mn>0</mml:mn><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mtext>as </mml:mtext><mml:mi>T</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi mathvariant="normal">∞</mml:mi><mml:mo>.</mml:mo></mml:math></span></div> <p>After we estimate <em>ω</em><sub>1</sub>, it is possible to estimate consistently the second location parameter, <em>ω</em><sub>2</sub>, which has sufficient distance from the first location, as: </p><div class="formula" id="j_jtse-2018-0024_eq_028"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:msub><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy="false">ˆ</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mi>T</mml:mi></mml:mfrac><mml:mtext>arg</mml:mtext><mml:munder><mml:mrow><mml:munder><mml:mrow><mml:mtext>max</mml:mtext></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:munder></mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy="false">ˆ</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>≥</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mi>I</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>.</mml:mo></mml:math></span></div> <p>For general <em>k</em>, we can estimate <span class="inline-formula" id="j_jtse-2018-0024_ineq_134"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math></span></span> sequentially, by applying the above procedure.</p> <p> <a href="#j_jtse-2018-0024_ref_020" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_020" data-bs-toggle="tooltip" title="Hidalgo, J., and P. Soulier. 2004. “Estimation of the Location and Exponent of the Spectral Singularity of a Long Memory Process.” Journal of Time Series Analysis 25: 55–81.10.1111/j.1467-9892.2004.00337.xSearch in Google Scholar">Hidalgo and Soulier (2004)</a> modified the GPH estimator of <a href="#j_jtse-2018-0024_ref_015" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_015" data-bs-toggle="tooltip" title="Geweke, J., and S. Porter-Hudak. 1983. “The Estimation and Application of Long-Memory Time Series Models.” Journal of Time Series Analysis 4: 87–104.10.1111/j.1467-9892.1983.tb00371.xSearch in Google Scholar">Geweke and Porter-Hudak (1983)</a>, which was originally suggested to estimate the long memory parameter, <em>d</em>, using a log-periodogram regression, in order to estimate <em>d</em><sub><em>l</em></sub> at the Gegenbauer frequency <em>ω</em><sub><em>l</em></sub>. In order to identify the number of location frequencies, <em>k</em>, we follow the approach of <a href="#j_jtse-2018-0024_ref_020" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_020" data-bs-toggle="tooltip" title="Hidalgo, J., and P. Soulier. 2004. “Estimation of the Location and Exponent of the Spectral Singularity of a Long Memory Process.” Journal of Time Series Analysis 25: 55–81.10.1111/j.1467-9892.2004.00337.xSearch in Google Scholar">Hidalgo and Soulier (2004)</a>, based on their modified GPH estimator for <span class="inline-formula" id="j_jtse-2018-0024_ineq_135"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>d</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math></span></span>, which is defined by: </p><div class="formula" id="j_jtse-2018-0024_eq_029"><span class="label">(16)</span><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:msub><mml:mover><mml:mi>d</mml:mi><mml:mo stretchy="false">ˆ</mml:mo></mml:mover><mml:mi>l</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:munder><mml:mrow><mml:mrow><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mi>j</mml:mi><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:munder></mml:mrow></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:msub><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy="false">ˆ</mml:mo></mml:mover><mml:mi>l</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>≤</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mi>ξ</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo form="prefix">ln</mml:mo><mml:mfenced open="{" close="}"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy="false">ˆ</mml:mo></mml:mover><mml:mi>l</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced><mml:mo>,</mml:mo></mml:math></span></div> <p>where <span class="inline-formula" id="j_jtse-2018-0024_ineq_136"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ξ</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>s</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>ζ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mover><mml:mi>ζ</mml:mi><mml:mo>ˉ</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math></span></span>, <span class="inline-formula" id="j_jtse-2018-0024_ineq_137"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ζ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>λ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mo