Vector autoregression with varied frequency data

The Vector Autoregression (VAR) model has been extensively applied in macroeconomics. A typical VAR requires its component variables being sampled at a uniformed frequency, regardless of the fact that some macro data are available monthly and some are only quarterly. Practitioners invariably align variables to the same frequency either by aggregation or imputation, regardless of information loss or noises gain. We study a VAR model with varied frequency data in a Bayesian context. Lower frequency (aggregated) data are essentially a linear combination of higher frequency (disaggregated) data. The observed aggregated data impose linear constraints on the autocorrelation structure of the latent disaggregated data. The perception of a constrained multivariate normal distribution is crucial to our Gibbs sampler. Furthermore, the Markov property of the VAR series enables a block Gibbs sampler, which performs faster for evenly aggregated data. Lastly, our approach is applied to two classic structural VAR analyses, one with long-run and the other with short-run identification constraints. These applications demonstrate that it is both feasible and sensible to use data of different frequencies in a new VAR model, the one that keeps the branding of the economic ideas underlying the structural VAR model but only makes minimum modification from a technical perspective.