Testing diffusion processes for non-stationarity

Financial data are often assumed to be generated by diffusions. Using recent results of Fan et al. (J Am Stat Assoc, 102:618–631, 2007; J Financ Econometer, 5:321–357, 2007) and a multiple comparisons procedure created by Benjamini and Hochberg (J R Stat Soc Ser B, 59:289–300, 1995), we develop a test for non-stationarity of a one-dimensional diffusion based on the time inhomogeneity of the diffusion function. The procedure uses a single sample path of the diffusion and involves two estimators, one temporal and one spatial. We first apply the test to simulated data generated from a variety of one-dimensional diffusions. We then apply our test to interest rate data and real exchange rate data. The application to real exchange rate data is of particular interest, since a consequence of the law of one price (or the theory of purchasing power parity) is that real exchange rates should be stationary. With the exception of the GBP/USD real exchange rate, we find evidence that interest rates and real exchange rates are generally non-stationary. The software used to implement the estimation and testing procedure is available on demand and we describe its use in the paper.