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<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Publishing DTD v1.1d1 20130915//EN" "http://jats.nlm.nih.gov/publishing/1.1d1/JATS-journalpublishing1.dtd">
<article xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" article-type="research-article" xml:lang="en">
<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">JEF</journal-id>
<journal-title-group>
<journal-title>Journal of Economic and Financial Sciences</journal-title>
</journal-title-group>
<issn pub-type="ppub">1995-7076</issn>
<issn pub-type="epub">2312-2803</issn>
<publisher>
<publisher-name>AOSIS</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">JEF-19-1108</article-id>
<article-id pub-id-type="doi">10.4102/jef.v19i1.1108</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Original Research</subject>
</subj-group>
</article-categories>
<title-group>
<article-title>Dynamic linkages between systemic risk and real-estate investment trusts in major emerging markets</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author" corresp="yes">
<contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-3297-6238</contrib-id>
<name>
<surname>De Wet</surname>
<given-names>Milan C.</given-names>
</name>
<xref ref-type="aff" rid="AF0001">1</xref>
</contrib>
<aff id="AF0001"><label>1</label>Department of Accounting, College of Business and Economics, University of Johannesburg, Johannesburg, South Africa</aff>
</contrib-group>
<author-notes>
<corresp id="cor1"><bold>Corresponding author:</bold> Milan De Wet, <email xlink:href="miland@uj.ac.za">miland@uj.ac.za</email></corresp>
</author-notes>
<pub-date pub-type="epub"><day>12</day><month>05</month><year>2026</year></pub-date>
<pub-date pub-type="collection"><year>2026</year></pub-date>
<volume>19</volume>
<issue>1</issue>
<elocation-id>1108</elocation-id>
<history>
<date date-type="received"><day>19</day><month>11</month><year>2025</year></date>
<date date-type="accepted"><day>25</day><month>02</month><year>2026</year></date>
</history>
<permissions>
<copyright-statement>&#x00A9; 2026. The Author</copyright-statement>
<copyright-year>2026</copyright-year>
<license license-type="open-access" xlink:href="https://creativecommons.org/licenses/by/4.0/">
<license-p>Licensee: AOSIS. This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.</license-p>
</license>
</permissions>
<abstract>
<sec id="st1">
<title>Orientation</title>
<p>Hedging against price risk is central to asset management, especially during instability. The rise of real-estate investment trusts (REITs) has increased the use of property-related portfolios.</p>
</sec>
<sec id="st2">
<title>Research purpose</title>
<p>This study tests whether systemic risk spillovers occur in REIT markets across major emerging countries and whether these linkages strengthen under stress.</p>
</sec>
<sec id="st3">
<title>Motivation for the study</title>
<p>Although REITs matter in emerging-market portfolios, limited evidence shows how domestic systemic risk affects REIT volatility in normal and extreme conditions. This limits guidance on when REITs diversify portfolios and when hedging effectiveness weakens.</p>
</sec>
<sec id="st4">
<title>Research approach/design and method</title>
<p>The study uses three econometric models: DCC-GARCH to capture volatility co-movements, Diebold&#x2013;Yilmaz FEVD to measure volatility transmission and an Asymmetric GARCH-Copula to assess tail dependence during extreme episodes.</p>
</sec>
<sec id="st5">
<title>Main findings</title>
<p>In normal periods, volatility transmission from systemic risk proxies to REIT volatility is low, with FEVD shares ranging from 0.01&#x0025; to 2.48&#x0025; for China and Brazil. The strongest channels are South Africa&#x2019;s yield share (10.16&#x0025;) and India&#x2019;s volatility-index share (12.51&#x0025;). Under stress, dependence rises markedly: tail dependence reaches 0.580&#x2013;0.619 for South Africa and 0.421 for India, showing that diversification benefits weaken when systemic risk is elevated.</p>
</sec>
<sec id="st6">
<title>Practical/managerial implications</title>
<p>REIT hedging performance is market- and regime-dependent. Asset managers should apply conditional hedging and stress testing, with greater vigilance in South Africa and India.</p>
</sec>
<sec id="st7">
<title>Contribution/value-add</title>
<p>The study shows that normal-period spillover estimates can understate crisis-period dependence and provides a multi-model benchmark for monitoring REIT hedging effectiveness under systemic stress.</p>
</sec>
</abstract>
<kwd-group>
<kwd>hedging</kwd>
<kwd>non-linear modelling</kwd>
<kwd>portfolio management</kwd>
<kwd>REIT volatility</kwd>
<kwd>systemic risk</kwd>
<kwd>tail-risk spillovers</kwd>
</kwd-group>
<funding-group>
<funding-statement><bold>Funding information</bold> This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.</funding-statement>
</funding-group>
</article-meta>
</front>
<body>
<sec id="s0001">
<title>Introduction</title>
<p>Property markets play a key role in many local economies and are fundamental to financial stability in a country. The rise in real-estate investment trusts (REITs) has increased the reliance of asset managers on property-related assets as a hedging mechanism in the portfolio construction process.</p>
<p>Real-estate investment trusts are positioned at the intersection of financial markets and the real economy. Their liquidity makes them an attractive vehicle to obtain real-estate exposure, and fund managers have increasingly incorporated REITs into portfolio hedging and risk-management strategies. However, because REITs are traded assets, they are sensitive to shifts in funding conditions, discount rates, and broader risk sentiment. This fact is particularly relevant today, given the recurrence of market stress episodes in recent years, during which risk premiums widen and financial conditions tighten, potentially altering how REIT volatility co-moves with systemic risk measures. This situation makes it important to assess whether REITs remain reliable hedges when systemic conditions deteriorate.</p>
<p>Hedging against price-movement risk in an asset portfolio is an essential part of effective asset management and plays a key role in the preservation of wealth during unstable periods. Part of effective hedging is to ensure that a hedging instrument acts as a hedge during volatile periods and downturns. It is essential to understand how property-related assets react to systemic risk in a country. Research on the dependence of property-related assets on systemic risk is required to determine whether these assets act as sufficient hedging instruments during periods of volatility. If the volatility of property-related assets has a strong dependence on systemic risk, then the hedging ability of these assets is limited. On the other hand, limited volatility dependence will justify the inclusion of these assets as a hedging mechanism in portfolios.</p>
<p>Complicating the issue is the growing evidence showing that the link between real-estate assets and systemic risk is not static. Evidence shows that the beta of real-estate assets shifts across regimes and crises (Voicu &#x0026; Seiler <xref ref-type="bibr" rid="CIT0036">2013</xref>). It also shows that volatility connectedness increases during periods of extreme global uncertainty and that negative shocks elevate volatility disproportionately (Abdullah et al. <xref ref-type="bibr" rid="CIT0001">2023</xref>). This finding suggests that property assets provide a reasonable hedge during normal periods, but their hedging ability might deteriorate during abnormal periods of uncertainty and volatility. There is not a single linear relationship between systemic risk and real-estate-related assets; therefore, studies should employ non-linear modelling techniques. Yet most of the literature in this field makes use of linear modelling techniques, see for example Zhu and Lizieri (<xref ref-type="bibr" rid="CIT0040">2024</xref>), Yong (<xref ref-type="bibr" rid="CIT0037">2015</xref>), Li (<xref ref-type="bibr" rid="CIT0027">2012</xref>), and Adams, F&#x00FC;ss and Schindler (<xref ref-type="bibr" rid="CIT0002">2015</xref>) and Gyourko and Nelling (<xref ref-type="bibr" rid="CIT0018">1996</xref>). Many studies in this field focus on REITs&#x2019; returns and not volatility. Yet, measuring volatility dependence typically provides a more risk-oriented and robust measure of an asset&#x2019;s ability to serve as a hedge, rather than just working with returns (Hsu, Tseng &#x0026; Wang <xref ref-type="bibr" rid="CIT0021">2008</xref>). Hsu et al. (<xref ref-type="bibr" rid="CIT0021">2008</xref>) provide evidence that volatility modelling outperforms traditional models in hedging. It is essential to research the non-linear volatility dependence of real-estate-related assets on systemic risk.</p>
<p>This focus is especially relevant in emerging markets in which systemic shocks can be transmitted rapidly through capital flows, exchange-rate movements, and domestic funding conditions (Bwowa, Mouton &#x0026; De Wet <xref ref-type="bibr" rid="CIT0008">2024</xref>; Zhang et al. <xref ref-type="bibr" rid="CIT0039">2025</xref>). In these economies, property-linked assets are a key channel because real estate is typically credit- and rate-sensitive; therefore, shifts in funding costs and risk premia associated with systemic risk can translate into volatility in listed property exposures (De Wet &#x0026; Botha <xref ref-type="bibr" rid="CIT0011">2019</xref>).</p>
<p>Furthermore, the listed REIT market in major emerging markets has expanded significantly. In South Africa, listed REIT market capitalisation increased from roughly South Africa Rands (ZAR) 25 billion in 2004 (circa 2004) to above ZAR200bn in 2024&#x2013;2025, indicating a substantially larger and more liquid investable universe (SA REIT Association <xref ref-type="bibr" rid="CIT0034">2025</xref>). In India and China, listed REIT markets are newer but have scaled rapidly: India&#x2019;s REIT market capitalisation increased from Indian Rupees (INR) 264bn in 2020 to INR1.6 trillion by 2025 (Jones Lang LaSalle [JLL] <xref ref-type="bibr" rid="CIT0023">2025</xref>), while China&#x2019;s public REIT market expanded from Renminbi (RMB) 31.4bn in 2021 to a total market value exceeding RMB200bn by June 2025 (Shanghai Stock Exchange <xref ref-type="bibr" rid="CIT0035">2025</xref>). These developments support the relevance of examining systemic risk dependence on property-linked assets as these markets deepen and mature. Given the growth of listed REIT markets in several BRICS (Brazil, Russia, India, China, and South Africa) countries, REITs are increasingly considered as liquid instruments for real-estate exposure and potential portfolio hedging, making it necessary to test whether they dampen or amplify systemic risk.</p>
<p>Despite the growth of REIT markets in BRICS economies, it remains unclear which domestic systemic-risk measures transmit into REIT volatility, and whether those relationships are linear through normal conditions and during periods of stress. The main aim of this study is to determine whether there are systemic risk spillovers to the REITs market in major emerging countries. Specifically, the study will focus on BRICS countries, except for Russia, because Russia does not have an established REITs market.</p>
<p>Systemic risk is the economy-wide risk that disruptions in financial markets or institutions trigger widespread instability and spillovers across many assets and sectors, rather than remaining confined to a single firm or market (Voicu &#x0026; Seiler <xref ref-type="bibr" rid="CIT0036">2013</xref>). While systemic risk cannot be fully diversified away, assets differ in how strongly they are exposed to it. The relevant question for investors is therefore not whether systemic risk can be eliminated, but whether assets such as REITs exhibit lower sensitivity to systemic risk or maintain more stable dependence under stress. If REIT volatility responds weakly to systemic risk measures, or if dependence does not strengthen in stress regimes, REITs can serve as a partial stabiliser that dampens portfolio vulnerability during systemic shocks. Conversely, if REITs amplify systemic risk through stronger volatility spillovers or tail dependence, their hedging usefulness is reduced.</p>
<p>The novelty of this study lies in providing a BRICS-wide, volatility-focused assessment of how domestic systemic risk transmits into REIT markets under both normal and extreme conditions, using one consistent empirical design. Existing work is largely focused on developed REIT markets or single emerging-market case studies, and it typically relies on a single modelling framework that cannot jointly capture time-varying co-movement, directional volatility transmission, and tail-risk amplification. By combining Dynamic conditional correlation&#x2013;Generalised autoregressive conditional heteroscedasticity (DCC-GARCH), Diebold&#x2013;Yilmaz vector autoregressive (DY-VAR) variance decompositions, and an asymmetric copula approach, the study separates these channels clearly and produces directly comparable evidence across REITs in BRICS markets. This study will produce decision-relevant results for portfolio construction and risk management.</p>
<p>The core debate in the REIT&#x2013;systemic risk literature is whether REITs function as reliable hedging and/or diversifying instruments against systemic risk shocks, or whether their risk characteristics deteriorate and spillovers intensify when markets enter stress regimes. The debate is around whether the REIT&#x2013;systemic risk relationship is stable, and whether the hedging benefit observed in normal conditions can weaken or reverse when dependence becomes stronger and more asymmetric during periods of heightened uncertainty. To evaluate this debate in the BRICS context, the study tests for time-varying co-movements, directional volatility spillovers, and tail-risk amplification, using a complementary multi-model framework. Research remains limited to the volatility dependence between systemic risk measures and property-related assets, such as REITs, in the BRICS context.</p>
<p>To address this issue, the study employs the DCC-GARCH model, the DY-VAR model, and an Asymmetric GARCH-Copula model. Each of these three models provides a different perspective on the spillovers from systemic risk to REITs&#x2019; volatility and provides a holistic analysis of this relationship. The DCC-GARCH model shows how the volatility of REITs co-moves with systemic risk and how this relationship evolves with regime shifts. The DY-VAR shows how much of the REIT volatility is explained by shocks in systemic risk. Lastly, the Asymmetric GARCH-Copula model focuses on extreme tail risks and indicates how systemic risk spills over to REIT volatility during extreme periods. To the best of my knowledge, this is the paper that considers systemic risk spillovers to REIT volatility in such a comprehensive way. This paper will provide asset managers with a comprehensive understanding of how REITs in BRICS countries react to systemic risk in various market conditions. This research, in turn, will show the ability as well as the limitations of REITs as a hedging instrument, and thereby provide the necessary information required to effectively use REITs as a hedging tool.</p>
</sec>
<sec id="s0002">
<title>Literature review</title>
