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Clustering space-time series: FSTAR as a flexible STAR approach

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Abstract

The STAR model is widely used to represent the dynamics of a certain variable recorded at several locations at the same time. Its advantages are often discussed in terms of parsimony with respect to space-time VAR structures because it considers a single coefficient for each time and spatial lag. This hypothesis can be very strong; we add a certain degree of flexibility to the STAR model, providing the possibility for coefficients to vary in groups of locations. The new class of models (called Flexible STAR–FSTAR) is compared to the classical STAR and the space-time VAR by simulations and an application.

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Notes

  1. Necessary and sufficient conditions are much more complicated, so, as in Arbia et al. (2011), we impose only the usual necessary conditions.

  2. We would like to thank an anonymous referee and the Associate Editor who called this problem to our attention.

  3. The number of spatial units might seem small, but this choice is consistent with the selection of balanced space-to-time ratios in STARMA models illustrated in Otranto and Gallo (1994). They show, simulating data from STAR(1,1) processes in regular lattices, that, in correspondence of a time span \(T=100\), a number of locations higher than 25 can cause an ill-conditioning of the covariance matrix of the estimators, with a determinant close-to-zero. This problem is even greater in the presence of spatial autocorrelation between the disturbances because the collinearity problems could arise also in presence of a large T.

  4. Other classical contiguity criteria are the bishop criterion (common vertex) and the queen criterion (common edge or vertex); see Anselin (1988), Sect. 3.1.2.

  5. We have also used the bishop contiguity matrix and the results are similar and in many cases worse than the queen case. Results available on request.

  6. In the wrong weight matrix case, we fix \(T=1000\) as in the previous subsection.

  7. The value of the variance is the estimated variance of the FSTAR model shown in Table 7; the coefficient \(\rho \) is the average of the 12 Moran indices (see Anselin 1988) of each spatial unit.

  8. We are very grateful to Francesco Giorgianni and Gianluca Trifirò who have produced and made available this data set.

  9. The other time series have a very similar behaviour; they are available on request.

  10. Moreover, given the relationship between STAR and VAR models, as shown in Eq. (2.3), we estimate also a Sparse VAR (SVAR) model, shrinking some coefficients toward 0. The results are not good, compared with respect to the spatial models, particularly in terms out-of-sample forecasting, so we do not report the corresponding results. Details are available on request.

  11. Data provided by Regione Campania—A.A.L. Caserta. The ageing index is calculated as the number of persons 60 years old or over per hundred persons under age 15. The old-age dependency ratio is the number of persons 65 years and over per one hundred persons 15–64 years.

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Authors and Affiliations

  1. Department of Economics and CRENoS, University of Messina, Piazza Pugliatti 1, 98122, Messina, Italy

    Edoardo Otranto

  2. Department of Economics, University of Messina, Piazza Pugliatti 1, 98122, Messina, Italy

    Massimo Mucciardi

Authors
  1. Edoardo Otranto
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  2. Massimo Mucciardi
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Correspondence to Massimo Mucciardi.

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Otranto, E., Mucciardi, M. Clustering space-time series: FSTAR as a flexible STAR approach. Adv Data Anal Classif 13, 175–199 (2019). https://doi.org/10.1007/s11634-018-0314-5

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  • DOI: https://doi.org/10.1007/s11634-018-0314-5

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