form="prefix">ln</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></span></span>, <span class="inline-formula" id="j_jtse-2018-0024_ineq_138"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mover><mml:mi>ζ</mml:mi><mml:mo>ˉ</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover></mml:mrow><mml:mi>ζ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math></span></span>, and <span class="inline-formula" id="j_jtse-2018-0024_ineq_139"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>s</mml:mi><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ζ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mover><mml:mi>ζ</mml:mi><mml:mo>ˉ</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></span></span>. <a href="#j_jtse-2018-0024_ref_020" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_020" data-bs-toggle="tooltip" title="Hidalgo, J., and P. Soulier. 2004. “Estimation of the Location and Exponent of the Spectral Singularity of a Long Memory Process.” Journal of Time Series Analysis 25: 55–81.10.1111/j.1467-9892.2004.00337.xSearch in Google Scholar">Hidalgo and Soulier (2004)</a> show that <span class="inline-formula" id="j_jtse-2018-0024_ineq_140"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mover><mml:mi>d</mml:mi><mml:mo stretchy="false">ˆ</mml:mo></mml:mover><mml:mi>l</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>l</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math></span></span> converges weakly to <em>N</em>(0, <em>π</em><sup>2</sup>/12), under the assumption of a Gaussian process. For the case <em>ω</em><sub><em>l</em></sub> = 0, <em>π</em>, <a href="#j_jtse-2018-0024_ref_020" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_020" data-bs-toggle="tooltip" title="Hidalgo, J., and P. Soulier. 2004. “Estimation of the Location and Exponent of the Spectral Singularity of a Long Memory Process.” Journal of Time Series Analysis 25: 55–81.10.1111/j.1467-9892.2004.00337.xSearch in Google Scholar">Hidalgo and Soulier (2004)</a> also show that the limiting distribution is <em>N</em> (0, <em>π</em><sup>2</sup>/6).</p> <p>The procedure of <a href="#j_jtse-2018-0024_ref_020" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_020" data-bs-toggle="tooltip" title="Hidalgo, J., and P. Soulier. 2004. “Estimation of the Location and Exponent of the Spectral Singularity of a Long Memory Process.” Journal of Time Series Analysis 25: 55–81.10.1111/j.1467-9892.2004.00337.xSearch in Google Scholar">Hidalgo and Soulier (2004)</a> consists of the following steps: (i) Find the largest periodogram ordinate; (ii) if the corresponding estimate of <em>d</em><sub><em>l</em></sub> is significant, add the respective Gegenbauer filter to the model, otherwise terminate the procedure; (iii) exclude the neighborhood of the last pole from the periodogram, and repeat the procedure from (i) onward. In the empirical analysis, we apply the method just described for identifying <em>k</em> and estimating <span class="inline-formula" id="j_jtse-2018-0024_ineq_141"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math></span></span> for the GG model.</p> </section> <section id="j_jtse-2018-0024_s_015"> <h3 class="subheading">A.2 Generation of GG Processes</h3> <p>We generate the GG process by the modified Durbin’s algorithm based on the theoretical covariance function for the whole sample, applying the discussion in <a href="#j_jtse-2018-0024_ref_013" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_013" data-bs-toggle="tooltip" title="Doornik J. A., and M. Ooms. 2003. “Computational Aspects of Maximum Likelihood Estimation of Autoregressive Fractionally Integrated Moving Average Models.” Computational Statistics & Data Analysis 42: 333–48.10.1016/S0167-9473(02)00212-8Search in Google Scholar">Doornik and Ooms (2003)</a>. In the following, we first explain the calculation of the coefficients of the MA(<span class="inline-formula" id="j_jtse-2018-0024_ineq_142"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">∞</mml:mi></mml:math></span></span>) representation of the general Gegenbauer process in eq. (<a href="#j_jtse-2018-0024_eq_004" class="link link-disp-formula" data-bs-target="j_jtse-2018-0024_eq_004">2</a>) in order to show the calculation of the autocovariance functions.</p> <p>Even for the simple Gegenbauer process with ARMA parameters, it is not easy to obtain explicit formulae for the coefficients for the MA(<span class="inline-formula" id="j_jtse-2018-0024_ineq_143"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">∞</mml:mi></mml:math></span></span>) representation, and the autocovariances, that are valid for all lags. Recently, McElroy and Holan (<a href="#j_jtse-2018-0024_ref_031" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_031" data-bs-toggle="tooltip" title="McElroy, T. S., and S. H. Holan. 