<p>The paradigm of empirical property market research has shifted considerably over the years. Initial research predominantly focused on the link between property prices and traditional economic indicators like Gross Domestic Product (GDP) growth, interest rates, the unemployment rate, and income levels (Bj&#x00F8;rnland &#x0026; Jacobsen <xref ref-type="bibr" rid="CIT0006">2010</xref>). Even though these economic fundamentals are important to the property market, they have limitations in explaining the high volatility of REIT markets (Alkay, Watkins &#x0026; Keskin <xref ref-type="bibr" rid="CIT0004">2018</xref>). Recently, the growing prominence of REITs as a hedging option in portfolio construction has attracted significant research interest related to the diversification potential of REIT markets (Bossman et al. <xref ref-type="bibr" rid="CIT0007">2022</xref>; Motegi &#x0026; Iitsuka <xref ref-type="bibr" rid="CIT0031">2023</xref>). A central debate in this literature is whether REITs maintain diversification and hedging characteristics when financial conditions deteriorate, or whether interdependence and spillovers intensify during stress and reduce those benefits. Hence, an increasing number of researchers focus on determining the interdependence between REIT markets and other markets such as stock markets, the commodity market, and the bond market, see for example, Ito (<xref ref-type="bibr" rid="CIT0022">2025</xref>) and Odusami (<xref ref-type="bibr" rid="CIT0033">2024</xref>).</p>
<p>The results from these studies are mixed, but several researchers provide evidence of non-linear relationships between REITs and other markets; for example, Ito (<xref ref-type="bibr" rid="CIT0022">2025</xref>), Alam et al. (<xref ref-type="bibr" rid="CIT0003">2023</xref>), and Mensi et al. (<xref ref-type="bibr" rid="CIT0029">2023</xref>) found that connectedness between REITs and traditional assets increases during periods of uncertainty. Ito (<xref ref-type="bibr" rid="CIT0022">2025</xref>) and Alam et al. (<xref ref-type="bibr" rid="CIT0003">2023</xref>) show that during calm periods, the relationship between REITs and traditional markets is broadly insignificant, yet during periods of uncertainty, the relationship between REITs and traditional markets becomes positive and statistically significant. Evidence from Armah and Amewu (<xref ref-type="bibr" rid="CIT0005">2024</xref>) shows that linkages between REITs and financial stress are asymmetric across market states, with stronger positive dependence during lower-tail conditions. These findings indicate that a linear estimate of linkages between REITs and other markets could overstate the diversification potential of REITs during extreme periods.</p>
<p>Another strand of research that is particularly relevant to understanding REITs as a potential diversifying asset focuses on systemic risk and volatility spillovers between REITs and other markets. The most recent literature related to this line of research includes work by Hadad, Le and Luong (<xref ref-type="bibr" rid="CIT0019">2024</xref>), Katyoka, Riberio and Ling (<xref ref-type="bibr" rid="CIT0024">2024</xref>) and Danila (<xref ref-type="bibr" rid="CIT0010">2024</xref>). Findings by these researchers highlight the complex and conditional nature of systemic risk spillover dynamics to REITs as well as volatility spillovers from other markets. Findings show that volatility spillovers between REITs and other markets are modest during normal conditions, but strong volatility spillovers exist during extreme market conditions and international uncertainty (Hadad et al. <xref ref-type="bibr" rid="CIT0019">2024</xref>; Katyoka et al. <xref ref-type="bibr" rid="CIT0024">2024</xref>). Interestingly, Hadad et al. (<xref ref-type="bibr" rid="CIT0019">2024</xref>) as well as Le and Luong (<xref ref-type="bibr" rid="CIT0019">2024</xref>) found that REITs are net receivers of volatility spillovers during normal periods but become transmitters during extreme periods. This phenomenon reduces REITs&#x2019; diversification characteristics when markets are extremely volatile. These findings support the argument that the volatility spillover dynamics form systemic risk to REITs, and volatility spillovers from other assets to REITs should not be considered a linear function but should be modelled with non-linear models. Recent work using copula-based and related tail-risk frameworks similarly highlights that diversification benefits can weaken when lower-tail dependence increases during crisis or high-uncertainty periods (Ito <xref ref-type="bibr" rid="CIT0022">2025</xref>; Odusami <xref ref-type="bibr" rid="CIT0033">2024</xref>).</p>
<p>Most of the research in this field focuses on REITs in developed economies. Where developing markets are considered, the evidence is more limited and usually market-specific. For example, Lekhuleni and Ndlovu (<xref ref-type="bibr" rid="CIT0026">2023</xref>) show that South African housing prices are cointegrated with GDP and inflation but remain vulnerable in the short run to shocks from mortgage rates, household debt, and foreign investment. Ndamanomhata (<xref ref-type="bibr" rid="CIT0032">2023</xref>) reaches a similar conclusion from a market-risk perspective, documenting that South African REITs display time-varying spillovers with equities, bonds, Foreign Exchange (FX), and commodities, and that they can switch between being net transmitters and net receivers depending on the regime. Evidence outside South Africa points to comparable state-dependence. Liu et al. (<xref ref-type="bibr" rid="CIT0028">2023</xref>), for instance, found that Hong Kong REITs are strongly interconnected with Chinese markets and global volatility (Volatility Index [VIX]), particularly during coronavirus disease 2019 (COVID-19), which supports the view that international shocks can materially shape REIT&#x2013;risk dynamics in developing-market settings. Related work also highlights that developing-market REIT volatility can respond asymmetrically to shocks and exhibit strong clustering (Danila <xref ref-type="bibr" rid="CIT0010">2024</xref>). Broader BRICS evidence on volatility transmission and connectedness reinforces this point, suggesting that these markets are exposed to time-varying contagion and that dependence is better treated as regime-dependent rather than constant (Zhang et al. <xref ref-type="bibr" rid="CIT0039">2025</xref>).</p>
<p>Although several studies have investigated REIT volatility and risk spillovers in developing markets, the existing literature remains fragmented with narrow methodological approaches. Existing research typically utilises a single econometric framework, such as GARCH for volatility clustering (Fateye, Ajayi &#x0026; Ajayi <xref ref-type="bibr" rid="CIT0017">2022</xref>), Vector Error Correction Model (VECM) for macro fundamentals (Lekhuleni &#x0026; Ndlovu <xref ref-type="bibr" rid="CIT0026">2023</xref>), or spillover indices (Danila <xref ref-type="bibr" rid="CIT0010">2024</xref>; Liu et al. <xref ref-type="bibr" rid="CIT0028">2023</xref>; Ndamanomhata <xref ref-type="bibr" rid="CIT0032">2023</xref>). These studies provide valuable insights, but they are limited in scope because a single model cannot concurrently capture directional volatility transmission, dynamic correlations, and the role of tail risks. In addition, the broader BRICS spillover literature tends to focus on equities, exchange rates, commodities, and policy uncertainty, rather than on REIT markets and the domestic systemic channels that are most relevant to property-linked risk, such as volatility indices and sovereign yields. As a result, the literature provides limited evidence that jointly measures time-varying dependence, directional spillover strength, and extreme tail co-movement between domestic systemic risk measures and REIT volatility across BRICS countries within a single, comparable empirical framework.</p>
<p>Given the complexities of the spillover dynamics between systemic risk and volatility to REITs, a study that comprehensively and consistently analyses these spillovers from multiple angles, across several key developing countries, is required. This study aims to contribute towards filling this research gap by analysing systemic risk spillovers to the REIT market in BRICS countries by means of employing multiple asymmetric models. This approach supports the use of a multi-model design to separate dynamic co-movements, directional spillovers, and tail behaviour within a BRICS setting.</p>
</sec>
<sec id="s0003">
<title>Research design</title>
<sec id="s20004">
<title>Methodology</title>
<p>The study used daily data with a time horizon from January 2016 to June 2025. A description of the variables, symbols, measurement, and data resources is provided in <xref ref-type="table" rid="T0001">Table 1</xref>. The data that support the findings of this study are available in the public domain. Real-estate investment trust index and stock market volatility data were obtained from Reuters Datastream (<ext-link ext-link-type="uri" xlink:href="https://www.reuters.com/markets/global-market-data/">https://www.reuters.com/markets/global-market-data/</ext-link>), while bond yield data were obtained from the Organisation for Economic Co-operation and Development (OECD) Data Repository (<ext-link ext-link-type="uri" xlink:href="https://data.oecd.org">https://data.oecd.org</ext-link>).</p>
<table-wrap id="T0001">
<label>TABLE 1</label>
<caption><p>Variables, symbols, measurement, and data resources.</p></caption>
<table frame="hsides" rules="groups">
<thead>
<tr>
<th valign="top" align="left">Variable</th>
<th valign="top" align="left">Symbol</th>
<th valign="top" align="left">Measurement and/or construction</th>
<th valign="top" align="left">Unit</th>
<th valign="top" align="left">Data resource</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left">REIT market (index level)</td>
<td align="left"><inline-formula id="I1"><alternatives><mml:math display="inline" id="IM1"><mml:mrow><mml:msubsup><mml:mi>P</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>R</mml:mi><mml:mi>E</mml:mi><mml:mi>I</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i001.tif"/></alternatives></inline-formula></td>
<td align="left"><italic>REIT index level</italic></td>
<td align="left">Index points</td>
<td align="left">Refinitiv Datastream (Reuters)</td>
</tr>
<tr>
<td align="left">REIT returns (input to GARCH/DCC)</td>
<td align="left"><inline-formula id="I2"><alternatives><mml:math display="inline" id="IM2"><mml:mrow><mml:msubsup><mml:mi>r</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>R</mml:mi><mml:mi>E</mml:mi><mml:mi>I</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i002.tif"/></alternatives></inline-formula></td>
<td align="left"><italic>r<sub>t</sub></italic> = 100 &#x00B7; <italic>&#x0394;</italic> In (<italic>P<sub>t</sub></italic>)</td>
<td align="left">&#x0025;</td>
<td align="left">Refinitiv Datastream (Reuters)</td>
</tr>
<tr>
<td align="left">Stock market volatility index (systemic risk proxy)</td>
<td align="left"><italic>VOL<sub>i, t</sub></italic></td>
<td align="left"><italic>Volatility index level</italic></td>
<td align="left">Index points</td>
<td align="left">Refinitiv Datastream (Reuters)</td>
</tr>
<tr>
<td align="left">Volatility index returns and/or changes (optional input)</td>
<td align="left"><inline-formula id="I3"><alternatives><mml:math display="inline" id="IM3"><mml:mrow><mml:msubsup><mml:mi>r</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>V</mml:mi><mml:mi>O</mml:mi><mml:mi>L</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i003.tif"/></alternatives></inline-formula></td>
<td align="left"><italic>r<sub>t</sub></italic> = 100 &#x00B7; <italic>&#x0394;</italic> In (<italic>VOL<sub>t</sub></italic>)</td>
<td align="left">&#x0025; (or points)</td>
<td align="left">Refinitiv Datastream (Reuters)</td>
</tr>
<tr>
<td align="left">10-year sovereign bond yield (systemic risk proxy)</td>
<td align="left"><italic>Y</italic> 10<sub><italic>i, t</italic></sub></td>
<td align="left">10 &#x2013; <italic>year government yield</italic></td>
<td align="left">&#x0025; yield</td>
<td align="left">OECD Data Repository</td>
</tr>
<tr>
<td align="left">Conditional volatility: REIT (GARCH output)</td>
<td align="left"><inline-formula id="I4"><alternatives><mml:math display="inline" id="IM4"><mml:mrow><mml:msubsup><mml:mi>&#x03C3;</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>R</mml:mi><mml:mi>E</mml:mi><mml:mi>I</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i004.tif"/></alternatives></inline-formula></td>
<td align="left"><italic>GARCH SD</italic> (<italic>REIT</italic>)</td>
<td align="left">&#x0025;</td>
<td align="left">Model-implied (constructed)</td>
</tr>
<tr>
<td align="left">Conditional volatility: systemic proxy (GARCH output)</td>
<td align="left"><inline-formula id="I5"><alternatives><mml:math display="inline" id="IM5"><mml:mrow><mml:msubsup><mml:mi>&#x03C3;</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>S</mml:mi><mml:mi>Y</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i005.tif"/></alternatives></inline-formula></td>
<td align="left"><italic>GARCH SD</italic> (<italic>SYS</italic>)</td>
<td align="left">&#x0025; (or pp)</td>
<td align="left">Model-implied (constructed)</td>
</tr>
<tr>
<td align="left">Time-varying correlation (DCC output)</td>
<td align="left"><italic>&#x03C1;<sub>i, t</sub></italic></td>
<td align="left"><italic>DCC corr</italic></td>
<td align="left">Correlation</td>
<td align="left">Model-implied (constructed)</td>
</tr>
<tr>
<td align="left">Spillover and/or connectedness measure (DY output)</td>
<td align="left"><italic>C<sub>t</sub></italic></td>
<td align="left"><italic>DY connectedness</italic></td>
<td align="left">&#x0025;</td>
<td align="left">Model-implied (constructed)</td>
</tr>
<tr>
<td align="left">Tail dependence (Copula output)</td>
<td align="left"><italic>&#x03BB;<sub>L</sub>, &#x03BB;<sub>U</sub></italic></td>
<td align="left"><italic>Copula tail dep</italic>.</td>
<td align="left">Tail dependence</td>
<td align="left">Model-implied (constructed)</td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<fn><p>REIT, real-estate investment trust; OECD, Organisation for Economic Co-operation and Development; DCC, dynamic conditional correlation; GARCH, generalised autoregressive conditional heteroscedasticity; DY, Diebold&#x2013;Yilmaz; dep., dependence.</p></fn>
</table-wrap-foot>
</table-wrap>
<p>In this study, the BRICS countries are chosen as a sample of major emerging markets, with the exception of Russia, because Russia does not have an established REITs market. In this study, two variables were used to proxy and measure systemic risk: the stock market volatility index of each country, as well as the 10-year sovereign bond yield for each country. These two measures are widely used in literature to proxy systemic risk, see, for example, Kumar, Yadav and Zymek (<xref ref-type="bibr" rid="CIT0025">2017</xref>); and Hansen and Timmermann (<xref ref-type="bibr" rid="CIT0020">2024</xref>). As outlined in the introduction of the study, three models were employed to analyse the systemic risk spillovers and dependencies to REITs in each country, namely, a DCC-GARC, DY-VAR, and an asymmetric GARCH-Copula model.</p>
</sec>
<sec id="s20005">
<title>Model specifications</title>
<p>This study follows a three-stage empirical strategy to measure: (1) time-varying volatility and correlation (DCC-GARCH), (2) shock transmission and/or connectedness (DY-VAR spillovers), and (3) distributional dependence (including tails) (copula).</p>
<p>Stage 1 (DCC-GARCH) produces two key outputs: (1) series-specific conditional volatility <italic>&#x03C3;</italic><sub>{<italic>i, t</italic>}</sub>, and (2) conditional correlation matrices <italic>R<sub>t</sub></italic> (or pairwise correlations), describing time-varying co-movement.</p>
