Clustering heteroskedastic time series by model-based procedures.

*(English)*Zbl 1452.62784Summary: Financial time series are often characterized by similar volatility structures. The detection of clusters of series displaying similar behavior could be important in understanding the differences in the estimated processes, without having to study and compare the estimated parameters across all the series. This is particularly relevant when dealing with many series, as in financial applications. The volatility of a time series can be characterized in terms of the underlying GARCH process. Using Wald tests and the Autoregressive metrics to measure the distance between GARCH processes, it is shown that it is possible to develop a clustering algorithm, which can provide three classifications (with increasing degree of deepness) based on the heteroskedastic patterns of the time series. The number of clusters is detected automatically and it is not fixed a priori or a posteriori. The procedure is evaluated by simulations and applied to the sector indices of the Italian market.

##### MSC:

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62H30 | Classification and discrimination; cluster analysis (statistical aspects) |

62-08 | Computational methods for problems pertaining to statistics |

##### Software:

itsmr
PDF
BibTeX
Cite

\textit{E. Otranto}, Comput. Stat. Data Anal. 52, No. 10, 4685--4698 (2008; Zbl 1452.62784)

Full Text:
DOI

##### References:

[1] | Agrawal, R.; Faloutsos, C.; Swami, A., Efficient similarity search in sequence databases, Lecture notes in computer science, 69-84, (1994) |

[2] | Arnott, R.; Fabozzi, F.J., Asset allocation: A handbook of portfolio policies, strategies and tactics, (1988), Probus Publishing Company Chicago, Illinois |

[3] | Billio, M.; Caporin, M.; Gobbo, M., Flexible dynamic conditional correlation multivariate GARCH models for asset allocation, Applied financial economics letters, 2, 123-130, (2006) |

[4] | Bollerslev, T., Generalized autoregressive conditional heteroskedasticity, Journal of econometrics, 31, 307-327, (1986) · Zbl 0616.62119 |

[5] | Bollerslev, T.; Engle, R.F.; Nelson, D., ARCH models, (), 2959-3038 |

[6] | Brockwell, P.J.; Davis, R.A., Introduction to time series and forecasting, (1996), Springer-Verlag New York · Zbl 0868.62067 |

[7] | Brooks, C.; Burke, S.P.; Persand, G., Benchmarks and the accuracy of GARCH model estimation, International journal of forecasting, 17, 45-56, (2001) |

[8] | Caiado, J.; Crato, N.; Peña, D., A periodogram-based metric for time series classification, Computational statistics and data analysis, 50, 2668-2684, (2006) · Zbl 1445.62222 |

[9] | Chenoweth, T.; Hubata, R.; St Louis, R., The power of tests for equivalent ARMA models: the implications for practitioners, Empirical economics, 29, 281-292, (2004) |

[10] | Corduas, M.; Piccolo, D., Time series clustering and classification by the autoregressive metric, Computational statistics and data analysis, 52, 1860-1872, (2008) · Zbl 1452.62624 |

[11] | Dalla Valle, L.; Giudici, P., A Bayesian approach to estimate the marginal loss distributions in operational risk management, Computational statistics and data analysis, 52, 3107-3127, (2008) · Zbl 1452.62788 |

[12] | Davies, D.; Bouldin, D., A cluster separation measure, IEEE transactions pattern recognition machine intelligent, 1, 224-227, (1979) |

[13] | Dunn, J., Well separated clusters and optimal fuzzy partitions, Journal of cybernetics, 4, 95-104, (1974) · Zbl 0304.68093 |

[14] | Engle, R., Autoregressive conditional heteroskedasticity with estimates of the variance of U K inflation, Econometrica, 50, 987-1008, (1982) |

[15] | Engle, R., Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models, Journal of business and economic statistics, 20, 339-350, (2002) |

[16] | Forbes, K.J.; Rigobon, R., No contagion, only interdependence: measuring stock market comovements, The journal of finance, 67, 2223-2261, (2002) |

[17] | Gallo, G.M.; Otranto, E., Volatility transmission across markets: A multi-chain Markov switching model, Applied financial economics, 17, 659-670, (2007) |

[18] | Gallo, G.M.; Otranto, E., Volatility spillovers, interdependence and comovements: A Markov switching approach, Computational statistics and data analysis, 58, 3011-3026, (2008) · Zbl 1452.62766 |

[19] | Gray, A.; Markel, J., Distance measures for speech processing, IEEE transactions on acoustic, speech and signal processing, ASSP-24, 380-391, (1976) |

[20] | Hubert, L.; Schultz, J., Quadratic assignment as a general data-analysis strategy, British journal of mathematics, statistics and psychology, 29, 190-241, (1976) · Zbl 0356.92027 |

[21] | Jain, A.; Murty, M.; Flynn, P., Data clustering: A review, ACM computational surveys, 31, 264-323, (1999) |

[22] | Liao, T., Clustering time series data: A survey, Pattern recognition, 38, 1857-1874, (2005) · Zbl 1077.68803 |

[23] | Luenberger, D.G., Linear and nonlinear programming, (1984), Addison-Wesley Reading, MA · Zbl 0571.90051 |

[24] | Maharaj, E.A., A significance test for classifying ARMA models, Journal of statistical computation and simulation, 54, 305-331, (1996) · Zbl 0899.62116 |

[25] | Maharaj, E.A., Comparison and classification of stationary multivariate time series, Pattern recognition, 32, 1129-1138, (1999) |

[26] | Maharaj, E.A., Clusters of time series, Journal of classification, 17, 297-314, (2000) · Zbl 1017.62079 |

[27] | Otranto, E., Classifying the markets volatility with ARMA distance measures, Quaderni di statistica, 6, 1-19, (2004) |

[28] | Otranto, E.; Triacca, U., Testing for equal predictability of stationary ARMA processes, Journal of applied statistics, 34, 1091-1108, (2007) |

[29] | Pattarin, F.; Paterlini, S.; Minerva, T., Clustering financial time series: an application to mutual funds style analysis, Computational statistics and data analysis, 47, 353-372, (2004) · Zbl 1429.62476 |

[30] | Piccolo, D., A distance measure for classifying ARIMA models, Journal of time series analysis, 11, 153-164, (1990) · Zbl 0691.62083 |

[31] | Piccolo, D., Statistical Issues on the AR Metric in Time Series Analysis. Proceedings of the SIS 2007 intermediate conference “Risk and Prediction”, 2007, pp. 221-232 |

[32] | Steece, B.; Wood, S., A test for the equivalence of k ARMA models, Empirical economics, 10, 1-11, (1985) |

[33] | Theodoridis, S.; Koutroumbas, K.K., Pattern recognition, (1998), Academic Press · Zbl 0954.68131 |

[34] | Xiong, Y., Yeung, D.-Y., Mixtures of ARMA models for model-based time series clustering. In: Proceedings of the IEEE International Conference on Data Mining, 2002 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.