Andrea Renzetti, Department of Economics, Alma Mater Studiorium Universit`a di Bologna, Piazza Scaravilli 2, 40126 BologNA, Italy. Theory coherent TVP-VAR can be re-written as: (https://cdn.hackernoon.com/images/fWZa4tUiBGemnqQfBGgCPf9594N2-hpd31y3.png)

Andrea Renzetti, Department of Economics, Alma Mater Studiorium Universit`a di Bologna, Piazza Scaravilli 2, 40126 Bologna, Italy.

A.1.1 Time Varying Parameters by dummy observations

Starting from:

we can write the TVP-VAR in static compact form as:

Suppose we want to specify independent RW stochastic processes for all the coefficients in Φ as:

This is just another way of writing:

A.1.2 Population moments

A.1.3 Integrating constant of the theory coherent prior

The integrating constant of the Normal-Inverse-Wishart prior

A.1.4 Conditional distribution of theory coherent prior

Considering the three first blocks we get

A.1.5 Marginal likelihood and fit-complexity trade off

The marginal likelihood is given by:

Following the same steps as in (Giannone et al. 2015) it can be re-written as :

A.1.6 Formulas with distinct λj for j = 1, . . . ,K

A.2 Small scale New Keynesian model for the forecasting exercise

A.2.1 Data

A.2.2 Competing models in the forecasting exercise

The competing models in the out of sample forecasting exercise in Section 3 are

• A constant parameters VAR with flat prior.

• A constant parameters VAR with Normal Inverse-Wishart prior.

• A TVP-VAR model

The VAR with Normal Inverse-Wishart prior is given by:

A.2.3 Prior for the DSGE parameters

A.2.4 Posterior estimate for the DSGE parameters and IRFs from the TC-TVPVAR

A.3 Medium scale New Keynesian model

The model is taken from Del Negro et al. (2015) and it is a version of the popular medium scale New Keynesian model in Smets et al. (2007). The set of log-linearized equilibrium conditions of the model is