This paper proposes a novel Bayesian approach to jointly model realized data and survey forecasts of the same variable in a vector autoregression (VAR). In particular, our method imposes a prior distribution on the consistency between the forecast implied by the VAR and the survey forecast for the same variable. When the prior is placed on unconditional forecasts from the VAR, the prior shapes the posterior of the reduced-form VAR coefficients. When the prior is placed on conditional forecasts (specifically, impulse responses), the prior shapes the posterior of the structural VAR coefficients.
To implement our prior, we combine importance sampling with a maximum entropy prior for forecast consistency to obtain posterior draws of VAR parameters at low computational cost. We use two empirical examples to illustrate some potential applications of our methodology: (i) the evolution of tail risks for inflation in a time-varying parameter VAR model and (ii) the identification of forward guidance shocks using sign and forecast-consistency restrictions in a monetary VAR model.