DSGE-VAR Modeling
==================

Description
-------------

The DSGE-VAR or elswhere known as BVAR-DSGE is a methodology that combines a BVAR and a DSGE model following the methodology
in :cite:`DelNegroSchorfheide2004` and :cite:`DelNegroSSW2007`.

There are two possible interpretations. One is that the DSGE is used as a prior for the BVAR model and
the other is that the BVAR serves to relax the tight restrictions in the DSGE model.

In the end we have four sub-models in one object:

	- The VAR model
	- The VAR approximation of the DSGE model
	- The DSGE model
	- The BVAR model i.e. the VAR model with the (VAR approximation of the) DSGE as prior

In RISE, the DSGE model can be a model with a simple instrument rule (e.g. Taylor rule) or an optimal policy
under commitment or under discretion.

The DSGE model can be stationary or nonstationary.

A quick-start example
----------------------

A simple New Keynesian DSGE model
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

::

	dsgemodel={
	'model:New Keynesian model'
	'@endogenous X "Output gap" R "interest rate" P "Inflation" G U'
	'@exogenous EG "Demand shock" EU "Monetary Policy shock"'
	'@parameters beta "discount factor" kappa "Phillips curve slope" sigu sigg rhou rhog psi'
	'@observables  P R'
	'@model'
	'	P = beta*P{+1}+kappa*X;'	
	'	X = X{+1}-(R-P{+1}-G);'
	'	R = psi*P+U;'
	'	U = rhou*U{-1} + sigu*EU;'
	'	G = rhog*G{-1} + sigg*EG;'
	    };


Setting up the BVAR-DSGE model
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

::

	nlags=4;
	
	constant=false;
	
	mdl=bvar_dsge(dsgemodel,nlags,constant);

Fixed parameters
~~~~~~~~~~~~~~~~~~~~~

::

	mdl=set(mdl,'parameters',{'beta',0.96});

Setting up and visualing the priors
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

::

	priors=struct();
	
	% ___priors on the DSGE parameters___
	priors.kappa={0.2,0.5,0.5,'gamma'};
	priors.psi={1.5,2,.5,'gamma'};
	priors.rhou={0.75,0.75,0.1,'beta'};
	priors.rhog={0.75,0.75,0.1,'beta'};
	priors.sigu={0.01,0.01,4,'sichisq'}; 
	priors.sigg={0.01,0.01,4,'sichisq'};
	
	% ___prior on the dsge model___
	priors.dsge_prior_weight={3,3,1,'gamma'};
	
	plotOpts=struct();
	plotOpts.prior_trunc=2e-3;
	rdist.plot(priors,plotOpts)

Collecting and transforming the data
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

::

	d=fetch_fred({'CPALTT01USQ661S','BOGZ1FL072052006Q'});
	
	db=struct();
	
	db.P=log(d(1).series/lag(d(1).series,1));
	
	db.R=d(2).series/100;

Maximizing the posterior
~~~~~~~~~~~~~~~~~~~~~~~~~

::

	mdlest=estimate(mdl,'priors',priors,'data_demean',true,'data',db,...
	    'estim_start_date','1960Q2','estim_end_date','2022Q3');

IRFs of the BVAR-DSGE at the maximized posterior
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
::

	myirfs_bvar_dsge=irf(mdlest);

IRFs of the DSGE model at the maximized posterior
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

::

	myirfs_dsge=irf(mdlest.dsge);

IRF comparison
~~~~~~~~~~~~~~~~~

::

	myirfs=ts.concatenator(myirfs_bvar_dsge,myirfs_dsge)
	
	quick_irfs(mdlest.dsge,myirfs,{'P','R'})		


Technical documentation for dsge_var objects
----------------------------------------------------

.. toctree::
   :maxdepth: 2
   :caption: Contents:

   bvar_dsge_properties_methods