Start from the reduced-form VAR(p):

The reduced-form residuals \(u_t\) are composites of the underlying structural shocks \(\varepsilon_t\). Define the structural model as \(B_0 y_t = \Gamma_0 + \Gamma_1 y_{t-1} + \cdots + \Gamma_p y_{t-p} + \varepsilon_t\), where \(B_0\) is invertible and \(E[\varepsilon_t \varepsilon_t'] = D\) (diagonal, typically \(I_n\)). Pre-multiplying by \(B_0^{-1}\) recovers the reduced form with \(u_t = B_0^{-1} \varepsilon_t\).

The covariance restriction is:

This system of equations has \(n(n+1)/2\) unique entries on the left (since \(\Sigma_u\) is symmetric) and \(n^2\) unknowns in \(B_0^{-1}\) on the right. The gap is \(n^2 - n(n+1)/2 = n(n-1)/2\) restrictions. For a bivariate system, that's 1 restriction. For a 4-variable system, 6. These must come from outside the data.

Without additional restrictions, the structural parameters are not identified. Any orthogonal matrix \(Q\) produces an observationally equivalent decomposition: if \(B_0^{-1}\) satisfies the covariance equation, so does \(\tilde{B}_0^{-1} = B_0^{-1} Q\) for any \(Q'Q = I_n\). The set of all such rotations forms the orthogonal group \(O(n)\), which has dimension \(n(n-1)/2\) -- exactly the identification deficit.

Recursive identification sets \(B_0^{-1}\) equal to the unique lower-triangular Cholesky factor of \(\Sigma_u\). The economic content of this choice is a causal ordering: the variable in position 1 does not respond contemporaneously to shocks hitting variables in positions 2 through \(n\). The variable in position 2 can respond to shock 1 but not to shocks 3 through \(n\). And so on.

For the monetary policy application, a standard ordering places output and prices before the interest rate, implying that the central bank sees current output and price data when setting the rate, but output and prices cannot react within the period to a monetary surprise. Christiano, Eichenbaum, and Evans (1999) used this ordering with monthly data, arguing that the production and pricing decisions that determine output and the price level within a month are made before the FOMC meets.

The Cholesky factor is computed as follows. Let \(\Sigma_u = P P'\) where \(P\) is lower triangular with positive diagonal entries. Then \(B_0^{-1} = P\), and the structural shocks are \(\varepsilon_t = P^{-1} u_t\). The \(j\)-th column of \(P\) gives the contemporaneous response of all variables to structural shock \(j\).

Sign restrictions replace zero restrictions with inequality constraints on the impulse-response function. The researcher specifies that certain responses have a known sign at certain horizons -- for example, a contractionary monetary policy shock raises the interest rate and does not increase output on impact. The method accepts all orthogonal rotations \(Q\) that produce IRFs satisfying these sign constraints.

The algorithm (Rubio-Ramirez, Waggoner, and Zha 2010) proceeds in three stages. First, estimate the reduced-form VAR and compute the Cholesky factor \(P\) of \(\Sigma_u\). Second, draw \(Q\) uniformly from the orthogonal group \(O(n)\) using the QR decomposition of a matrix of standard normal draws (this gives the Haar measure on \(O(n)\)). Third, compute \(\tilde{B}_0^{-1} = PQ\) and the associated IRFs; if all sign constraints are satisfied, keep the draw. Repeat many times to characterize the identified set.

Because sign restrictions are inequality constraints, they generically do not pin down a unique \(B_0^{-1}\). The result is set identification: for each accepted draw, the researcher gets a full IRF path, and the collection of accepted paths defines the identified set. Reporting conventions vary -- some papers show pointwise medians and percentile bands, while others report the entire identified set as a shaded region.

Proxy SVAR identifies a single structural shock using an external instrument \(z_t\) that correlates with the target shock but not with the others. Stock and Watson (2012) and Mertens and Ravn (2013) developed the econometric framework independently. The instrument does not need to enter the VAR -- it is used only in a second stage, applied to the estimated residuals.

The identification works through a two-step procedure. First, estimate the reduced-form VAR and collect the residuals \(\hat{u}_t\). Second, use \(z_t\) as an instrument in a regression of \(\hat{u}_{2t}\) (the residuals of the non-target variables) on \(\hat{u}_{1t}\) (the residual of the target variable). The IV estimator recovers the ratios of the structural impact coefficients -- the relative responses of all variables to the target shock -- up to a scale normalization.

The most influential application is Gertler and Karadi (2015), who used high-frequency changes in fed-funds futures within 30-minute windows around FOMC announcements as the external instrument for monetary policy shocks. The tight window ensures the futures move primarily because of the policy announcement, satisfying relevance. The exogeneity assumption is that these high-frequency moves are uncorrelated with other structural shocks (supply, demand) operating at lower frequencies.

Recursive identification fails silently when the causal ordering is wrong. The Cholesky factor always exists and always produces well-behaved IRFs -- there's no statistical test that rejects a bad ordering. The only check is economic plausibility and robustness to reordering. If swapping two variables in the ordering flips the sign or magnitude of the key impulse response, the recursive scheme is not informative about that shock.

Sign restrictions can produce very wide identified sets, especially in systems with weak dynamic interactions. If the IRF range spans both positive and negative values at the horizon of interest, the sign restrictions are too weak to resolve the identification problem. Tighter restrictions -- sign constraints imposed at multiple horizons, magnitude restrictions, or narrative constraints (Antolin-Diaz and Rubio-Ramirez 2018) -- narrow the set but require stronger assumptions.

Proxy SVAR inherits all the problems of instrumental-variable estimation. Weak instruments -- a first-stage F below 10 -- produce unreliable point estimates and undersized confidence intervals. The Montiel Olea, Stock, and Watson (2021) weak-instrument test is designed for the proxy SVAR setting. Exogeneity violations (the instrument correlates with non-target shocks) bias the IRFs, and there is no overidentification test when only one instrument is available for one shock.

All SVAR schemes assume the structural parameters are time-invariant over the sample. If the economy underwent a regime change (a shift in monetary policy rule, a financial crisis, a pandemic), the constant-parameter SVAR averages across regimes and the identified shocks mix pre- and post-break structural disturbances. TVP-VAR or Markov-switching SVAR models handle this, at the cost of substantially more estimation complexity.

Lag-length selection in the underlying VAR propagates directly into the structural analysis. Too few lags leave serial correlation in the residuals, which means \(u_t\) is not a proper innovation and the mapping \(u_t = B_0^{-1} \varepsilon_t\) is contaminated. Too many lags waste degrees of freedom and inflate estimation uncertainty. Standard information criteria (AIC, BIC, HQ) apply, but the researcher should also check residual autocorrelation directly via the Ljung-Box or Breusch-Godfrey test.