<p>Stage 2 (Diebold&#x2013;Yilmaz) uses a volatility proxy (e.g. REIT conditional volatility from Stage 1 and additional risk variables) in a vector autoregressive (VAR) and decomposes forecast-error variance into total, directional, and net spillovers.</p>
<p>Stage 3 (Copula) models the joint dependence structure between key variables (e.g. REIT&#x2013;risk vs a systemic factor), using parametric marginals and a copula to assess dependence beyond linear correlation, including tail dependence.</p>
<p>To improve readability, full derivations and extended forms (full log-likelihood, asymmetric dynamic conditional correlation (DCC) variants, Forecast Error Variance Decomposition (FEVD) algebra, and copula density expansion) are provided in <xref ref-type="app" rid="app001">Appendix 1</xref>, with cross-references in the relevant subsections.</p>
</sec>
<sec id="s20006">
<title>Dynamic conditional correlation&#x2013;generalised autoregressive conditional heteroscedasticity model specification (Stage 1)</title>
<sec id="s30007">
<title>Purpose and outputs</title>
<p>The DCC-GARCH model (Engle <xref ref-type="bibr" rid="CIT0015">2002</xref>) is used to estimate time-varying volatilities and time-varying correlations between the REIT series and systemic risk series. This stage outputs conditional standard deviations <italic>&#x03C3;</italic>{<italic>i, t</italic>} for each series and the conditional correlation matrix R_t, which summarises time-varying co-movement. There are two sets of hypotheses for this model. The first set relates to whether the conditional correlation is time-varying. The null hypothesis is that the conditional correlation between REIT volatility and the domestic systemic-risk proxy is constant over time (i.e. not time-varying). The alternative hypothesis is that the conditional correlation between REIT volatility and the domestic systemic-risk proxy is time-varying.</p>
<p>The second set relates to whether correlation dynamics are asymmetric in response to negative shocks (asymmetric dynamic conditional correlation [ADCC] extension). The null hypothesis is that the conditional correlation responds symmetrically to positive and negative shocks of similar magnitude. The alternative hypothesis is that the conditional correlation responds asymmetrically, such that negative shocks generate a stronger increase in conditional correlation than positive shocks do.</p>
</sec>
<sec id="s30008">
<title>Mean equation (univariate)</title>
<p>Let <italic>y</italic>{<italic>i, t</italic>} denote the return of series <italic>i</italic> at time <italic>t</italic>, where <italic>i</italic> = 1, &#x2026;, <italic>N</italic>. The conditional mean is specified as <xref ref-type="disp-formula" rid="FD1">Equation 1</xref>:</p>
<disp-formula id="FD1"><alternatives><mml:math display="block" id="M1"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>&#x03BC;</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>&#x03B5;</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>&#x03B5;</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>&#x03C3;</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo>&#x223C;</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e001.tif"/></alternatives><label>[Eqn 1]</label></disp-formula>
<p>Where;</p>
<list list-type="bullet">
<list-item><p><italic>y</italic><sub>{<italic>i, t</italic>}</sub>: return of series i at time t (e.g. REIT or a systemic risk factor)</p></list-item>
<list-item><p><italic>&#x03BC;<sub>i</sub></italic>: constant mean return of series i</p></list-item>
<list-item><p><italic>&#x03B5;</italic><sub>{<italic>i, t</italic>}</sub>: innovation (residual)</p></list-item>
<list-item><p><italic>&#x03C3;</italic><sub>{<italic>i, t</italic>}</sub>: conditional standard deviation</p></list-item>
<list-item><p><italic>z</italic><sub>{<italic>i, t</italic>}</sub>: standardised residual (i.i.d. with mean 0 and variance 1)</p></list-item>
</list>
<p>(If Student-t innovations are used, they are stated explicitly; the alternative distribution form is provided in <xref ref-type="sec" rid="s20034">Appendix 1-A1</xref>)</p>
</sec>
<sec id="s30009">
<title>Variance equation (GARCH[1,1])</title>
<p>Each series&#x2019; conditional variance follows a GARCH(1,1) (<xref ref-type="disp-formula" rid="FD2">Equation 2</xref>):</p>
<disp-formula id="FD2"><alternatives><mml:math display="block" id="M2"><mml:mrow><mml:msubsup><mml:mi>&#x03C3;</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mi>&#x03C9;</mml:mi><mml:mtext>i</mml:mtext></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mtext>i</mml:mtext></mml:msub><mml:msubsup><mml:mi>&#x03B5;</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mi>&#x03B2;</mml:mi><mml:mtext>i</mml:mtext></mml:msub><mml:msubsup><mml:mi>&#x03C3;</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e002.tif"/></alternatives><label>[Eqn 2]</label></disp-formula>
<p>with constraints <italic>&#x03C9;<sub>i</sub></italic> &#x003E; 0, <italic>&#x03B1;<sub>i</sub></italic> &#x2265; 0, <italic>&#x03B2;<sub>i</sub></italic> &#x2265; 0. Covariance-stationarity typically requires <italic>&#x03B1;<sub>i</sub></italic> + <italic>&#x03B2;<sub>i</sub></italic> &#x003C; 1.</p>
</sec>
<sec id="s30010">
<title>Correlation dynamics (dynamic conditional correlation)</title>
<p>Stack the standardised residuals into the <italic>N</italic> &#x00D7; 1 vector <italic>z<sub>t</sub></italic> = (<italic>z</italic><sub>{1, <italic>t</italic>}</sub>, &#x22EF;, <italic>z</italic><sub>{<italic>N, t</italic>}</sub>)<sup>T</sup>. The DCC recursion (<xref ref-type="disp-formula" rid="FD3">Equation 3</xref>) is:</p>
<disp-formula id="FD3"><alternatives><mml:math display="block" id="M3"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>a</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>&#x2009;</mml:mtext><mml:mover accent="true"><mml:mi>Q</mml:mi><mml:mo>&#x00AF;</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mi>a</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x007D;</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mrow><mml:msup><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mtext>T</mml:mtext></mml:msup></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>b</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e003.tif"/></alternatives><label>[Eqn 3]</label></disp-formula>
<p>Where;</p>
<list list-type="bullet">
<list-item><p><italic>Q<sub>t</sub></italic>: time-varying covariance matrix of standardised residuals</p></list-item>
<list-item><p><inline-formula id="I6"><alternatives><mml:math display="inline" id="IM6"><mml:mover accent="true"><mml:mi>Q</mml:mi><mml:mo>&#x00AF;</mml:mo></mml:mover></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i006.tif"/></alternatives></inline-formula>: unconditional covariance matrix of <italic>z_t</italic></p></list-item>
<list-item><p><italic>a</italic> &#x2265; 0: correlation ARCH parameter (reaction to new shocks)</p></list-item>
<list-item><p><italic>b</italic> &#x2265; 0: correlation GARCH parameter (persistence)</p></list-item>
<list-item><p>A typical stability condition is <italic>a</italic> + <italic>b</italic> &#x003C; 1.</p></list-item>
</list>
<p>Optional asymmetry (ADCC): If stronger correlation responses to negative shocks are modelled, the asymmetric term is reported in <xref ref-type="sec" rid="s20035">Appendix 1-A2</xref> (see for the ADCC specification).</p>
</sec>
<sec id="s30011">
<title>Convert Q_t to a correlation matrix</title>
<p>The conditional correlation matrix R_t is obtained by rescaling Q_t (<xref ref-type="disp-formula" rid="FD4">Equation 4</xref>):</p>
<disp-formula id="FD4"><alternatives><mml:math display="block" id="M4"><mml:mrow><mml:mi>R</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mi>d</mml:mi><mml:mi>i</mml:mi><mml:mi>a</mml:mi><mml:mi>g</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mi>Q</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x005E;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mtext>&#x2009;</mml:mtext><mml:mi>Q</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mi>t</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:mi>d</mml:mi><mml:mi>i</mml:mi><mml:mi>a</mml:mi><mml:mi>g</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mi>Q</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x005E;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e004.tif"/></alternatives><label>[Eqn 4]</label></disp-formula>
<p>This step ensures that <italic>R<sub>t</sub></italic> has ones on the diagonal and valid correlations off-diagonal.</p>
</sec>
<sec id="s30012">
<title>Conditional covariance matrix and estimation</title>
<p>Let <italic>D<sub>t</sub></italic> = <italic>diag</italic> (&#x03C3;<sub>{1,<italic>t</italic>}</sub>,&#x2026;, &#x03C3;<sub>{<italic>N,t</italic>}</sub>). The conditional covariance matrix of returns (<xref ref-type="disp-formula" rid="FD5">Equation 5</xref>) is:</p>
<disp-formula id="FD5"><alternatives><mml:math display="block" id="M5"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:msub><mml:mi>R</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:msub><mml:mi>D</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>d</mml:mi><mml:mi>i</mml:mi><mml:mi>a</mml:mi><mml:mi>g</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>&#x03C3;</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>&#x03C3;</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e005.tif"/></alternatives><label>[Eqn 5]</label></disp-formula>
<p>Estimation follows the standard two-step DCC procedure: univariate GARCH parameters are estimated first to obtain z_t, after which the correlation parameters (a, b) are estimated from the DCC recursion. For completeness, the multivariate Gaussian log-likelihood is provided in <xref ref-type="sec" rid="s20036">Appendix 1-A3</xref>.</p>
</sec>
</sec>
<sec id="s20013">
<title>Diebold&#x2013;Yilmaz vector autoregressive connectedness (Stage 2)</title>
<p>Stage 2 quantifies spillovers and/or connectedness among the risk proxies using the Diebold&#x2013;Yilmaz framework (Diebold &#x0026; Yilmaz <xref ref-type="bibr" rid="CIT0013">2012</xref>, <xref ref-type="bibr" rid="CIT0014">2014</xref>). The key input from Stage 1 is a volatility proxy for the REIT series, such as <inline-formula id="I7"><alternatives><mml:math display="inline" id="IM7"><mml:mrow><mml:msubsup><mml:mi>&#x03C3;</mml:mi><mml:mi>t</mml:mi><mml:mi>r</mml:mi></mml:msubsup><mml:mo>&#x2261;</mml:mo><mml:msub><mml:mi>&#x03C3;</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>R</mml:mi><mml:mi>E</mml:mi><mml:mi>I</mml:mi><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i007.tif"/></alternatives></inline-formula> (the conditional standard deviation from the GARCH in Stage 1) or realised volatility. There are two sets of hypotheses for this model. The first set relates to whether spillovers from the domestic systemic-risk proxies to REIT volatility are statistically significant. The null hypothesis is that spillovers from the domestic systemic-risk proxies (the domestic volatility index and the 10-year sovereign yield) to REIT volatility are not statistically significant. The alternative hypothesis is that spillovers from at least one domestic systemic-risk proxy (the domestic volatility index and/or the 10-year sovereign yield) to REIT volatility are statistically significant.</p>
<p>The second set relates to whether the two systemic-risk proxies differ in their spillover contribution to REIT volatility. The null hypothesis is that the spillover contribution of the domestic volatility index to REIT volatility is not statistically different from the spillover contribution of the 10-year sovereign yield to REIT volatility. The alternative hypothesis is that the spillover contribution of the domestic volatility index to REIT volatility is statistically different from the spillover contribution of the 10-year sovereign yield to REIT volatility.</p>
<sec id="s30014">
<title>State vector</title>
<p>Define the N &#x00D7; 1 state vector (<xref ref-type="disp-formula" rid="FD6">Equation 6</xref>):</p>
<disp-formula id="FD6"><alternatives><mml:math display="block" id="M6"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mi>&#x03C3;</mml:mi><mml:mi>t</mml:mi><mml:mi>r</mml:mi></mml:msubsup><mml:mo>,</mml:mo><mml:mi>V</mml:mi><mml:mi>I</mml:mi><mml:msubsup><mml:mi>X</mml:mi><mml:mi>t</mml:mi><mml:mi>c</mml:mi></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mrow><mml:mn>10</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>T</mml:mi></mml:msup></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e006.tif"/></alternatives><label>[Eqn 6]</label></disp-formula>
<p>where <inline-formula id="I8"><alternatives><mml:math display="inline" id="IM8"><mml:mrow><mml:msubsup><mml:mi>&#x03C3;</mml:mi><mml:mi>t</mml:mi><mml:mi>r</mml:mi></mml:msubsup></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i008.tif"/></alternatives></inline-formula> is the REIT volatility proxy, <inline-formula id="I9"><alternatives><mml:math display="inline" id="IM9"><mml:mrow><mml:mi>V</mml:mi><mml:mi>I</mml:mi><mml:msubsup><mml:mi>X</mml:mi><mml:mi>t</mml:mi><mml:mi>c</mml:mi></mml:msubsup></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i009.tif"/></alternatives></inline-formula> is the domestic volatility index, and y_<italic>t</italic><sup>10</sup> is the 10-year sovereign yield.</p>
</sec>
<sec id="s30015">
<title>Vector autoregressive(p) specification</title>
<p>The VAR(p) (<xref ref-type="disp-formula" rid="FD7">Equation 7</xref>) is:</p>
<disp-formula id="FD7"><alternatives><mml:math display="block" id="M7"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:msubsup><mml:mi>&#x03A3;</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>p</mml:mi></mml:msubsup><mml:msub><mml:mi>A</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>k</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>&#x223C;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>&#x03A3;</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e007.tif"/></alternatives><label>[Eqn 7]</label></disp-formula>
<p>Here, c is an intercept vector, <italic>A<sub>k</sub></italic> are autoregressive coefficient matrices, and <italic>&#x2211;<sub>u</sub></italic> is the innovation covariance matrix. Lag order p is chosen using information criteria.</p>
</sec>
<sec id="s30016">
<title>Generalised FEVD (order-invariant)</title>
<p>Connectedness is measured by using the generalised forecast-error variance decomposition, which is invariant to variable ordering. Because the FEVD expression is algebraically dense, the full formula and notation are presented in <xref ref-type="sec" rid="s20037">Appendix 1-A4</xref>. Denote the FEVD share as:</p>
<disp-formula id="FD8"><alternatives><mml:math display="block" id="M8"><mml:mrow><mml:msubsup><mml:mi>&#x03B8;</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>H</mml:mi></mml:msubsup><mml:mo>&#x2208;</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e008.tif"/></alternatives><label>[Eqn 8]</label></disp-formula>
<p>This equation (<xref ref-type="disp-formula" rid="FD8">Equation 8</xref>) is interpreted as the fraction of the H-step-ahead forecast-error variance of variable i attributable to shocks in variable j.</p>
</sec>
<sec id="s30017">
<title>Normalisation</title>
<p>Because generalised FEVD rows do not necessarily sum to 1, we normalise (<xref ref-type="disp-formula" rid="FD9">Equation 9</xref>):</p>
