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

(Also known in the literature as BVAR-DSGE; the modern factory
is ``dsge_var_model``.)

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

The DSGE-VAR (also known as BVAR-DSGE) is a methodology that
combines a BVAR and a DSGE model following Del Negro and
Schorfheide (2004) and Del Negro, Schorfheide, Smets and Wouters
(2007).

There are two possible interpretations: the DSGE is used as a
*prior* for the BVAR, or the BVAR serves to *relax* the tight
restrictions in the DSGE.

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

* the VAR model,
* the VAR approximation of the DSGE model,
* the DSGE model,
* the BVAR model -- the VAR with the (VAR approximation of the)
  DSGE as prior.

The DSGE can be a model with a simple instrument rule (e.g. a
Taylor rule) or an optimal policy under commitment or under
discretion. It can be stationary or non-stationary.

.. contents::
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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 = dsge_var_model(dsgemodel, ...
        lag_length    = nlags, ...
        constant_term = constant);

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

::

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

Priors
~~~~~~~

::

    priors = struct();

    % priors on the DSGE parameters
    priors.kappa = {0.2, 0.5, 0.5, 'gamma'};
    priors.psi   = {1.5, 2,   0.5, '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-prior weight
    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, ...
        estim_priors      = priors, ...
        data              = db, ...
        data_demean       = true, ...
        estim_start_date  = date2serial('1960Q2'), ...
        estim_end_date    = date2serial('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'});

Choosing the VAR representation
-------------------------------

A DSGE-VAR carries three reduced-form representations, and the
analytics path (``irf``, variance and historical decompositions)
draws from whichever you select with the ``which_var`` option:

* ``'var_dsge'`` *(default)* -- the Bayesian DSGE-VAR combination:
  the data VAR shrunk toward the DSGE-implied restrictions with
  weight ``dsge_prior_weight``. This is the BVAR-DSGE proper, and it
  is what ``irf(mdlest)`` returns above.
* ``'var'`` -- the pure data VAR (OLS), the
  ``dsge_prior_weight -> 0`` limit. Use it to see what the data say
  with no DSGE shrinkage.
* ``'var_approx'`` -- the DSGE-implied VAR, the
  ``dsge_prior_weight -> inf`` limit. Use it to see the DSGE's own
  finite-order VAR approximation.

Select a representation with ``set`` and run the analytics as
usual::

    myirfs_data_var = irf(set(mdlest, 'which_var', 'var'));
    myirfs_dsge_var = irf(set(mdlest, 'which_var', 'var_approx'));

The two limits bracket the Bayesian combination: ``'var'`` ignores
the DSGE prior, ``'var_approx'`` ignores the data, and the default
``'var_dsge'`` interpolates between them according to
``dsge_prior_weight``.