Testing for Unobserved Heterogeneity via k-means Clustering

Abstract

Clustering methods such as k-means have found widespread use in a variety of applications. This article proposes a split-sample testing procedure to determine whether a null hypothesis of a single cluster, indicating homogeneity of the data, can be rejected in favor of multiple clusters. The test is simple to implement, valid under mild conditions (including nonnormality, and heterogeneity of the data in aspects beyond those in the clustering analysis), and applicable in a range of contexts (including clustering when the time series dimension is small, or clustering on parameters other than the mean). We verify that the test has good size control in finite samples, and we illustrate the test in applications to clustering vehicle manufacturers and U.S. mutual funds.

Keywords:

• Classification methods
• Machine learning
• Model selection
• Overfitting

Supplementary Materials

The supplemental appendix contains additional theoretical and simulation results.

Acknowledgments

For helpful comments we thank the editor and two referees, as well as Tim Bollerslev and Jia Li.

Notes

1 Note that $c^{\star }$ depends on a variety of features of the problem: the true number of clusters $(G),$ the number of clusters used in the alternative ($\tilde{G}$), the separation of the true cluster means $(c),$ the dimension of the data $(d),$ and, for $d>1$, the within-variable correlation of the data $({\Sigma }_i),$ which can vary across $i=1,\dots ,n.$

2 An alternative approach for choosing a value of G to use in the alternative hypothesis is via cross-validation. In this approach, the sample is split into three subsamples: the first for estimation of cluster assignments across a range of values of $G;$ the second for choosing the optimal value ${\widehat{G}}^{\ast };$ the third for testing G = 1 versus $G={\widehat{G}}^{\ast }.$ Such an approach avoids the need for a Bonferroni adjustment, which can cost power, but involves using splitting the data across three subsamples rather than two, which can also cost power. We leave a detailed investigation of such an approach for future research.

3 The factor data is available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.