Using dynamic mode decomposition to extract cyclic behavior in the stock market

The presence of cyclic expansions and contractions in the economy has been known for over a century. The work reported here searches for similar cyclic behavior in stock valuations. The variations are subtle and can only be extracted through analysis of price variations of a large number of stocks. Koopman mode analysis is a natural approach to establish such collective oscillatory behavior. The difficulty is that even non-cyclic and stochastic constituents of a finite data set may be interpreted as a sum of periodic motions. However, deconvolution of these irregular dynamical facets may be expected to be non-robust, i.e., to depend on specific data set. We propose an approach to differentiate robust and non-robust features in a time series; it is based on identifying robust features with reproducible Koopman modes, i.e., those that persist between distinct sub-groupings of the data. Our analysis of stock data discovered four reproducible modes, one of which has period close to the number of trading days/year. To the best of our knowledge these cycles were not reported previously. It is particularly interesting that the cyclic behaviors persisted through the great recession even though phase relationships between stocks within the modes evolved in the intervening period.