Long range dependence in daily stock returns

The tests of Robinson (Journal of the American Statistical Association, 89, 1420-37, 1994a) are used to analyse the degree of dependence in the intertemporal structure of daily stock returns (defined as the first difference of the logarithm of stock prices, where the series being considered is the S&P500 index). These tests have several distinguishing features compared with other procedures for testing for unit (or fractional) roots. In particular, they have a standard null limit distribution and they are the most efficient ones when carried out against the appropriate alternatives. In addition, they allow the incorporation of the Bloomfield (Biometrika, 60, 217-226, 1973) exponential spectral model for the underlying I(0) disturbances. The full sample, which comprises 17 000 observations, is first divided in 10 subsamples of 1700 observations each. These are then grouped two by two, and five by five; finally, the whole sample is considered. The results indicate that the degree of dependence remains relatively constant over time, with the order of integration of stock returns fluctuating slightly above or below zero. On the whole, there is very little evidence of fractional integration, despite the length of the series. Therefore, it appears that the standard practice of taking first differences when modelling stock returns might be adequate.