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.