<disp-formula id="FD9"><alternatives><mml:math display="block" id="M9"><mml:mrow><mml:msubsup><mml:mover accent="true"><mml:mi>&#x03B8;</mml:mi><mml:mo>&#x02DC;</mml:mo></mml:mover><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>H</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mi>&#x03B8;</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>H</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>N</mml:mi></mml:msubsup><mml:mrow><mml:msubsup><mml:mi>&#x03B8;</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>H</mml:mi></mml:msubsup></mml:mrow></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>N</mml:mi></mml:msubsup><mml:mrow><mml:msubsup><mml:mover accent="true"><mml:mi>&#x03B8;</mml:mi><mml:mo>&#x02DC;</mml:mo></mml:mover><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>H</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mstyle></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e009.tif"/></alternatives><label>[Eqn 9]</label></disp-formula>
</sec>
<sec id="s30018">
<title>Spillover measures</title>
<p>Using the normalised FEVD matrix (<xref ref-type="disp-formula" rid="FD10">Equation 10</xref>): <inline-formula id="I10"><alternatives><mml:math display="inline" id="IM10"><mml:mrow><mml:msup><mml:mover accent="true"><mml:mi>&#x03B8;</mml:mi><mml:mo>&#x02DC;</mml:mo></mml:mover><mml:mi>H</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msubsup><mml:mover accent="true"><mml:mi>&#x03B8;</mml:mi><mml:mo>&#x02DC;</mml:mo></mml:mover><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>H</mml:mi></mml:msubsup></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i010.tif"/></alternatives></inline-formula>,</p>
<p>Total spillover index</p>
<disp-formula id="FD10"><alternatives><mml:math display="block" id="M10"><mml:mrow><mml:mi>T</mml:mi><mml:msup><mml:mi>S</mml:mi><mml:mi>H</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mn>100</mml:mn><mml:mo>&#x00D7;</mml:mo><mml:mfrac><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>N</mml:mi></mml:msubsup><mml:mrow><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>&#x2260;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>N</mml:mi></mml:msubsup><mml:mrow><mml:msubsup><mml:mover accent="true"><mml:mi>&#x03B8;</mml:mi><mml:mo>&#x02DC;</mml:mo></mml:mover><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>H</mml:mi></mml:msubsup></mml:mrow></mml:mstyle></mml:mrow></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>N</mml:mi></mml:mfrac></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e010.tif"/></alternatives><label>[Eqn 10]</label></disp-formula>
<p>Directional spillovers received by <italic>i</italic> (from others to <italic>i</italic>) (<xref ref-type="disp-formula" rid="FD11">Equation 11</xref>):</p>
<disp-formula id="FD11"><alternatives><mml:math display="block" id="M11"><mml:mrow><mml:msubsup><mml:mi>S</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x2190;</mml:mo><mml:mo>&#x2022;</mml:mo><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>H</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:mn>100</mml:mn><mml:mo>&#x00D7;</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>&#x2260;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>N</mml:mi></mml:msubsup><mml:mrow><mml:msubsup><mml:mover accent="true"><mml:mi>&#x03B8;</mml:mi><mml:mo>&#x02DC;</mml:mo></mml:mover><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>H</mml:mi></mml:msubsup></mml:mrow></mml:mstyle></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e011.tif"/></alternatives><label>[Eqn 11]</label></disp-formula>
<p>Directional spillovers transmitted by <italic>i</italic> (from <italic>i</italic> to others) (<xref ref-type="disp-formula" rid="FD12">Equation 12</xref>):</p>
<disp-formula id="FD12"><alternatives><mml:math display="block" id="M12"><mml:mrow><mml:mi>S</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mo>&#x007B;</mml:mo><mml:mo>&#x2022;</mml:mo><mml:mo>&#x2190;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x007D;</mml:mo><mml:mo>&#x005E;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>100</mml:mn><mml:mo>&#x00D7;</mml:mo><mml:mstyle displaystyle="true"><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mo>&#x005F;</mml:mo><mml:mo>&#x007B;</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>&#x2260;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x007D;</mml:mo><mml:mo>&#x005E;</mml:mo><mml:mi>N</mml:mi><mml:mover accent="true"><mml:mi>&#x03B8;</mml:mi><mml:mo>&#x02DC;</mml:mo></mml:mover><mml:mo>&#x005F;</mml:mo><mml:mo>&#x007B;</mml:mo><mml:mi>j</mml:mi><mml:mi>i</mml:mi><mml:mo>&#x007D;</mml:mo><mml:mo>&#x005E;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mstyle></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e012.tif"/></alternatives><label>[Eqn 12]</label></disp-formula>
<p>Net spillovers (<xref ref-type="disp-formula" rid="FD13">Equation 13</xref>):</p>
<disp-formula id="FD13"><alternatives><mml:math display="block" id="M13"><mml:mrow><mml:mi>N</mml:mi><mml:msubsup><mml:mi>S</mml:mi><mml:mi>i</mml:mi><mml:mi>H</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mi>S</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mo>&#x2022;</mml:mo><mml:mo>&#x2190;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>H</mml:mi></mml:msubsup><mml:mo>&#x2212;</mml:mo><mml:msubsup><mml:mi>S</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x2190;</mml:mo><mml:mo>&#x2022;</mml:mo><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>H</mml:mi></mml:msubsup></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e013.tif"/></alternatives><label>[Eqn 13]</label></disp-formula>
<p>A positive <inline-formula id="I11"><alternatives><mml:math display="inline" id="IM11"><mml:mrow><mml:mi>N</mml:mi><mml:msubsup><mml:mi>S</mml:mi><mml:mi>i</mml:mi><mml:mi>H</mml:mi></mml:msubsup></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i011.tif"/></alternatives></inline-formula> indicates that variable i is a net transmitter of shocks, while a negative value indicates a net receiver. (If rolling-window connectedness is used, window length and implementation details are reported in <xref ref-type="sec" rid="s20038">Appendix 1-A5</xref>.)</p>
</sec>
</sec>
<sec id="s20019">
<title>Copula model specification (Stage 3)</title>
<p>Copulas are used to model dependence beyond correlation, including asymmetric co-movement in extremes. In this study, copulas are applied to key series (e.g. REIT&#x2013;risk vs a systemic risk factor), using either raw returns or standardised innovations (e.g. <italic>Z</italic><sub>{<italic>i, t</italic>}</sub> from Stage 1), depending on the study design. The copula stage proceeds in two steps: (1) estimate marginals, and (2) estimate copula dependence.</p>
<p>There are two sets of hypotheses for this model. The first set relates to whether REIT volatility and the domestic systemic-risk proxy exhibit statistically significant dependence at the tails. The null hypothesis is that REIT volatility and the domestic systemic-risk proxy exhibit no statistically significant tail dependence. The alternative hypothesis is that REIT volatility and the domestic systemic-risk proxy exhibit statistically significant tail dependence.</p>
<p>The second set relates to whether tail dependence is symmetric across upper- and lower-tail states. The null hypothesis is that upper-tail dependence and lower-tail dependence are not statistically different (tail dependence is symmetric). The alternative hypothesis is that upper-tail dependence and lower-tail dependence are statistically different (tail dependence is asymmetric).</p>
<sec id="s30020">
<title>Marginal models</title>
<p>Let <italic>X</italic><sub>{<italic>i, t</italic>}</sub> denote the observed variable used in the copula step for series <italic>i</italic> &#x2208; {1, 2}. The marginal CDF (<xref ref-type="disp-formula" rid="FD14">Equation 14</xref>) is:</p>
<disp-formula id="FD14"><alternatives><mml:math display="block" id="M14"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo>&#x223C;</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>&#x22C5;</mml:mo><mml:mo>;</mml:mo><mml:msub><mml:mi>&#x03B8;</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e014.tif"/></alternatives><label>[Eqn 14]</label></disp-formula>
<p>Transform data to uniforms by using the probability integral transform (<xref ref-type="disp-formula" rid="FD15">Equation 15</xref>):</p>
<disp-formula id="FD15"><alternatives><mml:math display="block" id="M15"><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo>;</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>&#x03B8;</mml:mi><mml:mo>&#x005E;</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo>&#x2208;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e015.tif"/></alternatives><label>[Eqn 15]</label></disp-formula>
</sec>
<sec id="s30021">
<title>Gaussian copula (joint CDF)</title>
<p>The Gaussian copula with correlation parameter &#x03C1; has (<xref ref-type="disp-formula" rid="FD16">Equation 16</xref>):</p>
<disp-formula id="FD16"><alternatives><mml:math display="block" id="M16"><mml:mrow><mml:mi>C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo>;</mml:mo><mml:mi>&#x03C1;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>&#x03A6;</mml:mi><mml:mrow><mml:mi>&#x03C1;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>&#x03A6;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>&#x03A6;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msub></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e016.tif"/></alternatives><label>[Eqn 16]</label></disp-formula>
<p>where <italic>&#x03A6;<sub>&#x03C1;</sub></italic> is the bivariate standard normal CDF with correlation <italic>&#x03C1;</italic>, and <italic>&#x03A6;</italic><sup>&#x2013;1</sup> is the standard normal quantile function.</p>
</sec>
<sec id="s30022">
<title>Copula estimation (inference functions for margins)</title>
<p>Inference Functions are used for Margins (IFM): estimate &#x03B8;_i for marginals first, then estimate &#x03C1; by maximising the copula log-likelihood (<xref ref-type="disp-formula" rid="FD17">Equation 17</xref>):</p>
<disp-formula id="FD17"><alternatives><mml:math display="block" id="M17"><mml:mrow><mml:msub><mml:mi>&#x2113;</mml:mi><mml:mrow><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C1;</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mi>T</mml:mi></mml:msubsup><mml:mrow><mml:mi>ln</mml:mi><mml:mi>c</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow></mml:msub><mml:mo>;</mml:mo><mml:mi>&#x03C1;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e017.tif"/></alternatives><label>[Eqn 17]</label></disp-formula>
<p>where <italic>c</italic>(&#x00B7;) is the copula density. The explicit density expression is placed in <xref ref-type="sec" rid="s20039">Appendix 1-A6</xref>.</p>
</sec>
<sec id="s30023">
<title>Dependence measures and tail dependence</title>
<p>Dependence is summarised by using Kendall&#x2019;s tau (<xref ref-type="disp-formula" rid="FD18">Equation 18</xref>):</p>
<disp-formula id="FD18"><alternatives><mml:math display="block" id="M18"><mml:mrow><mml:mi>&#x03C4;</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mn>2</mml:mn><mml:mi>&#x03C0;</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>arcsin</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C1;</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e018.tif"/></alternatives><label>[Eqn 18]</label></disp-formula>
<p>Tail dependence coefficients are defined as <xref ref-type="disp-formula" rid="FD19">Equation 19</xref>:</p>
<disp-formula id="FD19"><alternatives><mml:math display="block" id="M19"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mi>&#x03BB;</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mi>U</mml:mi><mml:mo>=</mml:mo><mml:mi>lim</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mo>&#x007B;</mml:mo><mml:mi>&#x03B1;</mml:mi><mml:mo>&#x2192;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x005E;</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mo>&#x007D;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mn>2</mml:mn><mml:mi>&#x03B1;</mml:mi><mml:mo>+</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03B1;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#x03B1;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>&#x03B1;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi>&#x03BB;</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mi>lim</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mo>&#x007B;</mml:mo><mml:mi>&#x03B1;</mml:mi><mml:mo>&#x2192;</mml:mo><mml:mn>0</mml:mn><mml:mo>&#x005E;</mml:mo><mml:mo>+</mml:mo><mml:mo>&#x007D;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03B1;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#x03B1;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>&#x03B1;</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e019.tif"/></alternatives><label>[Eqn 19]</label></disp-formula>
<p>(For the Gaussian copula, tail dependence is theoretically zero unless extensions are used; interpret results accordingly. Further notes are provided in <xref ref-type="sec" rid="s20034">Appendix 1-A1</xref> to <xref ref-type="sec" rid="s20039">Appendix 1-A6</xref>).</p>
<p>Extreme tail risks, which reflect a stress regime, are defined in this study as periods when the systemic-risk proxy lies in its upper-tail, specifically, observations are classified as stressed when <italic>S</italic>_<italic>t</italic> exceeds the 95th percentile (upper 5&#x0025;) of its country-specific sample distribution, and as normal otherwise (<xref ref-type="disp-formula" rid="FD20">Equation 20</xref>).</p>
<p>Formally:</p>
<disp-formula id="FD20"><alternatives><mml:math display="block" id="M20"><mml:mrow><mml:mi>I</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x005E;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi><mml:mi>s</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>&#x003E;</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0.95</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e020.tif"/></alternatives><label>[Eqn 20]</label></disp-formula>
<list list-type="bullet">
<list-item><p><inline-formula id="I12"><alternatives><mml:math display="inline" id="IM12"><mml:mrow><mml:msubsup><mml:mi>I</mml:mi><mml:mi>t</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i012.tif"/></alternatives></inline-formula>: stress-regime indicator at time t (equals 1 in stress periods, 0 otherwise).</p></list-item>
<list-item><p>1(&#x00B7;): indicator function, equal to 1 if the condition in brackets holds and 0 otherwise.</p></list-item>
<list-item><p><italic>S<sub>t</sub></italic>: systemic-risk proxy at time t (e.g. the domestic volatility index measure or sovereign yield measure used in the study).</p></list-item>
<list-item><p><italic>Q<sub>s</sub></italic>(0.95): the 95th percentile (upper 5&#x0025; threshold) of the country-specific sample distribution of <italic>S<sub>t</sub></italic>.</p></list-item>
<list-item><p><italic>t</italic>: time index.</p></list-item>
</list>
</sec>
</sec>
<sec id="s20024">
<title>Frequency-domain connectedness decomposition</title>
<p>As a robustness check, the study implements a frequency-domain (spectral) connectedness decomposition to assess whether spillovers are concentrated in short-run (high-frequency) dynamics or long-run (low-frequency) dynamics. This robustness check is not a primary objective of the study; rather, it verifies whether the connectedness remains stable once spillovers are decomposed by time horizon (<xref ref-type="disp-formula" rid="FD21">Equation 21</xref>):</p>
