Structural VAR (SVAR) Model

A reduced-form VAR estimates a set of forecast-error covariance matrices and moving-average coefficients, but those objects are silent about causation. The residual vector \($u_t$\) bundles every contemporaneous structural disturbance into a single composite innovation for each variable. Sims (1980) proposed the VAR framework to let the data speak, but extracting economic stories - a monetary policy shock, an oil supply disruption, a technology impulse - requires additional restrictions that map the reduced-form residuals \($u_t$\) into orthogonal structural shocks \($\varepsilon_t$\). That mapping is the identification problem. This model extends the reduced-form VAR, already documented in the VAR page of this platform.

The core issue is algebraic. A VAR with \(n\) variables has a reduced-form covariance matrix \($\Sigma_u$\) with \(n(n+1)/2\) unique elements. The structural impact matrix \($B_0^{-1}$\) that maps \($\varepsilon_t$\) into \($u_t$\) via \($u_t = B_0^{-1} \varepsilon_t$\) has \($n^2$\) free parameters. The orthogonality assumption \($E[\varepsilon_t \varepsilon_t'] = I_n$\) delivers \($\Sigma_u = B_0^{-1} B_0^{-1\prime}$\), which supplies \(n(n+1)/2\) equations for \($n^2$\) unknowns. The gap is \(n(n-1)/2\) restrictions that must come from economic theory, institutional knowledge, or external data. Every identification scheme - recursive ordering, sign restrictions, narrative instruments, long-run neutrality, heteroskedasticity-based approaches - fills exactly this gap.

Recursive identification (Cholesky decomposition of \($\Sigma_u$\)) imposes a causal ordering: variables ordered earlier cannot respond contemporaneously to shocks hitting variables ordered later. Sims (1980) and Christiano, Eichenbaum, and Evans (1999) used recursive structures to study monetary policy. Sign restrictions (Uhlig 2005; Faust 1998) replace zero restrictions with inequality constraints on impulse-response signs over specified horizons, accepting set identification rather than point identification. External instruments / proxy SVAR (Stock and Watson 2012; Mertens and Ravn 2013) use an outside variable correlated with one structural shock but uncorrelated with the others, recovering the column of \($B_0^{-1}$\) associated with that shock via instrumental variables on the reduced-form residuals.

Central banks lean on SVAR identification daily. The Federal Reserve Bank of New York publishes oil-price shock decompositions using Kilian's (2009) recursive SVAR. The ECB's monetary policy analysis unit estimates sign-restricted SVARs for euro-area transmission. Academic work on fiscal multipliers (Blanchard and Perotti 2002) uses institutional timing to identify government spending shocks. Proxy SVARs have become the default for identifying monetary policy shocks after Gertler and Karadi (2015) demonstrated the approach using high-frequency futures surprises as instruments.

The starting point is an estimated reduced-form VAR: \($y_t = c + A_1 y_{t-1} + \cdots + A_p y_{t-p} + u_t$\) with \($E[u_t u_t'] = \Sigma_u$\). Everything about SVAR identification concerns the relationship \($u_t = B_0^{-1} \varepsilon_t$\), where \($\varepsilon_t$\) is a vector of mutually uncorrelated structural shocks with unit variance. The structural impact matrix \($B_0^{-1}$\) (sometimes written \($A_0^{-1}$\) or \(S\)) maps structural shocks into the observable residuals. Each column of \($B_0^{-1}$\) gives the instantaneous response of every variable to one structural shock.

The identification scheme determines which \($B_0^{-1}$\) (or which set of matrices) is admissible. Recursive identification sets \($B_0^{-1}$\) equal to the lower-triangular Cholesky factor of \($\Sigma_u$\), imposing \(n(n-1)/2\) zero restrictions on the upper triangle. Sign restrictions constrain the signs of selected entries of the impulse-response function \($\Theta_h = \Phi_h B_0^{-1}$\) at horizons \(h = 0, 1, $\ldots, H$\), where \($\Phi_h$\) is the reduced-form MA coefficient at horizon \(h\). Because sign restrictions typically do not pin down a unique \($B_0^{-1}$\), the identified set is characterized by rotating through all orthogonal matrices \(Q\) such that \(B_0^{-1} = \tilde{P} Q\) satisfies the sign constraints, where \(\tilde{P}\) is the Cholesky factor of \($\Sigma_u$\). Proxy SVAR uses an external instrument \($z_t$\) satisfying relevance \($E[z_t \varepsilon_{1t}] \neq 0$\) and exogeneity \($E[z_t \varepsilon_{jt}] = 0$\) for \(j $\neq 1$\), and recovers the first column of \($B_0^{-1}$\) up to scale via two-stage least squares on the reduced-form residuals.

Structural impulse-response functions are the primary output. The response of variable \(i\) to structural shock \(j\) at horizon \(h\) is \($\theta_{ij,h} = e_i$' $\Phi_h B_0^{-1} e_j$\). Forecast error variance decompositions attribute the \(h\)-step-ahead forecast error of each variable to each structural shock. Historical decompositions allocate the realized path of each variable to the cumulative contribution of each structural shock over the sample period. Confidence intervals come from bootstrap methods - the recursive-design bootstrap and the wild bootstrap are standard for recursive SVAR; the Bayesian approach of drawing from the posterior over \(Q\) is standard for sign restrictions.

The Federal Reserve system uses SVAR identification extensively. Kilian (2009) decomposed oil-price fluctuations into supply, aggregate-demand, and oil-specific-demand shocks using a recursive SVAR with production-ordering restrictions, producing the most cited oil-shock decomposition in the literature. Gertler and Karadi (2015) pioneered the proxy SVAR for monetary policy using high-frequency fed-funds-futures surprises around FOMC announcements as external instruments, and this design has become the benchmark for monetary policy identification at the Fed, ECB, and Bank of England. Blanchard and Perotti (2002) exploited institutional features of the US tax and transfer system to identify fiscal shocks, establishing the timing-restriction approach that underpins most fiscal-multiplier estimates used in policy analysis.

Sign-restricted SVARs are standard at institutions that need to remain agnostic about specific causal orderings. The ECB's structural analysis uses Bayesian sign-restricted VARs to decompose euro-area GDP into demand, supply, and monetary-policy contributions without committing to a particular Cholesky ordering. The Bank of Canada and Reserve Bank of Australia publish scenario analyses built on sign-identified technology and demand shocks. Researchers use sign restrictions when the economic theory provides directional predictions (an expansionary monetary shock raises output and lowers the interest rate) but not zero-impact restrictions.

SVAR breaks down when the number of variables is large relative to the sample, because the reduced-form VAR itself becomes unreliable. For systems with more than 8-10 variables, factor-augmented VARs (FAVAR) or Bayesian VARs with Minnesota-type priors are preferred. The method also struggles when structural breaks or time-varying parameters are present - the fixed-coefficient assumption means the identified shocks blend pre- and post-break behavior. If the researcher suspects regime-dependent transmission, a Markov-switching SVAR or TVP-VAR is more appropriate.

The choice between identification schemes depends on what the researcher is willing to assume. Recursive ordering is clean and delivers point identification but requires a defensible causal ordering - a strong demand if the timing of contemporaneous interactions is genuinely ambiguous. Sign restrictions avoid ordering assumptions but deliver set identification, meaning the impulse responses are intervals rather than point estimates, and these intervals can be wide. Proxy SVAR identifies a single shock without restricting the rest of the system but requires an external variable that satisfies both relevance and exogeneity, which is hard to find outside of financial-market-surprise settings.