Temporal aggregation of random walk processes and implications for economic analysis

This paper examines the impact of time averaging and interval sampling data assuming that the data generating process for a given series follows a random walk with iid errors. We provide exact expressions for the corresponding variances, and covariances, for both levels and higher order differences of the aggregated series, as well as that for the variance ratio, demonstrating exactly how the degree of temporal aggregation impacts these properties. We empirically investigate this issue on exchange rates and find that the values of the variance ratios and autocorrelation coefficients at different frequencies are consistent with our theoretical results. We also conduct a simulation exercise that illustrates the potential effect that conditional heteroskedasticity and fat tails may have on the temporal aggregation of a random walk and of a highly persistent autoregressive process.