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Principal Component Models for Generating Large GARCH Covariance Matrices

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  • Carol Alexander

Abstract

type="main" xml:lang="en"> The implementation of multivariate GARCH models in more than a few dimensions is extremely difficult: because the model has many parameters, the likelihood function becomes very flat, and consequently the optimization of the likelihood becomes practicably impossible. There is simply no way that full multivariate GARCH models can be used to estimate directly the very large covariance matrices that are required to net all the risks in a large trading book. This paper begins by describing the principal component GARCH or ‘orthogonal GARCH’ (O-GARCH) model for generating large GARCH covariance matrices that was first introduced in Alexander and Chibumba (1996) and subsequently developed in Alexander (2000, 2001b). The O-GARCH model is an accurate and efficient method for generating large covariance matrices that only requires the estimation of univariate GARCH models. Hence, it has many practical advantages, for example in value–at–risk models. It works best in highly correlated systems, such as term structures. The purpose of this paper is to show that, if sufficient care is taken with the initial calibration of the model, equities and foreign exchange rates can also be included in one large covariance matrix. Simple conditions for the final covariance matrix to be positive semi-definite are derived. (J.E.L.: C32, C53, G19, G21, G28).

Suggested Citation

  • Carol Alexander, 2002. "Principal Component Models for Generating Large GARCH Covariance Matrices," Economic Notes, Banca Monte dei Paschi di Siena SpA, vol. 31(2), pages 337-359, July.
  • Handle: RePEc:bla:ecnote:v:31:y:2002:i:2:p:337-359
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    File URL: http://hdl.handle.net/10.1111/1468-0300.00089
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    Cited by:

    1. repec:eee:empfin:v:45:y:2018:i:c:p:243-268 is not listed on IDEAS
    2. Claudio, Morana, 2018. "Regularized semiparametric estimation of high dimensional dynamic conditional covariance matrices," Working Papers 382, University of Milano-Bicocca, Department of Economics, revised 04 Jun 2018.
    3. Martin Vojtek, 2004. "Calibration of Interest Rate Models - Transition Market Case," Finance 0410015, University Library of Munich, Germany.
    4. Claudio Morana, 2014. "New insights on the US OIS spreads term structure during the recent financial turmoil," Applied Financial Economics, Taylor & Francis Journals, vol. 24(5), pages 291-317, March.
    5. Anthony H. Tu & Cathy Yi-Hsuan Chen, 2016. "What Derives the Bond Portfolio Value-at-Risk: Information Roles of Macroeconomic and Financial Stress Factors," SFB 649 Discussion Papers SFB649DP2016-006, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    6. Redouane Elkamhia & Denitsa Stefanova, 2011. "Dynamic Correlation or Tail Dependence Hedging for Portfolio Selection," Tinbergen Institute Discussion Papers 11-028/2/DSF10, Tinbergen Institute.
    7. Márcio Poletti Laurini & Roberto Baltieri Mauad & Fernando Antonio Lucena Aiube, 2016. "Multivariate Stochastic Volatility-Double Jump Model: an application for oil assets," Working Papers Series 415, Central Bank of Brazil, Research Department.
    8. Claudio, Morana, 2015. "Semiparametric Estimation of Multivariate GARCH Models," Working Papers 317, University of Milano-Bicocca, Department of Economics, revised 10 Dec 2015.
    9. Anshul Verma & Riccardo Junior Buonocore & Tiziana di Matteo, 2017. "A cluster driven log-volatility factor model: a deepening on the source of the volatility clustering," Papers 1712.02138, arXiv.org, revised May 2018.

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