Asymptotic Behavior of Temporal Aggregation in Mixed-Frequency Datasets

Here, we present an unexplored issue regarding temporal aggregation. When a model contains frequency-dependent coefficients, such as a distinct long- and short-term coefficient, temporal aggregation leads to inconsistent least squares estimates. Because the sub-sampled variable's spectrum is equal to its folded original spectrum, the low-frequency variable may exhibit a mixture of distinct linear relations for a given frequency. We propose a new method to disentangle the frequencies superposition based on band spectrum regression, thus avoiding the inconsistency problem. As a result, we can test for the presence of frequency-dependent coefficients. We use stationary and non-stationary linear semi-parametric models to demonstrate our findings. Our Monte Carlo simulations show good finite sample size and power properties. Finally, our empirical study rejects the presence of a single coefficient for all frequencies between quarterly GDP and monthly US indicators.