Costationarity and stationarity tests for stock index returns

We present a new analysis of the FTSE and SP500 stock index log return series and provide evidence that they are not stationary. We then discover two time-varying linear combinations of the FTSE and SP500 series that are stationary and hence declare the two series to be costationary. The stationary combinations are themselves worthy of study using classical time series methods. The existence of costationarity between time series implies an interesting and interpretable stochastic relationship between the series. We exhibit and interpret this relation for the FTSE and SP500 series. Given two (or more) locally stationary time series we introduce a method to discover, possibly time-varying, linear combinations that are stationary (costationary series). To determine costationarity we develop two new bootstrap stationarity tests and demonstrate their good empirical performance. Our method searches for the simplest time-varying linear combination of two (or more) series that results in a stationary series. The search is carried out via a staged numerical optimization procedure that starts with simple time-varying linear combinations and progressively increase the complexity until stationarity is discovered. Additionally, we introduce direct methods for estimating the evolutionary cross-spectrum and cross-covariance of locally stationary wavelet processes. ∗C1, Math. Dept., University of Bristol, UK. Email: {a.cardinali,g.p.nason}@bristol.ac.uk