2012. “On the Computation of Autocovariances for Generalized Gegenbauer Processes.” Statistica Sinica 22: 1661–87.10.5705/ss.2010.186Search in Google Scholar">2012</a>, <a href="#j_jtse-2018-0024_ref_032" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_032" data-bs-toggle="tooltip" title="McElroy, T. S., and S. H. Holan. 2016. “Computation of the Autocovariances for Time Series with Multiple Long-Range Persistencies.” Computational Statistics & Data Analysis 101: 44–56.10.1016/j.csda.2016.02.004Search in Google Scholar">2016</a>) developed a computationally efficient method for calculating these values. The spectral density of the general Gegenbauer process is given by (<a href="#j_jtse-2018-0024_eq_007" class="link link-disp-formula" data-bs-target="j_jtse-2018-0024_eq_007">5</a>). For convenience, we define <em>κ</em>(<em>z</em>) so that <span class="inline-formula" id="j_jtse-2018-0024_ineq_144"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>λ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mi>κ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></span></span>. Then, <em>κ</em>(<em>z</em>) takes the form <span class="inline-formula" id="j_jtse-2018-0024_ineq_145"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:munder><mml:mo>∏</mml:mo><mml:mi>l</mml:mi></mml:munder></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mi>ζ</mml:mi><mml:mi>l</mml:mi></mml:msub><mml:mi>z</mml:mi><mml:mrow><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mrow></mml:msup></mml:mrow></mml:math></span></span> for (possibly complex) reciprocal roots, <em>ζ</em><sub><em>l</em></sub>, of the moving average and autoregressive polynomials, where <em>p</em><sub><em>l</em></sub> is one if <em>l</em> corresponds to a moving average root, and minus one if <em>l</em> corresponds to an autoregressive root. We set <em>α</em> = max{<em>d</em><sub><em>l</em></sub>} for notational convenience.</p> <p>Define: </p><div class="formula" id="j_jtse-2018-0024_eq_030"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:mtable columnalign="left" rowspacing="4pt" columnspacing="1em"><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mrow><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mi>l</mml:mi></mml:munder></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mrow><mml:msubsup><mml:mi>ζ</mml:mi><mml:mi>l</mml:mi><mml:mi>j</mml:mi></mml:msubsup></mml:mrow><mml:mi>j</mml:mi></mml:mfrac></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mfrac><mml:mfenced open="{" close="}"><mml:mrow><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mrow><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>k</mml:mi></mml:munderover></mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mrow></mml:mrow><mml:mo form="prefix">cos</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mrow><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>ψ</mml:mi><mml:mo stretchy="false">˜</mml:mo></mml:mover></mml:mrow><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mfrac><mml:mrow><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>l</mml:mi></mml:munderover></mml:mrow><mml:mi>m</mml:mi><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>ψ</mml:mi><mml:mo stretchy="false">˜</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>−</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo><mml:mspace width="1em"></mml:mspace><mml:mspace width="1em"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>ψ</mml:mi><mml:mo stretchy="false">˜</mml:mo></mml:mover></mml:mrow><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mn>1.</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:math></span></div><a href="#j_jtse-2018-0024_ref_031" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_031" data-bs-toggle="tooltip" title="McElroy, T. S., and S. H. Holan. 2012. “On the Computation of Autocovariances for Generalized Gegenbauer Processes.” Statistica Sinica 22: 1661–87.10.5705/ss.2010.186Search in Google Scholar">McElroy and Holan (2012)</a> showed that the MA(<span class="inline-formula" id="j_jtse-2018-0024_ineq_146"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:mi mathvariant="normal">∞</mml:mi></mml:math></span></span>) representation of eq. (<a href="#j_jtse-2018-0024_eq_004" class="link link-disp-formula" data-bs-target="j_jtse-2018-0024_eq_004">2</a>) is given by: <div class="formula" id="j_jtse-2018-0024_eq_031"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:msub><mml:mi>y</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>μ</mml:mi><mml:mo>+</mml:mo><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant="normal">∞</mml:mi></mml:mrow></mml:munderover></mml:mrow><mml:msub><mml:mover><mml:mi>ψ</mml:mi><mml:mo