A discrete-time model of high-frequency stock returns

A model of intraday financial time series is developed. The model is a dynamic factor model consisting of two equations. First, a rate of return of a 'stock' in a single day is assumed to be generated by several common factors plus some additive error ('intraday equation'). Secondly, the joint distribution of those common factors is assumed to depend on the hidden state of the day, which fluctuates according to a Markov chain ('day-by-day equation'). Together the equations compose a hidden Markov model. We investigate properties of the model. Among them is a central limit theorem for cumulative returns, which agrees with the well-known empirical phenomenon in the stock markets that the distributions of longer-horizon returns are closer to the normal. We propose a two-step procedure consisting of the method of principal components and the EM algorithm to estimate the model parameters as well as the unobservable states. In addition, we propose a procedure for predicting intraday returns. Finally, the model is fitted to empirical data, the StandardtkPoors 500 Index 5 min return data, to see if the model is capable of describing intraday movements of the index.