<disp-formula id="FD21"><alternatives><mml:math display="block" id="M21"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mi>T</mml:mi><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>100</mml:mn><mml:mo>&#x00D7;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>N</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mstyle displaystyle="true"><mml:msub><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>&#x2260;</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mo>&#x222B;</mml:mo><mml:mrow><mml:msub><mml:mi>&#x03C9;</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>&#x03C9;</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:msubsup><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>d</mml:mi><mml:mi>&#x03C9;</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</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:msubsup><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mo>&#x222B;</mml:mo><mml:mrow><mml:msub><mml:mi>&#x03C9;</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>&#x03C9;</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:msubsup><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>d</mml:mi><mml:mi>&#x03C9;</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext>&#x2003;&#x2003;&#x2003;&#x2003;</mml:mtext><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>&#x03C9;</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>&#x03C9;</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e021.tif"/></alternatives><label>[Eqn 21]</label></disp-formula>
<p>where <italic>f</italic><sub>(<italic>ij</italic>)(<italic>&#x03C9;</italic>)</sub> is the generalised causation spectrum implied by the VAR. Everything defining <italic>f</italic><sub>(<italic>ij</italic>)(<italic>&#x03C9;</italic>)</sub>,<italic>&#x03A8;</italic><sub>(<italic>&#x03C9;</italic>)</sub>, and <italic>Sx</italic><sub>(<italic>&#x03C9;</italic>)</sub> can be viewed in Appendix 1-A9.</p>
</sec>
<sec id="s20025">
<title>Ethical considerations</title>
<p>Ethical clearance to conduct this study was obtained from the University of Johannesburg Research Ethics Committee (Ref. No. SAREC20251021/02).</p>
</sec>
</sec>
<sec id="s0026">
<title>Results</title>
<p>This section starts with the results from the DDC-GARCH model, which shows how volatilities between two time series move together over time. The results from the DY-VAR and Asymmetric GARCH-Copula models then provide additional insights on the volatility relationship by identifying the direction and magnitude of volatility transmission, as well as non-linear and tail-risk dependence.</p>
<p><xref ref-type="table" rid="T0002">Table 2</xref> shows the results from the DDC-GARCH model. The coefficients, <italic>Endog &#x03B2;</italic><sub>1</sub> and <italic>Endog &#x03B2;</italic><sub>2</sub>, indicate that there is significant volatility persistence in the endogenous, REIT, and exogenous Yield and volatility variables themselves. Importantly for this study, though, are the <italic>dcca</italic><sub>1</sub> and <italic>dccb</italic><sub>1</sub> coefficients as they point to the time-varying correlation characteristics between the REITs and the risk measures.</p>
<table-wrap id="T0002">
<label>TABLE 2</label>
<caption><p>Dynamic conditional correlation&#x2013;generalised autoregressive conditional heteroscedasticity.</p></caption>
<table frame="hsides" rules="groups">
<thead>
<tr>
<th valign="top" align="left">Country</th>
<th valign="top" align="left">Exogenous</th>
<th valign="top" align="center">Endog &#x03B2;1</th>
<th valign="top" align="center">Exog &#x03B1;1</th>
<th valign="top" align="center">Exog &#x03B2;2</th>
<th valign="top" align="center">dcca1</th>
<th valign="top" align="center">dccb1</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left">China</td>
<td align="left">Vol index</td>
<td align="center">0.947<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.457<xref ref-type="table-fn" rid="TFN0002">&#x002A;&#x002A;</xref></td>
<td align="center">0.944<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.009</td>
<td align="center">0.823<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
<tr>
<td align="left">China</td>
<td align="left">Yield</td>
<td align="center">0.947<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.526<xref ref-type="table-fn" rid="TFN0002">&#x002A;&#x002A;</xref></td>
<td align="center">0.936<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.018</td>
<td align="center">0.602<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
<tr>
<td align="left">Brazil</td>
<td align="left">Vol index</td>
<td align="center">0.903<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.429<xref ref-type="table-fn" rid="TFN0002">&#x002A;&#x002A;</xref></td>
<td align="center">0.942<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.000</td>
<td align="center">0.910<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
<tr>
<td align="left">Brazil</td>
<td align="left">Yield</td>
<td align="center">0.903<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.215<xref ref-type="table-fn" rid="TFN0001">&#x002A;</xref></td>
<td align="center">0.725<xref ref-type="table-fn" rid="TFN0002">&#x002A;&#x002A;</xref></td>
<td align="center">0.043</td>
<td align="center">0.887<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
<tr>
<td align="left">South Africa</td>
<td align="left">Vol index</td>
<td align="center">0.884<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.135<xref ref-type="table-fn" rid="TFN0001">&#x002A;</xref></td>
<td align="center">0.606<xref ref-type="table-fn" rid="TFN0002">&#x002A;&#x002A;</xref></td>
<td align="center">0.000</td>
<td align="center">0.997<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
<tr>
<td align="left">South Africa</td>
<td align="left">Yield</td>
<td align="center">0.884<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.773<xref ref-type="table-fn" rid="TFN0002">&#x002A;&#x002A;</xref></td>
<td align="center">0.827<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.022</td>
<td align="center">0.939<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
<tr>
<td align="left">India</td>
<td align="left">Vol index</td>
<td align="center">0.892<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.521<xref ref-type="table-fn" rid="TFN0002">&#x002A;&#x002A;</xref></td>
<td align="center">0.825<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.016</td>
<td align="center">0.933<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
<tr>
<td align="left">India</td>
<td align="left">Yield</td>
<td align="center">0.892<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.359<xref ref-type="table-fn" rid="TFN0002">&#x002A;&#x002A;</xref></td>
<td align="center">0.962<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.004</td>
<td align="center">0.888<xref ref-type="table-fn" rid="TFN0003">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<fn><p>Vol index, volatility index.</p></fn>
<fn id="TFN0001"><label>&#x002A;</label><p>, Statistical significance at a 90&#x0025; confidence level;</p></fn>
<fn id="TFN0002"><label>&#x002A;&#x002A;</label><p>, Statistical significance at a 95&#x0025; confidence level;</p></fn>
<fn id="TFN0003"><label>&#x002A;&#x002A;&#x002A;</label><p>, Statistical significance at a 99&#x0025; confidence level.</p></fn>
</table-wrap-foot>
</table-wrap>
<p>The <italic>dcca</italic><sub>1</sub> coefficient shows how strongly a new shock in standardised residuals adjusts the next period&#x2019;s correlation, thus, if co-movements are shock-dependent. In this case, there is no evidence suggesting that the correlation between the risk measures and the volatility in the REITs of any of these countries is shock-dependent. However, the <italic>dccb</italic><sub>1</sub> coefficient indicates that the co-movement of volatilities is long-lasting once established. Thus, once a relationship is established between the REITs and the risk measures, the relationship tends to exhibit statistically significant persistence. In this light, from a hedging perspective, REITs might sufficiently hedge against some shocks in all these countries, but if a relationship develops, the hedging ability might erode. However, these results tell only a part of the story and should be paired with the results from the next two models.</p>
<p><xref ref-type="table" rid="T0003">Table 3</xref> shows the results from the DY-VAR model and shows how much of the REIT volatility in each country is explained by volatility in the risk measures of each corresponding country. The results show that a very low percentage of REIT volatility in both China and Brazil is explained by either of the systemic risk measures. Similarly, the South African volatility index explains a very low percentage of South African REIT volatility, and Indian yields explain a very low percentage of REIT volatility in India. On the other hand, the South African yields explain approximately 10&#x0025; of South African REIT volatility, and the Indian volatility index explains approximately 12.5&#x0025; of REIT volatility in India.</p>
<table-wrap id="T0003">
<label>TABLE 3</label>
<caption><p>Diebold&#x2013;Yilmaz vector autoregressive results.</p></caption>
<table frame="hsides" rules="groups">
<thead>
<tr>
<th valign="top" align="left">Country</th>
<th valign="top" align="center">Volatility index</th>
<th valign="top" align="center">Yield</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left">China</td>
<td align="center">0.01</td>
<td align="center">0.25</td>
</tr>
<tr>
<td align="left">Brazil</td>
<td align="center">2.48</td>
<td align="center">0.88</td>
</tr>
<tr>
<td align="left">South Africa</td>
<td align="center">1.47</td>
<td align="center">10.16</td>
</tr>
<tr>
<td align="left">India</td>
<td align="center">12.51</td>
<td align="center">0.39</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>Even though 10&#x0025; and 12.5&#x0025;, respectively, are still relatively low, it does show that some of these systemic risk measures do have some explanatory power over REIT volatility in these countries and that asset managers should consider these variables in their hedging process. However, in general, these results show that the systemic risk measures considered in this study explain a very small portion of the volatility in REITs across the BRICS countries. However, the DY-VAR model is linear and, therefore, reflects dependencies during normal periods, not extreme periods. This is valuable information for portfolio construction during normal periods; yet, as argued previously, it is essential to measure this relationship at extreme levels to inform effective hedging. In this light, the results from the asymmetric GARCH-Copula model, shown in <xref ref-type="table" rid="T0004">Table 4</xref>, are now considered.</p>
<table-wrap id="T0004">
<label>TABLE 4</label>
<caption><p>Annotated copula dependence table.</p></caption>
<table frame="hsides" rules="groups">
<thead>
<tr>
<th valign="top" align="left">Country</th>
<th valign="top" align="left">Variable</th>
<th valign="top" align="center">Kendall Tau</th>
<th valign="top" align="center">Lower_Tail_5&#x0025;</th>
<th valign="top" align="center">Upper_Tail_95&#x0025;</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left" rowspan="2" valign="top">China</td>
<td align="left">Volatility index</td>
<td align="center">0.014<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.622<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.040<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
<tr>
<td align="left">Yield</td>
<td align="center">0.004<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.535<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.066<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
<tr>
<td align="left" rowspan="2" valign="top">Brazil</td>
<td align="left">Volatility index</td>
<td align="center">&#x2212;0.045<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.790<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.360<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
<tr>
<td align="left">Yield</td>
<td align="center">0.032<xref ref-type="table-fn" rid="TFN0004">&#x002A;&#x002A;</xref></td>
<td align="center">0.636<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.522<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
<tr>
<td align="left" rowspan="2" valign="top">South Africa</td>
<td align="left">Volatility index</td>
<td align="center">0.087<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.638<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.580<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
<tr>
<td align="left">Yield</td>
<td align="center">0.251<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.836<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.619<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
<tr>
<td align="left" rowspan="2" valign="top">India</td>
<td align="left">Volatility index</td>
<td align="center">0.339<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.684<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.421<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
<tr>
<td align="left">Yield</td>
<td align="center">0.051<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.7440<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
<td align="center">0.330<xref ref-type="table-fn" rid="TFN0005">&#x002A;&#x002A;&#x002A;</xref></td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<fn id="TFN0004"><label>&#x002A;&#x002A;</label><p>, Statistical significance at a 95&#x0025; confidence level;</p></fn>
<fn id="TFN0005"><label>&#x002A;&#x002A;&#x002A;</label><p>, Statistical significance at a 99&#x0025; confidence level.</p></fn>
</table-wrap-foot>
</table-wrap>
<p>Starting with the Kendall Tau output, we show that for most BRICS countries, the rank dependence during normal conditions is positive and very weak. For example, for China, the volatility on REITs is only 1.4 percentage points more likely to move in the same rank order as the volatility index, rather than in the opposite order, and 0.4 percentage points in the case of yields. For Brazil, REIT volatility is only 3.2 percentage points more likely to move in the same rank order as yields and 4.5&#x0025; less likely to move in the same rank order as the volatility index. For South African yields and the Indian Volatility Index, the probabilities are much higher. In the case of India, REIT volatility is 34 percentage points more likely to move in the same rank order as the volatility index, rather than in the opposite order. Furthermore, South African REIT volatility is 25 percentage points more likely to move in the same rank order as yields. This finding corresponds with the results from the DY-VAR model, which shows that spillovers from yields to South African REIT volatility and Indian spillovers from the Indian volatility index to Indian REITs are much more likely than in other BRICS countries.</p>
<p>Very important from a hedging perspective is the volatility spillovers from the systemic risk measures to REITs during extreme periods. In this light, the 95&#x0025; upper-tail results show how likely it is for REIT volatility to be extremely high when systemic risk is extremely high. In contrast, the 5&#x0025; lower-tail results show how likely it is for REIT volatility to be extremely low when systemic risk is extremely low. The results from the 95&#x0025; upper-tail show that only the volatility in Chinese REIT has a very low dependence on systemic risk at extreme levels. The results show that there is only a 4-percentage point chance that REIT volatility is extreme when the Volatility Index is extremely high and a 6.6-percentage point chance of being extremely high when yields are extremely high. South African REITs have the highest systemic risk dependence during extreme periods. The results show that there is a 58-percentage point chance that South African REIT volatility is extreme when the Volatility index is extremely high and a 62-percentage point chance of being extremely high when yields are extremely high. Furthermore, the results show that there is a 36-percentage point chance that Brazilian REIT volatility is extreme when the Volatility index is extremely high and a 52-percentage point chance of being extremely high when yields are extremely high. Lastly, there is a 42-percentage point chance that Brazilian REIT volatility is extreme when the Volatility index is extremely high and a 33-percentage point chance of being extremely high when yields are extremely high. On the other end of the spectrum, the results show that for all countries, it is likely that REIT volatility is extremely low when systemic risk measures are at extremely low levels.</p>