stretchy="false">˜</mml:mo></mml:mover><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mi>η</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></span></div> <p>and the autocovariances of <em>h</em><sub><em>t</em></sub> for <em>l</em> ≥ 0 are given by: </p><div class="formula" id="j_jtse-2018-0024_eq_032"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:msub><mml:mi>γ</mml:mi><mml:mi>l</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>J</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover></mml:mrow><mml:msub><mml:mover><mml:mi>ψ</mml:mi><mml:mo stretchy="false">˜</mml:mo></mml:mover><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mover><mml:mi>ψ</mml:mi><mml:mo stretchy="false">˜</mml:mo></mml:mover><mml:mrow><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>J</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>l</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:math></span></div> <p>where </p><div class="formula" id="j_jtse-2018-0024_eq_033"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:msub><mml:mi>R</mml:mi><mml:mi>J</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>l</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mfenced open="{" close="}"><mml:mrow><mml:msup><mml:mi>J</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>α</mml:mi></mml:mrow></mml:msup><mml:mfrac><mml:mrow><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mi>α</mml:mi><mml:mo>;</mml:mo><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mi>α</mml:mi><mml:mo>;</mml:mo><mml:mo>−</mml:mo><mml:mi>l</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>J</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mi mathvariant="normal">Γ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mi>α</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>o</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo fence="false" stretchy="false">}</mml:mo><mml:mo>,</mml:mo></mml:math></span></div> <p>and <em>F</em>(<em>a</em>, <em>b</em>; <em>c</em>; <em>z</em>) is the hypergeometric function evaluated at <em>z</em>. In the above, <em>K</em><sub><em>l</em></sub> is a component depending on the structure of (<a href="#j_jtse-2018-0024_eq_007" class="link link-disp-formula" data-bs-target="j_jtse-2018-0024_eq_007">5</a>), and our GG processes have </p><div class="formula" id="j_jtse-2018-0024_eq_034"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:mtable columnalign="left" rowspacing="4pt" columnspacing="1em"><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo form="prefix">cos</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>l</mml:mi><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:msup><mml:mfenced open="[" close="]"><mml:mrow><mml:mn>2</mml:mn><mml:mfenced open="{" close="}"><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mo form="prefix">cos</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mfenced open="[" close="]"><mml:mrow><mml:mn>2</mml:mn><mml:mfenced open="{" close="}"><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mo form="prefix">cos</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mrow></mml:msup></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mspace width="thinmathspace"></mml:mspace><mml:mo>×</mml:mo><mml:mrow><mml:msup><mml:mfenced open="[" close="]"><mml:mrow><mml:mn>2</mml:mn><mml:mfenced open="{" close="}"><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mo form="prefix">cos</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced><mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mfenced open="[" close="]"><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mi>ϕ</mml:mi><mml:mo form="prefix">cos</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></span></div> <p>with <em>k</em> = 2, <em>d</em><sub>0</sub> = 0, <span class="inline-formula" id="j_jtse-2018-0024_ineq_147"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>d</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>≥</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></span></span>. Note that <span class="inline-formula" id="j_jtse-2018-0024_ineq_148"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>γ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>γ</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:math></span></span>. <a href="#j_jtse-2018-0024_ref_031" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_031" data-bs-toggle="tooltip" title="McElroy, T. S., and S. H. Holan. 2012. “On the Computation of Autocovariances for Generalized Gegenbauer Processes.” Statistica Sinica 22: 1661–87.10.5705/ss.2010.186Search in Google Scholar">McElroy and Holan (2012)</a> recommend using the cutoff value <em>J</em> ≥ 2000, and we set <em>J</em> = 20000 in this paper.