7. Proxy (instrumental) SVAR Modeling

A proxy_svar object is a structural VAR identified with external instruments (“proxies”): observed series that are correlated with a structural shock of interest but with no other shock. It extends the svar object, so data handling, estimation, forecasting and the various decompositions are exactly as described in Structural VAR Modeling and Reduced-form VAR Modeling; this page covers what is specific to the proxy case.

7.1. The model

The structural VAR is as usual,

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

and each proxy \(m_{t}\) is linked to one structural shock by

\[m_{t}=\beta _{m}\left( r_{t}\right) \varepsilon _{y,t}+\sigma _{m}\left( r_{t}\right) \varepsilon _{m,t},\]

with \(r_{t}=1,2,...,h\) and transition probabilities \(p_{r_{t},r_{t+1}}\left( I_{t}\right)\). Here \(\varepsilon_{y,t}\) is the structural shock the proxy instruments, \(\varepsilon_{m,t}\) is proxy measurement noise, \(\beta_{m}\) the relevance coefficient and \(\sigma_{m}\) the noise scale. The proxy series themselves are part of the time-series database passed to estimate.

7.2. Creating a proxy SVAR

The proxy structure is the first argument; the rest of the signature is the svar one:

mdl = proxy_svar(proxies, varlist)
mdl = proxy_svar(proxies, varlist, exog, nlags, constant, markov_chains)

proxies is an array of structs, one per instrument, with fields:

  • var – name of the endogenous variable whose shock the proxy instruments;

  • eqtn – the proxy-equation number (matching the declaration order of var);

  • coef – the name of the proxy relevance coefficient (\(\beta_{m}\) above);

  • shock_eqtn – the equation whose structural shock is related to the proxy.

For example, to instrument the monetary-policy shock (the shock in the R equation) with a high-frequency surprise series mp_surprise:

prox        = struct();
prox.var    = 'R';
prox.eqtn   = 1;
prox.coef   = 'beta_mp';
prox.shock_eqtn = 'R';

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

mdl = proxy_svar(prox, endog, {}, 4, true);

(varlist must be ordered consistently with the proxy specification.)

7.3. Estimation, IRFs, decompositions, forecasting

These are called exactly as for an svar – estimate (with an optional prior and any additional identifying restrictions on a0/a1/…), print_structural_form, irf, variance_decomposition, historical_decomposition, forecast, bootstrap – the proxy relevance and noise parameters (beta_*, sigma_*) are estimated alongside the VAR coefficients, and the proxy equations supply the identification of the instrumented shock(s). See the structural- and reduced-form VAR chapters for the call patterns and the plotting helpers.

7.4. Adding regime switching

A Markov-chain structure can be passed as the last argument, as for the svar object; the proxy relevance and noise (\(\beta_{m}\), \(\sigma_{m}\)) and/or the VAR coefficients can be made regime-dependent through controlled_parameters, and time-varying transition probabilities are specified exactly as in the reduced-form VAR chapter.

7.5. Technical documentation for proxy_svar objects

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