<p>As a robustness check, the study constructed a special frequency-domain connectedness decomposition. The frequency-domain connectedness decomposition provides a robustness check on the DY-VAR results by testing whether spillovers are primarily a short-run (high-frequency) phenomenon &#x2013; consistent with transitory shocks and rapid repricing &#x2013; or whether they reflect long-run (low-frequency) transmission &#x2013; consistent with more persistent macro and/or financial linkages. <xref ref-type="table" rid="T0005">Table 5</xref> depicts the Frequency-domain connectedness decomposition results.</p>
<table-wrap id="T0005">
<label>TABLE 5</label>
<caption><p>Frequency-domain connectedness decomposition results.</p></caption>
<table frame="hsides" rules="groups">
<thead>
<tr>
<th valign="top" align="left" rowspan="2">Country</th>
<th valign="top" align="left" rowspan="2">Band</th>
<th valign="top" align="center" rowspan="2">TS (&#x0025;)</th>
<th valign="top" align="center" colspan="3">REIT_ret (&#x0025;)<hr/></th>
<th valign="top" align="center" colspan="3">VIX_ret (&#x0025;)<hr/></th>
<th valign="top" align="center" colspan="3">Y10_d (&#x0025;)<hr/></th>
</tr>
<tr>
<th valign="top" align="center">Rec</th>
<th valign="top" align="center">Trn</th>
<th valign="top" align="center">Net</th>
<th valign="top" align="center">Rec</th>
<th valign="top" align="center">Trn</th>
<th valign="top" align="center">Net</th>
<th valign="top" align="center">Rec</th>
<th valign="top" align="center">Trn</th>
<th valign="top" align="center">Net</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left" rowspan="2" valign="top">China</td>
<td align="left">Long (0, &#x03C0;/2]</td>
<td align="center">0.39</td>
<td align="center">0.04</td>
<td align="center">0.14</td>
<td align="center">0.10</td>
<td align="center">0.56</td>
<td align="center">0.45</td>
<td align="center">&#x2212;0.11</td>
<td align="center">0.58</td>
<td align="center">0.59</td>
<td align="center">0.01</td>
</tr>
<tr>
<td align="left">Short (&#x03C0;/2, &#x03C0;]</td>
<td align="center">0.50</td>
<td align="center">0.15</td>
<td align="center">0.08</td>
<td align="center">&#x2212;0.07</td>
<td align="center">0.61</td>
<td align="center">0.76</td>
<td align="center">0.15</td>
<td align="center">0.75</td>
<td align="center">0.68</td>
<td align="center">&#x2212;0.07</td>
</tr>
<tr>
<td align="left" rowspan="2" valign="top">Brazil</td>
<td align="left">Long (0, &#x03C0;/2]</td>
<td align="center">11.03</td>
<td align="center">8.33</td>
<td align="center">6.42</td>
<td align="center">&#x2212;1.91</td>
<td align="center">8.80</td>
<td align="center">21.09</td>
<td align="center">12.30</td>
<td align="center">15.97</td>
<td align="center">5.58</td>
<td align="center">&#x2212;10.39</td>
</tr>
<tr>
<td align="left">Short (&#x03C0;/2, &#x03C0;]</td>
<td align="center">5.33</td>
<td align="center">4.54</td>
<td align="center">3.61</td>
<td align="center">&#x2212;0.94</td>
<td align="center">8.62</td>
<td align="center">6.33</td>
<td align="center">&#x2212;2.29</td>
<td align="center">2.83</td>
<td align="center">6.06</td>
<td align="center">3.23</td>
</tr>
<tr>
<td align="left" rowspan="2" valign="top">South Africa</td>
<td align="left">Long (0, &#x03C0;/2]</td>
<td align="center">8.89</td>
<td align="center">13.00</td>
<td align="center">12.40</td>
<td align="center">&#x2212;0.60</td>
<td align="center">1.35</td>
<td align="center">0.38</td>
<td align="center">&#x2212;0.97</td>
<td align="center">12.31</td>
<td align="center">13.89</td>
<td align="center">1.58</td>
</tr>
<tr>
<td align="left">Short (&#x03C0;/2, &#x03C0;]</td>
<td align="center">7.49</td>
<td align="center">10.42</td>
<td align="center">11.53</td>
<td align="center">1.11</td>
<td align="center">0.24</td>
<td align="center">0.50</td>
<td align="center">0.26</td>
<td align="center">11.81</td>
<td align="center">10.44</td>
<td align="center">&#x2212;1.37</td>
</tr>
<tr>
<td align="left" rowspan="2" valign="top">India</td>
<td align="left">Long (0, &#x03C0;/2]</td>
<td align="center">12.36</td>
<td align="center">18.39</td>
<td align="center">17.09</td>
<td align="center">&#x2212;1.30</td>
<td align="center">17.54</td>
<td align="center">18.46</td>
<td align="center">0.92</td>
<td align="center">1.16</td>
<td align="center">1.53</td>
<td align="center">0.38</td>
</tr>
<tr>
<td align="left">Short (&#x03C0;/2, &#x03C0;]</td>
<td align="center">13.32</td>
<td align="center">18.91</td>
<td align="center">20.11</td>
<td align="center">1.20</td>
<td align="center">20.09</td>
<td align="center">19.26</td>
<td align="center">&#x2212;0.83</td>
<td align="center">0.95</td>
<td align="center">0.58</td>
<td align="center">&#x2212;0.37</td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<fn><p>REIT, real-estate investment trust; TS, Total Spillover; ret, return; VIX, Volatility Index; Y10_d, 10-year sovereign bond yield (first difference); Rec, received; Trn, transmitted.</p></fn>
</table-wrap-foot>
</table-wrap>
<p>The main cross-country conclusions remain unchanged. China exhibits negligible connectedness in both bands, confirming the minimal spillovers found in the DY-VAR. For the countries where connectedness is present, the frequency split adds useful detail. South Africa shows meaningful connectedness in both the short and the long-run, but the results point to a more persistent component linked to yields. In the long-run band, total connectedness is slightly higher, and yields emerge as the more structurally important channel, consistent with the DY-VAR result that South African yields explain the largest share of South African REIT volatility. Interpreted economically, this result indicates that the yield&#x2013;REIT volatility relationship is not only a short-lived shock effect; rather, it is supported by slower-moving, longer-horizon dynamics (e.g. sustained repricing of discount rates and funding conditions). The short-run band still shows non-trivial connectedness, implying that yields can also transmit more immediate risk during volatility episodes, but the stronger long-run support reinforces yields as a strategic hedging consideration in South Africa.</p>
<p>India displays strong connectedness in both frequency bands, and the volatility index remains the key risk measure while the yield channel stays negligible &#x2013; fully consistent with the DY-VAR findings. The presence of connectedness in the short-run band suggests that Indian REIT volatility responds to fast-moving equity-market risk sentiment and sudden volatility spikes. At the same time, the persistence of connectedness in the long-run band implies that this relationship is not confined to short-lived stress events; rather, it extends into longer-horizon co-movement in which elevated risk conditions can remain embedded over time. This relationship strengthens the interpretation that the volatility index is a relevant conditioning variable for both tactical (short-run) and strategic (long-run) risk management for Indian REIT exposure.</p>
<p>Brazil remains comparatively modest in overall connectedness, but the decomposition indicates that any linkage is more pronounced at lower frequencies than at higher frequencies. This pattern suggests that spillovers &#x2013; where they exist &#x2013; are less about rapid, short-term shock transmission and more about persistent dynamics that accumulate over longer horizons (e.g. gradual repricing of risk conditions). This nuance aligns with the DY-VAR&#x2019;s generally low explanatory shares, while clarifying that Brazil&#x2019;s limited connectedness is relatively more &#x2018;long-horizon&#x2019; in nature rather than purely transitory.</p>
<p>Overall, the frequency-domain robustness check supports the baseline results from the study. Connectedness is weakest for China, most economically relevant for South Africa (via yields) and India (via the volatility index), and modest for Brazil, with the added insight that South Africa and Brazil&#x2019;s linkages contain a stronger long-run component, while India&#x2019;s connectedness is clearly present in both short- and long-run dynamics.</p>
</sec>
<sec id="s0027">
<title>Discussion</title>
<p>The empirical results support the recent view that REIT&#x2013;risk linkages are conditional and can look limited in normal periods but become more relevant under stress. The DCC-GARCH results show strong persistence once co-movement develops (the dccb1 terms are large and statistically significant across all country&#x2013;proxy pairs), which corresponds with Liow and Chen (2007) and Voicu and Seiler (<xref ref-type="bibr" rid="CIT0036">2013</xref>), who argue that REIT dependence and systematic risk exposure can shift across regimes rather than remain constant. In the same light, the DY-VAR FEVD results indicate that, for most BRICS markets, domestic systemic proxies explain only a small share of REIT volatility in normal conditions (China: 0.01&#x2013;0.25; Brazil: 0.88&#x2013;2.48; South Africa volatility index: 1.47; India yields: 0.39). This result is similar to what Fasanya and Oyewole (<xref ref-type="bibr" rid="CIT0016">2023</xref>) and Danila (<xref ref-type="bibr" rid="CIT0010">2024</xref>) show in related spillover settings, where linear or average connectedness measures often appear modest outside of stress periods. Where a clearer normal-times channel does appear, it is economically interpretable and consistent with established mechanisms.</p>
<p>For South Africa, the yield channel accounts for the largest FEVD share (about 10.16&#x0025;), which matches the rate and discount-rate transmission logic emphasised by Bj&#x00F8;rnland and Jacobsen (<xref ref-type="bibr" rid="CIT0006">2010</xref>) and Yong (<xref ref-type="bibr" rid="CIT0037">2015</xref>), and also fits Caballero, Farhi and Gourinchas (<xref ref-type="bibr" rid="CIT0009">2017</xref>), who stress that yield movements can reflect broader system-level repricing. This finding aligns with Lekhuleni and Ndlovu (<xref ref-type="bibr" rid="CIT0026">2023</xref>) and Ndamanomhata (<xref ref-type="bibr" rid="CIT0032">2023</xref>), who document that South African property and REIT-related series are vulnerable to macro-financial shocks and can switch between transmitter and receiver roles. For India, the volatility index explains about 12.51&#x0025; of REIT volatility, which corresponds with Katyoka et al. (<xref ref-type="bibr" rid="CIT0024">2024</xref>), who highlight volatility conditions as an important transmission channel, and is consistent with Hansen and Timmermann (<xref ref-type="bibr" rid="CIT0020">2024</xref>), who link volatility and yield conditions to broader risk cycles.</p>
<p>Once extreme dependence is considered, the results line up more directly with the stress-regime literature that argues diversification can weaken when it is needed most. The copula results show that normal dependence (Kendall&#x2019;s tau) is generally weak for China and Brazil but becomes economically meaningful for South Africa-yield (tau about 0.251) and India-volatility index (tau about 0.339). This result is similar to Mensi, Nekhili and Kang (<xref ref-type="bibr" rid="CIT0030">2022</xref>), who show that REIT spillovers differ by quantile and are stronger in adverse states, and it corresponds with Hadad et al. (<xref ref-type="bibr" rid="CIT0019">2024</xref>), who report that REIT volatility spillovers intensify under extreme conditions and can change direction across regimes. The tail results reinforce the same point: South Africa shows very high upper-tail dependence (about 0.580 with the volatility index and 0.619 with yields), and India shows meaningful upper-tail dependence (about 0.421 with the volatility index), which is consistent with Abdullah et al. (<xref ref-type="bibr" rid="CIT0001">2023</xref>) and Mensi et al. (<xref ref-type="bibr" rid="CIT0029">2023</xref>) in showing stronger dependence and spillover behaviour during uncertainty episodes.</p>
<p>By contrast, China shows negligible spillovers in DY-VAR and very low upper-tail dependence (about 0.040&#x2013;0.066), which provides a clear contrast to the stronger REIT volatility linkages reported by Liu et al. (<xref ref-type="bibr" rid="CIT0028">2023</xref>) for Hong Kong REITs during COVID-19. The frequency-domain robustness check supports the same interpretation: where connectedness exists, it is not purely transitory, with South Africa&#x2019;s yield channel showing a more persistent component and India&#x2019;s volatility-index channel appearing in both short- and long-run bands, which is consistent with the idea that risk can transmit through both fast-moving sentiment shocks and slower-moving discount-rate and funding dynamics (Bj&#x00F8;rnland &#x0026; Jacobsen <xref ref-type="bibr" rid="CIT0006">2010</xref>; Caballero et al. <xref ref-type="bibr" rid="CIT0009">2017</xref>; Ndamanomhata <xref ref-type="bibr" rid="CIT0032">2023</xref>).</p>
</sec>
<sec id="s0028">
<title>Conclusion</title>
<p>This study provides a comprehensive examination of systemic risk spillovers to REIT volatility in BRICS countries by employing the DCC-GARCH, DY-VAR, and Asymmetric GARCH-Copula models. The findings show that volatility spillovers from systemic risk measures to REITs are generally limited, but spillovers increase during extreme periods, and this relationship exhibits strong persistence once established. Importantly, the results from the DY-VAR provide evidence that systemic risk measures explain only a small percentage of REIT volatility during normal periods. Systemic risk in South Africa and India has the highest explanatory power over REITs relative to the other countries, but the explanatory power remains fairly low. However, the asymmetric GARCH-Copula results highlight that dependencies become substantially more pronounced during extreme market conditions, particularly in South Africa and India, where REIT volatility exhibits strong co-movements with systemic risk in both tails.</p>
<p>In quantitative terms, the DY-VAR results indicate that the domestic systemic risk proxies explain a small share of REIT volatility in normal conditions across most BRICS markets, with China and Brazil remaining close to zero, while South Africa&#x2019;s yield share (about 10.16&#x0025;) and India&#x2019;s volatility-index share (about 12.51&#x0025;) stand out as the largest normal-times channels. By contrast, the copula results show that dependence is materially stronger in extremes than in normal ranks: for South Africa, the upper-tail dependence rises to about 0.580 (volatility index) and 0.619 (yields), and for India, the upper-tail dependence is about 0.421 (volatility index). This difference between &#x2018;normal-times&#x2019; spillover shares and &#x2018;extreme-times&#x2019; dependence provides clear evidence that hedging conclusions drawn from linear averages can understate the risk relationship that matters most during stress episodes.</p>