</p> </section> <section id="j_jtse-2018-0024_s_016"> <h3 class="subheading">A.3 Estimation of Cointegrating Vectors</h3> <p>For the modified ECM in (<a href="#j_jtse-2018-0024_eq_012" class="link link-disp-formula" data-bs-target="j_jtse-2018-0024_eq_012">8</a>), we obtain the estimate of <em>α</em> as <span class="inline-formula" id="j_jtse-2018-0024_ineq_149"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>α</mml:mi><mml:mo stretchy="false">ˆ</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">v</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">v</mml:mi></mml:mrow><mml:mi>r</mml:mi></mml:msub><mml:mo stretchy="false">]</mml:mo></mml:math></span></span>, where <span class="inline-formula" id="j_jtse-2018-0024_ineq_150"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">v</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:math></span></span> is the eigenvector corresponding to the eigenvalue <em>λ</em><sub><em>i</em></sub> (<em>i</em> = 1, …, <em>r</em>), which satisfies the normalization <em>V</em><sup>'</sup><em>S</em><sub>11</sub><em>V</em> = <em>I</em>, with <span class="inline-formula" id="j_jtse-2018-0024_ineq_151"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>V</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">v</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">v</mml:mi></mml:mrow><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">]</mml:mo></mml:math></span></span>. Simultaneously, we obtain <span class="inline-formula" id="j_jtse-2018-0024_ineq_152"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>γ</mml:mi><mml:mo stretchy="false">ˆ</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>01</mml:mn></mml:mrow></mml:msub><mml:mover><mml:mi>α</mml:mi><mml:mo stretchy="false">ˆ</mml:mo></mml:mover></mml:math></span></span>. In the decomposition (<a href="#j_jtse-2018-0024_eq_013" class="link link-disp-formula" data-bs-target="j_jtse-2018-0024_eq_013">9</a>), we need to estimate <em>γ</em><sub>⊥</sub> and <em>α</em><sub>⊥</sub>. <a href="#j_jtse-2018-0024_ref_016" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_016" data-bs-toggle="tooltip" title="Gonzalo, J., and C. W. J. Granger. 1995. “Estimation of Common Long-Memory Components in Cointegrated System.” Journal of Business & Economic Statistics 13: 27–35.Search in Google Scholar">Gonzalo and Granger (1995)</a> proved that the ML estimator is obtained by the following procedure. First, solve the equation: </p><div class="formula" id="j_jtse-2018-0024_eq_035"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mi>λ</mml:mi><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>00</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>01</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mi>S</mml:mi><mml:mrow><mml:mn>11</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>10</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></span></div> <p>which yields eigenvalues <span class="inline-formula" id="j_jtse-2018-0024_ineq_153"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>λ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>≥</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>≥</mml:mo><mml:mo>⋯</mml:mo><mml:mo>≥</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math></span></span> and eigenvectors <span class="inline-formula" id="j_jtse-2018-0024_ineq_154"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">m</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">m</mml:mi></mml:mrow><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">]</mml:mo></mml:math></span></span>, normalized such that <em>M</em><sup>'</sup><em>S</em><sub>00</sub><em>M</em> = <em>I</em>. Using these eigenvalues and eigenvectors, we obtain the ML estimator, <span class="inline-formula" id="j_jtse-2018-0024_ineq_155"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mover><mml:mi>γ</mml:mi><mml:mo stretchy="false">ˆ</mml:mo></mml:mover><mml:mrow><mml:mi mathvariant="normal">⊥</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">m</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">m</mml:mi></mml:mrow><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">]</mml:mo></mml:math></span></span>, and <span class="inline-formula" id="j_jtse-2018-0024_ineq_156"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mover><mml:mi>α</mml:mi><mml:mo stretchy="false">ˆ</mml:mo></mml:mover><mml:mrow><mml:mi mathvariant="normal">⊥</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>10</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mover><mml:mi>γ</mml:mi><mml:mo stretchy="false">ˆ</mml:mo></mml:mover><mml:mrow><mml:mi mathvariant="normal">⊥</mml:mi></mml:mrow></mml:msub></mml:math></span></span>.