6.1. Modelling structural VARs

An svar object represents a (possibly Markov-switching) structural vector autoregression – the structural form is parameterised and estimated directly, so there is no separate “estimate a reduced form, then rotate it” step. Almost everything else (data handling, estimation options, forecasting, the various decompositions, fan charts, bootstrap, Bayesian estimation, adding regime switching) works exactly as for reduced-form VARs; see Reduced-form VAR Modeling for the full worked example. This page covers what is specific to the structural case.

6.1.1. The model

\[A_{0}\left( r_{t}\right) y_{t}=C\left( r_{t}\right) x_{t}+A_{1}\left( r_{t}\right) y_{t-1}+...+A_{p}\left( r_{t}\right) y_{t-p}+\varepsilon _{t}\]

with \(r_{t}=1,2,...,h\) and transition probabilities \(p_{r_{t},r_{t+1}}\left( I_{t}\right)\), and structural shocks \(\varepsilon_{t}\sim N\left(0,I\right)\).

\(A_{0}\left(r_{t}\right)\) is the contemporaneous-impact matrix. It is normalised to have a unit diagonal (a0_i_i = 1), which fixes the scale of each structural shock.
\(A_{1},...,A_{p}\) are the lag-coefficient matrices, \(C\) the coefficients on the deterministic / exogenous block \(x_{t}\) (a constant, any declared exogenous regressors, and lags of \(y\)).

The estimated parameters are named by analogy with the matrices:

a0(row,col) – the contemporaneous coefficients (off-diagonal entries of \(A_{0}\); the diagonal is fixed at 1);
a1(row,col), …, ap(row,col) – the lag coefficients (a(row,col), with the lag index omitted, refers to all lags);
c(row,col) – the coefficients on the exogenous / constant block;

where row and col are integers or endogenous-variable names, e.g. a0(R,PAI) or a1(2,3).

6.1.2. Creating an SVAR

endog = {'PAIOIL','GROWTH','PAI','R','EXRATE'};

exog  = {};

nlags = 4;

const = true;

mdl = svar(endog, exog, nlags, const);

(The constructor signature mirrors rfvar’s: svar(varlist, exog, nlags, constant, markov_chains), the last two being optional.)

6.1.3. Identification

A structural VAR with \(n\) endogenous variables has \(n(n-1)/2\) more free parameters in \(A_{0}\) than the data can pin down, so identifying restrictions must be supplied. In RISE these are ordinary parameter restrictions on a0 (and, if you wish, on the lag coefficients), passed to estimate in the linear-restrictions cell array – the same mechanism used for block-exogeneity restrictions in the reduced-form VAR chapter.

A recursive (Cholesky-type) identification orders the variables and zeroes the entries of \(A_{0}\) above the diagonal, so that variable 1 is not affected contemporaneously by any other shock, variable 2 only by shock 1, and so on:

linres = {};
for ii = 1:numel(endog)
    for jj = ii+1:numel(endog)
        linres{end+1,1} = sprintf('a0(%s,%s)=0', endog{ii}, endog{jj}); %#ok<SAGROW>
    end
end

Exclusion (zero) restrictions can equally be imposed one at a time, e.g. 'a0(PAIOIL,R)=0' (“the policy-rate shock has no contemporaneous effect on oil-price inflation”), and you can mix them with restrictions on the lag coefficients ('a1(PAIOIL,GROWTH)=0', …).

6.1.4. Estimation

Estimation is exactly as for a reduced-form VAR – pass the model, a database of time series, the estimation sample, an (optional) prior, and the identifying restrictions:

data_range = {db.GROWTH.start, db.GROWTH.finish};

mdlest = estimate(mdl, db, data_range, [], linres);     % classical

% Bayesian: build a prior with svar.prior_template(), e.g.
var_prior        = svar.prior_template();
var_prior.type   = 'sz';                                % Sims-Zha prior
prior            = struct('var', var_prior);
mdlest_bayes     = estimate(mdl, db, data_range, prior, linres);

After estimation, inspect the structural form with:

print_structural_form(mdlest)

6.1.5. Impulse responses, variance and historical decompositions, forecasting

Because the model is already structural, no identification function is needed: the shock names are simply the structural shocks (one per endogenous variable, by RISE’s naming convention), and irf / variance_decomposition / historical_decomposition / forecast are called directly:

myirfs = irf(mdlest);                       % all shocks, default horizon
myirfs = irf(mdlest, shock_names, 40);

vd = variance_decomposition(mdlest);

hd = historical_decomposition(mdlest);

fkst = forecast(mdlest, db, '2003Q1');

For plotting (quick_irfs, plot_fanchart, plot_decomp, …), parameter uncertainty via bootstrap, Bayesian posterior sampling, fan charts of the decompositions and IRFs, and conditional forecasting, see the reduced-form VAR chapter – the calls are identical, with a0/a1/… parameters in place of the reduced-form b1/b2/… ones, and without the extra Rfunc (identification) argument.

6.1.6. Adding regime switching

As for the reduced-form VAR, pass a Markov-chain structure as the fifth argument and list the parameters it controls – for an SVAR these are typically a0 (switching contemporaneous transmission) and/or a1, …, ap:

mc = struct();
mc.name                      = 'policy';
mc.number_of_states          = 2;
mc.controlled_parameters     = {'a0(R,:)'};
mc.endogenous_probabilities  = [];
mc.probability_parameters    = [];

mdl = svar(endog, exog, nlags, const, mc);

Time-varying transition probabilities are specified exactly as in the reduced-form VAR chapter (an endogenous_probabilities definition plus the probability_parameters that enter it), and the switching parameters are given priors through prior.nonvar.

Proxy / instrumental SVARs are handled by the related proxy SVAR object, and panels of (structural) VARs by the panel VAR object.