<p>These findings have several important implications for portfolio managers. Firstly, the hedging benefits of REITs depend largely on the state of the market, as well as the country. Therefore, the hedging benefits of REITs should not be considered homogeneous across all the BRICS markets. In normal periods, REITs may serve as partial diversifiers, given the relatively weak dependence on systemic risk. However, during crises, their volatility is more likely to amplify systemic shocks, especially in South Africa and India, thereby limiting their effectiveness as a hedging asset. This situation underlines the need to employ dynamic hedging strategies that incorporate regime shifts and tail risks, rather than relying solely on linear models of risk transmission. It is therefore recommended by the researcher that asset managers adopt a conditional hedging approach when using REITs as a hedging instrument. Stress-testing should be used to model extreme tail risks and be paired with the portfolio construction process. Additionally, managers should exercise caution in assuming homogeneity across BRICS, as the spillover dynamics differ significantly across countries. Tail-risk management should therefore be country-specific, with greater vigilance in South Africa and India, where systemic risk spillovers are most pronounced. By integrating these insights, portfolio managers can enhance diversification benefits during normal periods while mitigating the erosion of hedging effectiveness during systemic crises.</p>
<p>The scientific value addition of the study lies in the consistent, multi-model evidence of how systemic risk transmits into REIT volatility in emerging markets. Most prior work relies on a single framework or a single market, which makes it difficult to separate co-movement, directional spillovers, and tail amplification within a comparable setting. By combining DCC-GARCH (time-varying correlation), DY-VAR (directional variance shares), and an asymmetric copula approach (tail dependence), the study provides a more complete assessment of hedging effectiveness across BRICS REIT markets. The results also provide an applied benchmark: when normal-times spillover shares remain below roughly 1&#x0025; &#x2013; 3&#x0025; (as in China and Brazil), the case for REITs as partial diversifiers is stronger, but when tail dependence approaches the levels observed for South Africa and India, portfolio risk controls should be tightened because diversification can deteriorate during stress.</p>
</sec>
</body>
<back>
<ack>
<title>Acknowledgements</title>
<p>The author hereby declares that the Codex tool in ChatGPT was used to construct the Python codes to estimate this study&#x2019;s models in Python. Furthermore, ChatGPT was used to refine and check the accuracy of the formulas presented in this section. Lastly, Grammarly was used for language and grammar editing of this document.</p>
<sec id="s20029" sec-type="COI-statement">
<title>Competing interests</title>
<p>The author declares that no financial or personal relationships inappropriately influenced the writing of this article.</p>
</sec>
<sec id="s20030">
<title>CRediT authorship contribution</title>
<p>Milan C. De Wet: Conceptualisation, Data curation, Formal analysis, Methodology, Software, Writing &#x2013; original draft. The author confirms that this work is entirely their own, has reviewed the article, approved the final version for submission and publication, and takes full responsibility for the integrity of its findings.</p>
</sec>
<sec id="s20031" sec-type="data-availability">
<title>Data availability</title>
<p>The data that support the findings of this study are available in the public domain. Real-estate investment trust index and stock market volatility data were obtained from Reuters Datastream (<ext-link ext-link-type="uri" xlink:href="https://www.reuters.com/markets/global-market-data/">https://www.reuters.com/markets/global-market-data/</ext-link>), while bond yield data were obtained from the Organisation for Economic Co-operation and Development (OECD) Data Repository (<ext-link ext-link-type="uri" xlink:href="https://data.oecd.org">https://data.oecd.org</ext-link>).</p>
</sec>
<sec id="s20032">
<title>Disclaimer</title>
<p>The views and opinions expressed in this article are those of the author and are the product of professional research. They do not necessarily reflect the official policy or position of any affiliated institution, funder, agency or that of the publisher. The author is responsible for this article&#x2019;s results, findings, and content.</p>
</sec>
</ack>
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</ref-list>
<app-group>
<app id="app001">
<title>Appendix 1</title>
<sec id="s20034">
<title>A1</title>
<p>Innovation distribution options for <xref ref-type="disp-formula" rid="FDA1">Equation (A1)</xref></p>
<p>Baseline (Gaussian) specification:</p>
<disp-formula id="FDA1"><alternatives><mml:math display="block" id="MA1"><mml:mrow><mml:msub><mml:mi>&#x03B5;</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>&#x03C3;</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>Z</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>&#x223C;</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e022.tif"/></alternatives><label>[Eqn A1]</label></disp-formula>
<p>where:</p>
<list list-type="bullet">
<list-item><p><italic>&#x03B5;<sub>i,t</sub></italic>: <italic>innovation (residual) of series i at time t</italic></p></list-item>
<list-item><p><italic>&#x03C3;<sub>i,t</sub></italic>: <italic>conditional standard deviation of series i at time t</italic></p></list-item>
<list-item><p><italic>z<sub>i,t</sub></italic>: <italic>standardised residual, i. i. d. N</italic> (0,1)</p></list-item>
</list>
<p>Optional robustness (Student-t) specification (<xref ref-type="disp-formula" rid="FDA2">Equation A2)</xref>:</p>
<disp-formula id="FDA2"><alternatives><mml:math display="block" id="MA2"><mml:mrow><mml:msub><mml:mi>&#x03B5;</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>&#x03C3;</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>Z</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>&#x223C;</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mrow><mml:mi>v</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e023.tif"/></alternatives><label>[Eqn A2]</label></disp-formula>
<p>where:</p>
<list list-type="bullet">
<list-item><p><italic>t<sub>v</sub></italic><sub>(0,1)</sub>: <italic>Standardised Student &#x2013; t distribution</italic></p></list-item>
<list-item><p><italic>v</italic> &#x003E; 2: <italic>degrees of freedom parameter controlling tail thickness</italic></p></list-item>
</list>
</sec>
<sec id="s20035">
<title>A2</title>
<p>Asymmetric dynamic conditional correlation specification (optional extension to <xref ref-type="disp-formula" rid="FDA3">Equation [A3]</xref>)</p>
<p>If asymmetric correlation responses to negative shocks are modelled, the ADCC recursion is:</p>
<disp-formula id="FDA3"><alternatives><mml:math display="block" id="MA3"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>a</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>b</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>g</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mover accent="true"><mml:mi>Q</mml:mi><mml:mo>&#x00AF;</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mi>a</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:msub><mml:mi>z</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mtext>&#x2009;</mml:mtext><mml:msubsup><mml:mi>z</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>T</mml:mi></mml:msubsup><mml:mo>+</mml:mo><mml:mi>b</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>g</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mtext>&#x2009;</mml:mtext><mml:msubsup><mml:mi>n</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>T</mml:mi></mml:msubsup></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e024.tif"/></alternatives><label>[Eqn A3]</label></disp-formula>
<p>where:</p>
<list list-type="bullet">
<list-item><p><italic>Q<sub>t</sub></italic>: <italic>time &#x2013; varying covariance matrix of standardised residuals</italic></p></list-item>
<list-item><p><inline-formula id="I13"><alternatives><mml:math display="inline" id="IM13"><mml:mover accent="true"><mml:mi>Q</mml:mi><mml:mo>&#x00AF;</mml:mo></mml:mover></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i013.tif"/></alternatives></inline-formula>: <italic>unconditional covariance matrix of standardised residuals</italic></p></list-item>
<list-item><p><italic>a</italic> &#x2265; 0: <italic>shock effect</italic> (<italic>correlation ARCH term</italic>)</p></list-item>
<list-item><p><italic>b</italic> &#x2265; 0: <italic>persistence effect</italic> (<italic>correlation GARCH term</italic>)</p></list-item>
<list-item><p><italic>g</italic> &#x2265; 0: <italic>asymmetry parameter</italic> (<italic>extra response to negative shocks</italic>).</p></list-item>
<list-item><p><italic>z<sub>t</sub></italic><sub>&#x2013;1</sub>: <italic>vector of standardised residuals at time t</italic> &#x2013; 1</p></list-item>
<list-item><p><italic>n<sub>t</sub></italic><sub>&#x2013;1</sub>: <italic>vector capturing negative components of z</italic><sub><italic>t</italic> &#x2013; 1</sub></p></list-item>
</list>
<p>The negative-shock vector can be defined elementwise as (<xref ref-type="fig" rid="E25">Equation A4</xref>):</p>
<fig id="E25">
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e025.tif"/>
</fig>
<p>where:</p>
<list list-type="bullet">
<list-item><p>&#x2299; : <italic>Hadamard</italic> (<italic>elementwise</italic>) <italic>product</italic></p></list-item>
<list-item><p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i019.tif"/> (<italic>z<sub>t</sub></italic> &#x003C; 0): <italic>indicator vector equal to</italic> 1 <italic>where z<sub>t</sub> is negative</italic> 0 <italic>otherwise Restrictions are a</italic> &#x2265; 0, <italic>b</italic> &#x2265; 0, <italic>g</italic> &#x2265; 0, and <italic>a</italic> + <italic>b</italic> + <italic>g</italic> &#x003C; 1.</p></list-item>
</list>
</sec>
<sec id="s20036">
<title>A3</title>
<p>Full multivariate log-likelihood for DCC-GARCH (supports <xref ref-type="disp-formula" rid="FDA5">Equation [A5]</xref>)</p>
<p>Let <italic>&#x03B5;<sub>t</sub></italic> = (<italic>&#x03B5;</italic><sub>1,<italic>t</italic></sub>, &#x2026;, <italic>&#x03B5;<sub>N,t</sub></italic>)<italic><sup>T</sup></italic> and <italic>H<sub>t</sub></italic> = <italic>D<sub>t</sub> R<sub>t</sub> D<sub>t</sub></italic> Under conditional normality, the log-likelihood is:</p>
<disp-formula id="FDA5"><alternatives><mml:math display="block" id="MA5"><mml:mrow><mml:mi>&#x2113;</mml:mi><mml:mo>=</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</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:msubsup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>N</mml:mi><mml:mi>ln</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>&#x03C0;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>ln</mml:mi><mml:mo>&#x007C;</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>&#x007C;</mml:mo><mml:mo>+</mml:mo><mml:msubsup><mml:mi>&#x03B5;</mml:mi><mml:mi>t</mml:mi><mml:mi>T</mml:mi></mml:msubsup><mml:msubsup><mml:mi>H</mml:mi><mml:mi>t</mml:mi><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mi>&#x03B5;</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e026.tif"/></alternatives><label>[Eqn A5]</label></disp-formula>
<p>where:</p>
<list list-type="bullet">
<list-item><p><italic>N</italic>: <italic>number of series in the multivariate system</italic>.</p></list-item>
<list-item><p><italic>T</italic>: <italic>number of observations</italic>.</p></list-item>
<list-item><p><italic>H<sub>t</sub></italic>: <italic>conditional covariance matrix</italic></p></list-item>
<list-item><p><italic>D<sub>t</sub></italic>: <italic>diagonal matrix of conditional standard deviations</italic></p></list-item>
<list-item><p><italic>R<sub>t</sub></italic>: <italic>conditional correlation matrix</italic></p></list-item>
</list>
</sec>
<sec id="s20037">
<title>A4</title>
<p>Generalised FEVD: MA representation and full expression (supports <xref ref-type="disp-formula" rid="FDA8">Equation [A8]</xref>)</p>
<p>A.4.1 Wold/MA representation of the VAR</p>
<p>For the VAR(p) in <xref ref-type="disp-formula" rid="FDA6">Equation (A6)</xref>:</p>
<disp-formula id="FDA6"><alternatives><mml:math display="block" id="MA6"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>p</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:mstyle></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e027.tif"/></alternatives><label>[Eqn A6]</label></disp-formula>
<p>where:</p>
<list list-type="bullet">
<list-item><p><italic>x<sub>t</sub></italic>: <italic>N&#x00D7;1 state vector</italic></p></list-item>
<list-item><p><italic>c</italic>: <italic>N&#x00D7;1 intercept vector</italic></p></list-item>
<list-item><p><italic>A<sub>k</sub></italic>: <italic>N&#x00D7;N coefficient matrices</italic></p></list-item>
<list-item><p><italic>u<sub>t</sub></italic>: <italic>N&#x00D7;1 innovation vector with covariance &#x03A3;<sub>u</sub></italic></p></list-item>
</list>
<p>The moving-average representation is (<xref ref-type="disp-formula" rid="FDA7">Equation A7)</xref>:</p>
<disp-formula id="FDA7"><alternatives><mml:math display="block" id="MA7"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x03BC;</mml:mi><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>&#x221E;</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mi>&#x03A6;</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>h</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>&#x03A6;</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>I</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:mrow></mml:mstyle></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e028.tif"/></alternatives><label>[Eqn A7]</label></disp-formula>
<p>where:</p>
<list list-type="bullet">
<list-item><p><italic>&#x03BC;</italic>: <italic>unconditional mean of x<sub>t</sub></italic></p></list-item>
<list-item><p><italic>&#x03A6;<sub>h</sub></italic>: <italic>N&#x00D7;N MA coefficient matrices</italic></p></list-item>
<list-item><p><italic>I<sub>N</sub></italic>: <italic>N&#x00D7;N identity matrix</italic></p></list-item>
</list>
<p>A recursion for <italic>&#x03A6;<sub>h</sub></italic> is (<xref ref-type="disp-formula" rid="FDA8">Equation A8)</xref>:</p>
<disp-formula id="FDA8"><alternatives><mml:math display="block" id="MA8"><mml:mrow><mml:msub><mml:mi>&#x03A6;</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>p</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msub><mml:mi>&#x03A6;</mml:mi><mml:mrow><mml:mi>h</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mtext>&#x2009;</mml:mtext><mml:mi>f</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:mi>h</mml:mi><mml:mo>&#x2265;</mml:mo><mml:mn>1</mml:mn><mml:mo>;</mml:mo><mml:msub><mml:mi>&#x03A6;</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mtext>&#x2009;</mml:mtext><mml:mi>f</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:mi>h</mml:mi><mml:mo>&#x003C;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mstyle></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e029.tif"/></alternatives><label>[Eqn A8]</label></disp-formula>
<p>A.4.2 Selection vectors</p>
<p>Let <italic>e<sub>i</sub></italic> be an <italic>N &#x00D7;</italic> 1 selection vector with 1 in position i and 0 elsewhere (<xref ref-type="disp-formula" rid="FDA9">Equation A9)</xref>:</p>
<disp-formula id="FDA9"><alternatives><mml:math display="block" id="MA9"><mml:mrow><mml:msub><mml:mi>e</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>T</mml:mi></mml:msup></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e030.tif"/></alternatives><label>[Eqn A9]</label></disp-formula>
<p>where:</p>
<list list-type="bullet">