</p> <p>Based on eq. (<a href="#j_jtse-2018-0024_eq_012" class="link link-disp-formula" data-bs-target="j_jtse-2018-0024_eq_012">8</a>), we can modify the test on the cointegrating vector as suggested by <a href="#j_jtse-2018-0024_ref_024" class="link link-bibr" data-bs-target="j_jtse-2018-0024_ref_024" data-bs-toggle="tooltip" title="Johansen, S. 1991. “Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models.” Econometrica 59: 1551–80.10.2307/2938278Search in Google Scholar">Johansen (1991)</a>. The null hypothesis of the test is: </p><div class="formula" id="j_jtse-2018-0024_eq_036"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>:</mml:mo><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mi>G</mml:mi><mml:mi>φ</mml:mi><mml:mo>,</mml:mo></mml:math></span></div> <p>where <em>G</em> is an <em>m</em> × <em>s</em> matrix, and <em>φ</em> is an <em>s</em> × <em>r</em> matrix (<em>r</em> ≤ <em>s</em> ≤ <em>m</em>). The alternative hypothesis is that <em>α</em> consists of <em>r</em> cointegrating vectors without any restrictions. We first solve: </p><div class="formula" id="j_jtse-2018-0024_eq_037"><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mi>λ</mml:mi><mml:msup><mml:mi>G</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mi>G</mml:mi><mml:mo>−</mml:mo><mml:msup><mml:mi>G</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>10</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mi>S</mml:mi><mml:mrow><mml:mn>00</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mn>01</mml:mn></mml:mrow></mml:msub><mml:mi>G</mml:mi><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></span></div> <p>to obtain <span class="inline-formula" id="j_jtse-2018-0024_ineq_157"><span class="alternatives"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>λ</mml:mi><mml:mrow><mml:mi>g</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>≥</mml:mo><mml:mo>⋯</mml:mo><mml:mo>≥</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mrow><mml:mi>g</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math></span></span>. Under the null hypothesis <em>α</em> = <em>G</em>φ, the test statistic: </p><div class="formula" id="j_jtse-2018-0024_eq_038"><span class="label">(17)</span><span class="alternatives"><mml:math xmlns:env="http://degruyter.com/resources/metadata" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:dgror="http://degruyter.com/resources/fetched-ror-id" xmlns:m="http://degruyter.com/resources/metadata" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:tei="http://www.tei-c.org/ns/1.0"><mml:msub><mml:mi>Q</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>T</mml:mi><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>r</mml:mi></mml:munderover></mml:mrow><mml:mo form="prefix">log</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mrow><mml:mi>g</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced><mml:mo>,</mml:mo></mml:math></span></div> <p>is expected to have an asymptotic <em>χ</em><sup>2</sup> distribution with degrees of freedom given by (<em>m</em> – <em>s</em>)<em>r</em>.</p> </section> </div> </span> <span class="ref-list" id="j_jtse-2018-0024_reflist_001"> <h2 class="subheading">References</h2> <p class="reference" id="j_jtse-2018-0024_ref_001"><span class="reference-mixed-citation">Andersson, M. 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"Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates" <i>Journal of Time Series Econometrics</i>, vol. 12, no. 1, 2020, pp. 20180024. <a href='https://doi.org/10.1515/jtse-2018-0024'>https://doi.org/10.1515/jtse-2018-0024</a></div> <div class="tab-pane fade " id="APA" role="tabpanel" aria-labelledby="APA-tab">Asai, M., Peiris, S., McAleer, M. & Allen, D. (2020). Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates. <i>Journal of Time Series Econometrics</i>, <i>12</i>(1), 20180024. <a href='https://doi.org/10.1515/jtse-2018-0024'>https://doi.org/10.1515/jtse-2018-0024</a></div> <div class="tab-pane fade " id="Harvard" role="tabpanel" aria-labelledby="Harvard-tab">Asai, M., Peiris, S., McAleer, M. and Allen, D. (2020) Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates. Journal of Time Series Econometrics, Vol. 12 (Issue 1), pp. 20180024. <a href='https://doi.org/10.1515/jtse-2018-0024'>https://doi.org/10.1515/jtse-2018-0024</a></div> <div class="tab-pane fade " id="Chicago" role="tabpanel" aria-labelledby="Chicago-tab">Asai, Manabu, Peiris, Shelton, McAleer, Michael and Allen, David E.. "Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates" <i>Journal of Time Series Econometrics</i> 12, no. 1 (2020): 20180024. <a href='https://doi.org/10.1515/jtse-2018-0024'>https://doi.org/10.1515/jtse-2018-0024</a></div> <div class="tab-pane fade " id="Vancouver" role="tabpanel" aria-labelledby="Vancouver-tab">Asai M, Peiris S, McAleer M, Allen D. 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Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates