Published

2015-01-01

Identification of Common Factors in Multivariate Time Series Modeling

Identificación de factores comunes en la modelización multivariante de series temporales

DOI:

https://doi.org/10.15446/rce.v38n1.48812

Keywords:

Cointegration, Factor Analysis, Stationarity (en)
Cointegración, Estacionariedad, Factores comunes, Modelo factorial dinámico. (es)

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Authors

  • Mariano González Universidad CEU Cardenal Herrera, Valencia, Spain
  • Juan M. Nave Universidad de Castilla La Mancha, Cuenca, Spain

For multivariate time series modelling, it is essential to know the number of common factors that define the behaviour. The traditional approach to this problem is investigating the number of cointegration relations among the data by determining the trace and the maximum eigenvalue and obtaining the number of stationary long-run relations. Alternatively, this problem can be analyzed using dynamic factor models, which involves estimating the number of common factors, both stationary and not, that describe the behaviour of the data. In this context, we empirically analyze the power of such alternative approaches by applying them to time series that are simulated using known factorial models and to financial market data. The results show that when there are stationary common factors, when the number of observations is reduced and/or when the variables are part of more than one cointegration relation, the common factors test is more powerful than the usually applied cointegration tests. These results, together with the greater flexibility to identify the loading matrix of the data generating process, render dynamic factor models more suitable for use in multivariate time series analysis.

Para la modelización multivariante de series temporales no estacionarias es imprescindible conocer el número de factores comunes que definen el comportamiento de las series. La forma tradicional de abordar este problema es el estudio de las relaciones de cointegración entre los datos a través de las pruebas de la traza y el máximo valor propio, obteniendo el número de relaciones de largo plazo estacionarias. Como alternativa, se pueden emplear modelos factoriales dinámicos que estiman el número de factores comunes, estacionarios o no, que describen el comportamiento de los datos. En este contexto, analizamos empíricamente el resultado de aplicar tales métodos a series simuladas mediante modelos factoriales conocidos, y a datos reales de los mercados financieros. Los resultados muestran que cuando hay factores comunes estacionarios, cuando el número de observaciones se reduce y/o cuando las variables participan en más de una relación de cointegración, la prueba de factores comunes es más potente que las pruebas habituales de cointegración. Estos resultados, junto con la mayor flexibilidad para identificar la matriz de cargas del proceso generador de datos, hacen que los modelos de factores dinámicos sean más adecuados para su utilización en el análisis multivariante.

https://doi.org/10.15446/rce.v38n1.48812

Identification of Common Factors in Multivariate Time Series Modeling

Identificación de factores comunes en la modelización multivariante de series temporales

MARIANO GONZÁLEZ1, JUAN M. NAVE2

1Universidad CEU Cardenal Herrera, Facultad de Derecho y Empresa, Departamento de Economía y Empresa, Valencia, Spain. Tenure Professor. Email: mariano.gonzalez@uch.ceu.es
2Universidad de Castilla La Mancha, Facultad de Ciencias Sociales, Departamento de Análisis Económico y Finanzas, Cuenca, Spain. Professor. Email: juan.nave@uclm.es

Abstract

For multivariate time series modelling, it is essential to know the number of common factors that define the behaviour. The traditional approach to this problem is investigating the number of cointegration relations among the data by determining the trace and the maximum eigenvalue and obtaining the number of stationary long-run relations. Alternatively, this problem can be analyzed using dynamic factor models, which involves estimating the number of common factors, both stationary and not, that describe the behaviour of the data. In this context, we empirically analyze the power of such alternative approaches by applying them to time series that are simulated using known factorial models and to financial market data. The results show that when there are stationary common factors, when the number of observations is reduced and/or when the variables are part of more than one cointegration relation, the common factors test is more powerful than the usually applied cointegration tests. These results, together with the greater flexibility to identify the loading matrix of the data generating process, render dynamic factor models more suitable for use in multivariate time series analysis.

Key words: Cointegration, Factor Analysis, Stationarity.

Resumen

Para la modelización multivariante de series temporales no estacionarias es imprescindible conocer el número de factores comunes que definen el comportamiento de las series. La forma tradicional de abordar este problema es el estudio de las relaciones de cointegración entre los datos a travé de las pruebas de la traza y el máximo valor propio, obteniendo el número de relaciones de largo plazo estacionarias. Como alternativa, se pueden emplear modelos factoriales dinámicos que estiman el número de factores comunes, estacionarios o no, que describen el comportamiento de los datos. En este contexto, analizamos empíricamente el resultado de aplicar tales métodos a series simuladas mediante modelos factoriales conocidos, y a datos reales de los mercados financieros. Los resultados muestran que cuando hay factores comunes estacionarios, cuando el número de observaciones se reduce y/o cuando las variables participan en más de una relación de cointegración, la prueba de factores comunes es más potente que las pruebas habituales de cointegración. Estos resultados, junto con la mayor flexibilidad para identificar la matriz de cargas del proceso generador de datos, hacen que los modelos de factores dinímicos sean más adecuados para su utilización en el análisis multivariante.

Palabras clave: cointegración, estacionariedad, factores comunes, modelo factorial dinámico.

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References

1. Ahlgren, N. & Antell, J. (2008), 'Bootstrap and fast double bootstrap tests of cointegration rank with financial time series', Computational Statistics and Data Analysis 52(10), 4754-4767.

2. Baillie, R. T. & Bollerslev, T. (1994), 'Cointegration, fractional cointegration, and exchange rate dynamics', Journal of Finance 49(2), 737-745.

3. Banerjee, A., Dolado, J. J. & Mestre, R. (1998), 'Error correction mechanism tests for cointegration in a single-equation framework', Journal of Time Series Analysis 19(3), 267-283.

4. Bauer, D. & Wagner, M. (2009), 'Using subspace algorithm cointegration analysis: Simulation performance and application to term structure', Computational Statistics and Data Analysis 53(6), 1954-1973.

5. Bayer, C. & Hanck, C. (2013), 'Combining non-cointegration tests', Journal of Time Series Analysis 34(1), 83-95.

6. Cavaliere, G. & Taylor, A. M. (2006), 'Testing the null of co-integration in the presence of variance breaks', Journal of Time Series Analysis 27(4), 613-636.

7. Chen, Y. P., Huang, H. C. & Tu, I. P. (2010), 'A new approach for selecting the number of factors', Computational Statistics and Data Analysis 54(12), 2990-2998.

8. Cheung, Y. W. & Lay, K. S. (1993), 'Finite-sample sizes of Johansen's likelihood ratio tests for cointegration', Oxford Bulletin of Economics 55(3), 313-328.

9. Correal, M. E. & Peña, D. (2008), 'Thresold dynamic factor model', Revista Colombiana de Estadística 31(2), 183-192.

10. Cubadda, G. (2007), 'A unifying framework for analysing common cyclical features in cointegrated time series', Computational Statistics and Data Analysis 52(2), 896-906.

11. Davidson, J. & Monticini, A. (2007), 'Test for cointegration with structural breaks base on subsamples', Computational Statistics and Data Analysis 54(11), 2498-2511.

12. Diebold, F. X., Gardeazabal, J. & Yilmaz, K. (1994), 'On cointegration and exchange rate dynamics', Journal of Finance 49(2), 727-735.

13. Dittmann, I. (2000), 'Residual-based tests for fractional cointegration: A Monte Carlo study', Journal of Time Series Analysis 21(6), 615-647.

14. Doornik, J. A. & O'Brien, R. J. (2002), 'Numerically stable cointegration analysis', Computational Statistics and Data Analysis 41(1), 185-193.

15. Engle, R. F. & Granger, C. W. J. (1987), 'Cointegration and error correction: Representation, estimation and testing', Econometrica 55(2), 251-276.

16. Escribano, A. & Peña, D. (1994), 'Cointegration and common factors', Journal of Time Series Analysis 15(6), 577-586.

17. Forni, M., Hallin, M., Lippi, M. & Reichlin, L. (2005), 'The generalized dynamic-factor model: One-sided estimation and forecasting', Journal of the American Statistical Association 100(471), 577-586.

18. Gonzalo, J. (1994), 'Five alternative methods of estimating long-run equilibrium relationshisps', Journal of Econometrics 60(1-2), 203-233.

19. Gonzalo, J. & Granger, C. W. J. (1995), 'Estimation of common long-memory components in cointegrated systems', Journal of Business and Economic Statistics 13(1), 27-35.

20. Gonzalo, J. & Lee, T. H. (1998), 'Pitfalls in testing for long-run relationshisps', Journal of Econometrics 86(1), 129-154.

21. González, M. & Nave, J. M. (2010), 'Portfolio immunization using Independent Component Analysis', The Spanish Review of Financial Economics 21(1), 37-46.

22. Hallin, M. & Liska, R. (2007), 'Determining the number of factors in the general dynamic factor model', Journal of the American Statistical Association 102(478), 603-617.

23. Hu, Y. P. & Chou, R. J. (2003), 'A dynamic factor model', Journal of Time Series Analysis 24(5), 529-538.

24. Hu, Y. P. & Chou, R. J. (2004), 'On the Peña-Box model', Journal of Time Series Analysis 25(6), 811-830.

25. Jing, L. & Junsoo, L. (2010), 'ADL tests for threshold cointegration', Journal of Time Series Analysis 31(4), 241-254.

26. Kapetanios, G. & Marcellino, M. (2009), 'A parametric estimation method for dynamic factor models of large dimensions', Journal of Time Series Analysis 30(2), 208-238.

27. Lansangan, J. R. & Barrios, E. B. (2009), 'Principal components analysis of nonstationary time series data', Statistics and Computing 19(2), 173-187.

28. Li, Q., Pan, J. & Yao, Q. (2009), 'On determination of cointegration ranks', Statistics and Its Interface 2(1), 45-56.

29. Lopes, H. F., Gamerman, D. & Salazar, E. (2011), 'Generalized spatial dynamic factor models', Computational Statistics and Data Analysis 55(3), 1319-1330.

30. Lorenzo-Seva, U., Timmerman, M. E. & Kiers, H. A. (2011), 'The Hull method for selecting the number of common factors', Multivariate Behavioral Research 46(2), 340-364.

31. Miller, J. I. (2010), 'Cointegrating regressions with messy regressors and an application to mixed-frecuency series', Journal of Time Series Analysis 31(4), 255-277.

32. Pan, J. & Yao, Q. (2010), 'Modelling multiple time series via common factors', Biometrika 95(2), 365-379.

33. Park, B., Mammen, E., Hardle, W. & Borak, S. (2009), 'Modelling dynamic semiparametric factor models', Journal of the American Statistical Association 104(485), 284-298.

34. Park, S., Ahn, S. K. & Cho, S. (2011), 'Modelling dynamic semiparametric factor models', Computational Statistics and Data Analysis 55(9), 2065-2618.

35. Peña, D. & Box, G. E. P. (1987), 'Identifying a simplifying structure in time series', Journal of the American Statistical Association 82(399), 863-843.

36. Peña, D. & Poncela, P. (2006), 'Nonstationary dynamic factor analysis', Journal of Statistical Planning and Inference 136(4), 1237-1257.

37. Peña, D. & Sanchez, I. (2007), 'Measuring the advantages of multivariate versus univariate forecasts', Journal of Time Series Analysis 28(6), 886-909.

38. Pesavento, E. (2007), 'Residuals-based tests for the null of no-cointegration: An analytical comparision', Journal of Time Series Analysis 28(1), 111-137.

39. Stock, J. H. & Watson, M. W. (1988), 'Testing for common trends', Journal of the American Statistical Association 83(404), 1097-1107.

40. Trenkler, C., Saikkonen, P. & Lütkepohl, H. (2007), 'Testing for the cointegrating rank of a VAR process with level shift and trend break', Journal of Time Series Analysis 29(2), 331-358.

41. Westerlund, J. & Egderton, D. L. (2006), 'New improved tests for cointegration with structural breaks', Journal of Time Series Analysis 28(2), 188-224.

42. Widaman, K. F. (1993), 'Common Factors Analysis versus Principal Component Analysis: Differential bias in representing model', Multivariate Behavioral Research 28(3), 263-311.

43. Zhang, H. (2009), 'Comparación entre dos métodos de reducción de dimensionalidad en series de tiempo', Revista Colombiana de Estadística 32(2), 189-212.

[Recibido en diciembre de 2013. Aceptado en noviembre de 2014]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv38n1a12,
    AUTHOR  = {González, Mariano and Nave, Juan M.},
    TITLE   = {{Identification of Common Factors in Multivariate Time Series Modeling}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2015},
    volume  = {38},
    number  = {1},
    pages   = {219-238}
}

References

Ahlgren, N. & Antell, J. (2008), ‘Bootstrap and fast double bootstrap tests of cointegration rank with financial time series’, Computational Statistics and Data Analysis 52(10), 4754–4767.

Baillie, R. T. & Bollerslev, T. (1994), ‘Cointegration, fractional cointegration, and exchange rate dynamics’, Journal of Finance 49(2), 737–745.

Banerjee, A., Dolado, J. J. & Mestre, R. (1998), ‘Error correction mechanism tests for cointegration in a single-equation framework’, Journal of Time Series Analysis 19(3), 267–283.

Bauer, D. & Wagner, M. (2009), ‘Using subspace algorithm cointegration analysis: Simulation performance and application to term structure’, Computational Statistics and Data Analysis 53(6), 1954–1973.

Bayer, C. & Hanck, C. (2013), ‘Combining non-cointegration tests’, Journal of Time Series Analysis 34(1), 83–95.

