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On the Ordering of Dynamic Principal Components and the Implications for Portfolio Analysis

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Abstract

When principal component analysis is used on a rolling or conditional setting, ordering and incoherence issues may emerge. We provide empirical evidence supporting this claim and introduce an algorithm that allows dynamic reordering of the principal components (PCs). We provide additional results that shed light on the consequences of incoherence when analyzing the link between PCs and macroeconomic risk factors, with a focus on the COVID-19 pandemic period. When PCs are optimally reordered, the roles of factors emerge more clearly, with relevant implications for risk management.

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Notes

  1. An alternative to using the rolling application of sample moment estimation is represented by the estimation of a Multivariate Generalized AutoRegressive Conditional Heteroskedastic (MGARCH) model and the subsequent estimation of the spectral decomposition of the estimated conditional covariances. We discuss this approach in the following section.

  2. To simplify the notation, we do not report the dimension of the zero vector, which changes according to the dimensions of the loading vectors.

  3. Ordering issues may arise in all frameworks in which a model for the conditional covariance or conditional correlation is first estimated in the data and then used to recover conditional eigenvectors.

  4. In addition to the GARCH(1,1) specification, we can flexibly employ any other univariate GARCH specification, with the possibility of including exogenous regressors and leverage effects (Piero Aielli & Caporin, 2015).

  5. We describe the dataset employed in our study in Sect. 3.

  6. Our reordering algorithm builds on the idea of increasing the stability and persistence of the structure characterizing the so-called dominant cluster. Note that alternative approaches can be used to reorder the columns of the \(\widehat{\varvec{L}}_t\) matrix. For instance, as simpler method, we can adopt the mean relevance along the time interval [1, T] of the K sectors as a target (i.e. the mean of the \(\widehat{w}_{k,1,j},\widehat{w}_{k,2,j},\cdots , \widehat{w}_{k,T,j}\) weights, for \(k=1,\ldots ,K\)). Therefore, we can switch the columns of \(\widehat{\varvec{L}}_t\) to minimize the deviations around each sector mean, with improvements in terms of stability. We also implement such approach as a robustness check, finding that it underperforms our original algorithm. In particular, the alternative approach does not properly work in case of peaks (whose magnitude can be also relevant) that emerge during short time periods.

  7. The data were recovered from Refinitiv Eikon.

  8. The data on the variables from (i) to (vii) were recovered from the Kenneth R. French library at https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

  9. The data on the variables from (viii) to (xi), (xv), and (xvi) were recovered from Refinitiv Eikon, and the data on the variables from (xii) to (xiv) were recovered from the Federal Reserve Bank of St. Louis at https://fred.stlouisfed.org.

  10. We recovered the data on the variables from (xvii) to (xx) from the Office of Financial Research of the US Department of the Treasury at https://www.financialresearch.gov/financial-stress-index/.

  11. We recovered the data on the variables in (xxi) and (xxii) from https://www.policyuncertainty.com.

  12. We recovered the data on the variables from (xxiii) to (xxv) from the COVID-19 Data Repository of the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University at https://github.com/CSSEGISandData/COVID-19.

  13. We found that this setup fits the dynamics of the estimated components well.

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Acknowledgements

We thank the participants of the 9th International Conference on Risk Analysis (ICRA9) for the important comments and suggestions.

Funding

Massimiliano Caporin acknowledges financial support from the MIUR projects “PRIN 2017 - HiDEA: Advanced Econometrics for High-frequency Data, 2017RSMPZZ” and “PRIN 2022 - PRICE: A New Paradigm for High-Frequency Finance, 2022C799SX”. In addition, this study was funded by the European Union - NextGenerationEU, in the framework of the GRINS - Growing Resilient, INclusive and Sustainable project (GRINS PE00000018 – CUP C93C22005270001). The views and opinions expressed are solely those of the authors and do not necessarily reflect those of the European Union, nor can the European Union be held responsible for them.

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Author notes

  1. Giovanni Bonaccolto and Massimiliano Caporin these authors contributed equally to this work.

Authors and Affiliations

  1. Department of Economics and Law, “Kore” University of Enna, Cittadella Universitaria, Enna, 94100, Italy

    Giovanni Bonaccolto

  2. Department of Statistical Sciences, University of Padova, via Cesare Battisti 241, Padova, 35121, Italy

    Massimiliano Caporin

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  1. Giovanni Bonaccolto
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  2. Massimiliano Caporin
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Correspondence to Giovanni Bonaccolto.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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Appendix A List of the companies included in our dataset

Appendix A List of the companies included in our dataset

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Table 1 Companies and economic sector
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Bonaccolto, G., Caporin, M. On the Ordering of Dynamic Principal Components and the Implications for Portfolio Analysis. Ann Oper Res (2024). https://doi.org/10.1007/s10479-024-06030-4

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  • DOI: https://doi.org/10.1007/s10479-024-06030-4

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