<list-item><p><italic>e<sub>i</sub></italic>: <italic>selection vector that extracts i &#x2013; th element of a</italic> <inline-formula id="I14"><alternatives><mml:math display="inline" id="IM14"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>v</mml:mi><mml:mi>e</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mi>t</mml:mi><mml:mi>r</mml:mi><mml:mi>i</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i014.tif"/></alternatives></inline-formula> <italic>expression</italic></p></list-item>
</list>
<p>A.4.3 Generalised FEVD (Pesaran&#x2013;Shin)</p>
<p>The generalised H-step-ahead FEVD entry is (<xref ref-type="disp-formula" rid="FDA10">Equation A10)</xref>:</p>
<disp-formula id="FDA10"><alternatives><mml:math display="block" id="MA10"><mml:mrow><mml:msubsup><mml:mi>&#x03B8;</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mi>H</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mi>&#x03C3;</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>H</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mi>e</mml:mi><mml:mi>i</mml:mi><mml:mi>T</mml:mi></mml:msubsup><mml:msub><mml:mi>&#x03A6;</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:mstyle displaystyle="true"><mml:msub><mml:mo>&#x2211;</mml:mo><mml:mi>u</mml:mi></mml:msub><mml:mrow><mml:msub><mml:mi>e</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>H</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mi>e</mml:mi><mml:mi>i</mml:mi><mml:mi>T</mml:mi></mml:msubsup><mml:msub><mml:mi>&#x03A6;</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:mstyle displaystyle="true"><mml:msub><mml:mo>&#x2211;</mml:mo><mml:mi>u</mml:mi></mml:msub><mml:mrow><mml:msubsup><mml:mi>&#x03A6;</mml:mi><mml:mi>h</mml:mi><mml:mi>T</mml:mi></mml:msubsup><mml:msub><mml:mi>e</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e031.tif"/></alternatives><label>[Eqn A10]</label></disp-formula>
<p>where:</p>
<list list-type="bullet">
<list-item><p><italic>H</italic>: <italic>forecast horizon</italic></p></list-item>
<list-item><p><italic>&#x03A3;<sub>u</sub></italic>: <italic>covariance matrix of VAR innovations</italic></p></list-item>
<list-item><p><italic>&#x03C3;<sub>jj</sub></italic>: <italic>j-th diagonal element of &#x03A3;<sub>u</sub></italic></p></list-item>
<list-item><p><inline-formula id="I15"><alternatives><mml:math display="inline" id="IM15"><mml:mrow><mml:msubsup><mml:mi>&#x03B8;</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mi>H</mml:mi></mml:msubsup></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i015.tif"/></alternatives></inline-formula>: <italic>share of i&#x2019;s H-step-ahead forecast-error variance due to shocks in j (order-invariant)</italic></p></list-item>
</list>
</sec>
<sec id="s20038">
<title>A5</title>
<p>Gaussian copula density (supports <xref ref-type="disp-formula" rid="FDA17">Equation [A17]</xref>)</p>
<p>Define <inline-formula id="I16"><alternatives><mml:math display="inline" id="IM16"><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>&#x03A6;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i016.tif"/></alternatives></inline-formula> and <italic>z<sub>t</sub></italic> = (<italic>z</italic><sub>1, <italic>t</italic></sub>, <italic>z</italic><sub>2, <italic>t</italic></sub>,)<sup><italic>T</italic></sup> :</p>
<disp-formula id="FDA11"><alternatives><mml:math display="block" id="MA11"><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>&#x03A6;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e032.tif"/></alternatives><label>[Eqn A11]</label></disp-formula>
<p>where:</p>
<list list-type="bullet">
<list-item><p><italic>u<sub>i,t</sub></italic>: <italic>probability integral transform</italic> (<italic>PIT</italic>), <inline-formula id="I17"><alternatives><mml:math display="inline" id="IM17"><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>;</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>&#x03B8;</mml:mi><mml:mo>&#x005E;</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i017.tif"/></alternatives></inline-formula></p></list-item>
<list-item><p><italic>&#x03A6;</italic><sup>(&#x2013;1)(.)</sup>: <italic>inverse standard normal CDF</italic></p></list-item>
</list>
<p>Let R be the copula correlation matrix. The Gaussian copula density is (<xref ref-type="disp-formula" rid="FDA12">Equation A12)</xref>:</p>
<disp-formula id="FDA12"><alternatives><mml:math display="block" id="MA12"><mml:mrow><mml:mi>c</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>;</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>&#x007C;</mml:mo><mml:mi>R</mml:mi><mml:msup><mml:mo>&#x007C;</mml:mo><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:msup><mml:mi>exp</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msubsup><mml:mi>z</mml:mi><mml:mi>t</mml:mi><mml:mi>T</mml:mi></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>&#x2212;</mml:mo><mml:msub><mml:mi>I</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy="false">)</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:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e033.tif"/></alternatives><label>[Eqn A12]</label></disp-formula>
<p>where:</p>
<list list-type="bullet">
<list-item><p>|<italic>R</italic>|: <italic>determinant of R</italic></p></list-item>
<list-item><p><italic>I<sub>2</sub></italic>: <italic>2&#x00D7;2 identity matrix</italic></p></list-item>
</list>
<p>For the bivariate case with correlation parameter &#x03C1;, an explicit form is (<xref ref-type="disp-formula" rid="FDA13">Equation A13)</xref>:</p>
<disp-formula id="FDA13"><alternatives><mml:math display="block" id="MA13"><mml:mrow><mml:mi>c</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>;</mml:mo><mml:mi>&#x03C1;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:msup><mml:mi>&#x03C1;</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac><mml:mo>&#x00D7;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>&#x03C1;</mml:mi><mml:msub><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>&#x2212;</mml:mo><mml:msup><mml:mi>&#x03C1;</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mi>Z</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mi>Z</mml:mi><mml:mn>2</mml:mn><mml:mn>2</mml:mn></mml:msubsup><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:msup><mml:mi>&#x03C1;</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e034.tif"/></alternatives><label>[Eqn A13]</label></disp-formula>
<p>This density is used inside the copula log-likelihood (<xref ref-type="disp-formula" rid="FDA14">Equation A14)</xref>:</p>
<disp-formula id="FDA14"><alternatives><mml:math display="block" id="MA14"><mml:mrow><mml:msub><mml:mi>&#x2113;</mml:mi><mml:mrow><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C1;</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</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:msubsup><mml:mrow><mml:mtext>In&#x2009;</mml:mtext><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>;</mml:mo><mml:mi>&#x03C1;</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mstyle></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e035.tif"/></alternatives><label>[Eqn A14]</label></disp-formula>
</sec>
<sec id="s20039">
<title>A6</title>
<p>Tail dependence interpretation notes (supports <xref ref-type="disp-formula" rid="FDA19">Equation [A19]</xref>)</p>
<p>Tail dependence is defined as (<xref ref-type="disp-formula" rid="FDA15">Equation A15)</xref>:</p>
<disp-formula id="FDA15"><alternatives><mml:math display="block" id="MA15"><mml:mrow><mml:mfrac><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mi>&#x03BB;</mml:mi><mml:mrow><mml:mi>U</mml:mi><mml:mo>=</mml:mo><mml:munder><mml:mrow><mml:mi>lim</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x03B1;</mml:mi><mml:mo>&#x2192;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo></mml:mrow></mml:munder><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mn>2</mml:mn><mml:mi>&#x03B1;</mml:mi><mml:mo>+</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03B1;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#x03B1;</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>&#x03B1;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>&#x03BB;</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>lim</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x03B1;</mml:mi><mml:mo>&#x2192;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03B1;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#x03B1;</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mi>&#x03B1;</mml:mi></mml:mfrac></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e036.tif"/></alternatives><label>[Eqn A15]</label></disp-formula>
<p>For the Gaussian copula with |&#x03C1;| &#x003C; 1, asymptotic tail dependence is zero (<xref ref-type="disp-formula" rid="FDA16">Equation A16</xref>):</p>
<disp-formula id="FDA16"><alternatives><mml:math display="block" id="MA16"><mml:mrow><mml:msub><mml:mi>&#x03BB;</mml:mi><mml:mi>U</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>&#x03BB;</mml:mi><mml:mi>L</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e037.tif"/></alternatives><label>[Eqn A16]</label></disp-formula>
<p>Accordingly, dependence is primarily summarised via &#x03C1; and Kendall&#x2019;s &#x03C4; in the Gaussian copula case.</p>
</sec>
<sec id="s20040">
<title>Frequency-domain connectedness definitions (supports <xref ref-type="disp-formula" rid="FDA20">Equation [A20]</xref>)</title>
<disp-formula id="FDA17"><alternatives><mml:math display="block" id="MA17"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mi>T</mml:mi><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>100</mml:mn><mml:mo>&#x00D7;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>N</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mstyle displaystyle="true"><mml:msub><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x2260;</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mo>&#x222B;</mml:mo><mml:mrow><mml:msub><mml:mi>&#x03C9;</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>&#x03C9;</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:msubsup><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>d</mml:mi><mml:mi>&#x03C9;</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</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:msubsup><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mo>&#x222B;</mml:mo><mml:mrow><mml:msub><mml:mi>&#x03C9;</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>&#x03C9;</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:msubsup><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>d</mml:mi><mml:mi>&#x03C9;</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:mstyle></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext>&#x2003;&#x2003;&#x2003;&#x2003;</mml:mtext><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>&#x03C9;</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>&#x03C9;</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e038.tif"/></alternatives><label>[Eqn A17]</label></disp-formula>
<p>Where;</p>
<disp-formula id="FDA21"><alternatives><mml:math display="block" id="MA21"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e039.tif"/></alternatives></disp-formula>
<p>is the generalised causation spectrum implied by the VAR. The objects needed to define <italic>f_</italic>(<italic>ij</italic>)(<italic>&#x03C9;</italic>),<italic>&#x03A8;</italic>(<italic>&#x03C9;</italic>),<italic>and S_x</italic>(<italic>&#x03C9;</italic>) are given below.</p>
<p>Let the VAR(p) admit the moving-average representation <inline-formula id="I18"><alternatives><mml:math display="inline" id="IM18"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x03BC;</mml:mi><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>&#x221E;</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mi>&#x03A6;</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>h</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mstyle></mml:mrow></mml:math><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-i018.tif"/></alternatives></inline-formula> with innovation covariance &#x03A3;_u.</p>
<p>The frequency response function (Fourier transform of the MA coefficients) is (<xref ref-type="disp-formula" rid="FDA18">Equation A18)</xref>:</p>
<disp-formula id="FDA18"><alternatives><mml:math display="block" id="MA18"><mml:mrow><mml:mi>&#x03A8;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>&#x221E;</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mi>&#x03A6;</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mi>i</mml:mi><mml:mi>&#x03C9;</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo>&#x2208;</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mi>&#x03C0;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#x03C0;</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mstyle></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e040.tif"/></alternatives><label>[Eqn A18]</label></disp-formula>
<p>The spectral density matrix of x_t is (<xref ref-type="disp-formula" rid="FDA19">Equation A19)</xref>:</p>
<disp-formula id="FDA19"><alternatives><mml:math display="block" id="MA19"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>x</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>2</mml:mn><mml:mi>&#x03C0;</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>&#x03A8;</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>&#x03A3;</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mi>&#x03A8;</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x002A;</mml:mo></mml:mrow></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e041.tif"/></alternatives><label>[Eqn A19]</label></disp-formula>
<p>where (&#x00B7;)^<sub>&#x002A;</sub> denotes the conjugate transpose.</p>
<p>Let e_i be the N &#x00D7; 1 selection vector (1 in position i, 0 elsewhere), and let &#x03C3;_(jj) be the j-th diagonal element of &#x03A3;_u. The generalised causation spectrum is:</p>
<disp-formula id="FDA20"><alternatives><mml:math display="block" id="MA20"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mi>f</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>&#x03C3;</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x005E;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x007C;</mml:mo><mml:mi>e</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x005E;</mml:mo><mml:mi>T</mml:mi><mml:mi>&#x03A8;</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>&#x03A3;</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mi>u</mml:mi><mml:mtext>&#x2009;</mml:mtext><mml:mi>e</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mi>j</mml:mi><mml:mo>&#x007C;</mml:mo><mml:mo>&#x005E;</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext>&#x2003;&#x2003;&#x2003;&#x2003;&#x2009;&#x2009;&#x2003;</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mi>e</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#x005E;</mml:mo><mml:mi>T</mml:mi><mml:mi>S</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>e</mml:mi><mml:mo>&#x005F;</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JEF-19-1108-e042.tif"/></alternatives><label>[Eqn A20]</label></disp-formula>
<p>Integrating <italic>f</italic>_(<italic>ij</italic>)(<italic>&#x03C9;</italic>) over a frequency band <italic>d</italic> = (<italic>&#x03C9;</italic>_1, <italic>&#x03C9;</italic>_2) yields the band-specific variance shares used to compute TS(d) in (<xref ref-type="disp-formula" rid="FDA20">Equation A20)</xref>.</p>
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<fn><p><bold>How to cite this article:</bold> De Wet, M.C., 2026, &#x2018;Dynamic linkages between systemic risk and real-estate investment trusts in major emerging markets&#x2019;, <italic>Journal of Economic and Financial Sciences</italic> 19(1), a1108. <ext-link ext-link-type="uri" xlink:href="https://doi.org/10.4102/jef.v19i1.1108">https://doi.org/10.4102/jef.v19i1.1108</ext-link></p></fn>
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