Cavaliere, G. & Taylor, A. M. (2006), ‘Testing the null of co-integration in the presence of variance breaks’, Journal of Time Series Analysis 27(4), 613–636.

Chen, Y. P., Huang, H. C. & Tu, I. P. (2010), ‘A new approach for selecting the number of factors’, Computational Statistics and Data Analysis 54(12), 2990–2998.

Cheung, Y. W. & Lay, K. S. (1993), ‘Finite-sample sizes of Johansen’s likelihood ratio tests for cointegration’, Oxford Bulletin of Economics 55(3), 313–328.

Correal, M. E. & Peña, D. (2008), ‘Thresold dynamic factor model’, Revista Colombiana de Estadística 31(2), 183–192.

Cubadda, G. (2007), ‘A unifying framework for analysing common cyclical features in cointegrated time series’, Computational Statistics and Data Analysis 52(2), 896–906.

Davidson, J. & Monticini, A. (2007), ‘Test for cointegration with structural breaks base on subsamples’, Computational Statistics and Data Analysis 54(11), 2498–2511.

Diebold, F. X., Gardeazabal, J. & Yilmaz, K. (1994), ‘On cointegration and Exchange rate dynamics’, Journal of Finance 49(2), 727–735.

Dittmann, I. (2000), ‘Residual-based tests for fractional cointegration: A Monte Carlo study’, Journal of Time Series Analysis 21(6), 615–647.

Doornik, J. A. & O’Brien, R. J. (2002), ‘Numerically stable cointegration analysis’, Computational Statistics and Data Analysis 41(1), 185–193.

Engle, R. F. & Granger, C. W. J. (1987), ‘Cointegration and error correction: Representation, estimation and testing’, Econometrica 55(2), 251–276.

Escribano, A. & Peña, D. (1994), ‘Cointegration and common factors’, Journal of Time Series Analysis 15(6), 577–586.

Forni, M., Hallin, M., Lippi, M. & Reichlin, L. (2005), ‘The generalized dynamicfactor model: One-sided estimation and forecasting’, Journal of the American Statistical Association 100(471), 577–586.

González, M. & Nave, J. M. (2010), ‘Portfolio immunization using Independent Component Analysis’, The Spanish Review of Financial Economics 21(1), 37–46.

Gonzalo, J. (1994), ‘Five alternative methods of estimating long-run equilibrium relationshisps’, Journal of Econometrics 60(1-2), 203–233.

Gonzalo, J. & Granger, C. W. J. (1995), ‘Estimation of common long-memory components in cointegrated systems’, Journal of Business and Economic Statistics 13(1), 27–35.

Gonzalo, J. & Lee, T. H. (1998), ‘Pitfalls in testing for long-run relationshisps’, Journal of Econometrics 86(1), 129–154.

Hallin, M. & Liska, R. (2007), ‘Determining the number of factors in the general dynamic factor model’, Journal of the American Statistical Association 102(478), 603–617.

Hu, Y. P. & Chou, R. J. (2003), ‘A dynamic factor model’, Journal of Time Series Analysis 24(5), 529–538.

Hu, Y. P. & Chou, R. J. (2004), ‘On the Peña-Box model’, Journal of Time Series Analysis 25(6), 811–830.

Jing, L. & Junsoo, L. (2010), ‘ADL tests for threshold cointegration’, Journal of Time Series Analysis 31(4), 241–254.

Kapetanios, G. & Marcellino, M. (2009), ‘A parametric estimation method for dynamic factor models of large dimensions’, Journal of Time Series Analysis 30(2), 208–238.

Lansangan, J. R. & Barrios, E. B. (2009), ‘Principal components analysis of nonstationary time series data’, Statistics and Computing 19(2), 173–187.

Li, Q., Pan, J. & Yao, Q. (2009), ‘On determination of cointegration ranks’, Statistics and Its Interface 2(1), 45–56.

Lopes, H. F., Gamerman, D. & Salazar, E. (2011), ‘Generalized spatial dynamic factor models’, Computational Statistics and Data Analysis 55(3), 1319–1330.

Lorenzo-Seva, U., Timmerman, M. E. & Kiers, H. A. (2011), ‘The Hull method for selecting the number of common factors’, Multivariate Behavioral Research 46(2), 340–364.

Miller, J. I. (2010), ‘Cointegrating regressions with messy regressors and an application to mixed-frecuency series’, Journal of Time Series Analysis 31(4), 255–277.

Pan, J. & Yao, Q. (2010), ‘Modelling multiple time series via common factors’, Biometrika 95(2), 365–379.

Park, B., Mammen, E., Hardle, W. & Borak, S. (2009), ‘Modelling dynamic semiparametric factor models’, Journal of the American Statistical Association 104(485), 284–298.

Park, S., Ahn, S. K. & Cho, S. (2011), ‘Modelling dynamic semiparametric factor models’, Computational Statistics and Data Analysis 55(9), 2065–2618.

Peña, D. & Box, G. E. P. (1987), ‘Identifying a simplifying structure in time series’, Journal of the American Statistical Association 82(399), 863–843.

Peña, D. & Poncela, P. (2006), ‘Nonstationary dynamic factor analysis’, Journal of Statistical Planning and Inference 136(4), 1237–1257.

Peña, D. & Sanchez, I. (2007), ‘Measuring the advantages of multivariate versus univariate forecasts’, Journal of Time Series Analysis 28(6), 886–909.

Pesavento, E. (2007), ‘Residuals-based tests for the null of no-cointegration: An analytical comparision’, Journal of Time Series Analysis 28(1), 111–137.

Stock, J. H. & Watson, M. W. (1988), ‘Testing for common trends’, Journal of the American Statistical Association 83(404), 1097–1107.

Trenkler, C., Saikkonen, P. & Lütkepohl, H. (2007), ‘Testing for the cointegrating rank of a VAR process with level shift and trend break’, Journal of Time Series Analysis 29(2), 331–358.

Westerlund, J. & Egderton, D. L. (2006), ‘New improved tests for cointegration with structural breaks’, Journal of Time Series Analysis 28(2), 188–224.

Widaman, K. F. (1993), ‘Common Factors Analysis versus Principal Component Analysis: Differential bias in representing model’, Multivariate Behavioral Research 28(3), 263–311.

Zhang, H. (2009), ‘Comparación entre dos métodos de reducción de dimensionalidad en series de tiempo’, Revista Colombiana de Estadística 32(2), 189–212.

How to Cite

APA

González, M. and Nave, J. M. (2015). Identification of Common Factors in Multivariate Time Series Modeling. Revista Colombiana de Estadística, 38(1), 219–237. https://doi.org/10.15446/rce.v38n1.48812

ACM

[1]
González, M. and Nave, J.M. 2015. Identification of Common Factors in Multivariate Time Series Modeling. Revista Colombiana de Estadística. 38, 1 (Jan. 2015), 219–237. DOI:https://doi.org/10.15446/rce.v38n1.48812.

ACS

(1)
González, M.; Nave, J. M. Identification of Common Factors in Multivariate Time Series Modeling. Rev. colomb. estad. 2015, 38, 219-237.

ABNT

GONZÁLEZ, M.; NAVE, J. M. Identification of Common Factors in Multivariate Time Series Modeling. Revista Colombiana de Estadística, [S. l.], v. 38, n. 1, p. 219–237, 2015. DOI: 10.15446/rce.v38n1.48812. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/48812. Acesso em: 24 apr. 2024.

Chicago

González, Mariano, and Juan M. Nave. 2015. “Identification of Common Factors in Multivariate Time Series Modeling”. Revista Colombiana De Estadística 38 (1):219-37. https://doi.org/10.15446/rce.v38n1.48812.

Harvard

González, M. and Nave, J. M. (2015) “Identification of Common Factors in Multivariate Time Series Modeling”, Revista Colombiana de Estadística, 38(1), pp. 219–237. doi: 10.15446/rce.v38n1.48812.

IEEE

[1]
M. González and J. M. Nave, “Identification of Common Factors in Multivariate Time Series Modeling”, Rev. colomb. estad., vol. 38, no. 1, pp. 219–237, Jan. 2015.

MLA

González, M., and J. M. Nave. “Identification of Common Factors in Multivariate Time Series Modeling”. Revista Colombiana de Estadística, vol. 38, no. 1, Jan. 2015, pp. 219-37, doi:10.15446/rce.v38n1.48812.

Turabian

González, Mariano, and Juan M. Nave. “Identification of Common Factors in Multivariate Time Series Modeling”. Revista Colombiana de Estadística 38, no. 1 (January 1, 2015): 219–237. Accessed April 24, 2024. https://revistas.unal.edu.co/index.php/estad/article/view/48812.

Vancouver

1.
González M, Nave JM. Identification of Common Factors in Multivariate Time Series Modeling. Rev. colomb. estad. [Internet]. 2015 Jan. 1 [cited 2024 Apr. 24];38(1):219-37. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/48812

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2. Giovanna Capizzi. (2016). Discussion of “Statistical process monitoring of time-dependent data”. Quality Engineering, 28(1), p.146. https://doi.org/10.1080/08982112.2015.1100476.

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