svarIdentify#

Purpose#

Find a structural rotation satisfying sign restrictions for a single VAR or BVAR estimate.

Format#

sr = svarIdentify(result, ctl)#

Parameters:

result (struct) – an instance of a varResult or bvarResult structure.

ctl (struct) –

an instance of an svarControl structure with sign restrictions defined. An instance is initialized by calling svarControlCreate() and the following members can be set:

adv.zero_restr

Nx3 matrix, zero restrictions on impulse responses. Each row specifies one restriction with columns:

1	Variable index (1 to m) – the responding variable.
2	Shock index (1 to m) – the structural shock.
3	Horizon (0 = impact, 1 = one step ahead, etc.).

Zero restrictions are satisfied exactly by construction using the ARW2018 null-space algorithm. When zero_restr is non-empty, the algorithm is automatically set to ARW2018. Default = {} (no zero restrictions).

adv.narrative_restr

Nx6 matrix, narrative restrictions on the historical decomposition. Each row specifies one restriction with columns:

1	Type: 1 = shock_sign, 2 = shock_dominance, 3 = decomposition_sign.
2	Variable index (1 to m).
3	Shock index (1 to m).
4	Date 1: observation index (1-indexed), or start of range.
5	Date 2: end observation (0 if unused).
6	Sign: 1 for positive, -1 for negative.

When narrative_restr is non-empty, the v3 narrative engine is used automatically. Default = {} (no narrative restrictions).

adv.algorithm

Scalar, algorithm selection. Default = 0 (auto-detect).

0	Auto: uses RRW2010 for pure sign, ARW2018 if `zero_restr` is non-empty, v3 narrative engine if `narrative_restr` is non-empty.
1	Accept-reject (RRW2010). Efficient for pure sign restrictions. Cannot handle zero restrictions.
2	ARW2018 null-space construction. Required for zero restrictions. Also works for pure sign restrictions.

adv.max_tries

Scalar, maximum rotation attempts per posterior draw. Default = 10000.

adv.seed

Scalar, RNG seed for reproducibility. Default = 42.

adv.quiet

Scalar, set to 1 to suppress printed output. Default = 0.

Returns:

sr (struct) –

An instance of an svarResult structure containing:

sr.p	mxm matrix, structural impact matrix P such that \(\Sigma = PP'\).
sr.irf	`irfResult` struct, identified impulse responses under this rotation.
sr.n_tries	Scalar, number of rotation attempts needed to find a valid rotation.
sr.m	Scalar, number of variables.
sr.var_names	Mx1 string array, variable names.
sr.shock_names	Mx1 string array, shock labels.

Examples#

Monetary Policy SVAR#

new;
library timeseries;

// Load data — ordering determines Cholesky structure
fname = getGAUSSHome("pkgs/timeseries/examples/data/us_macro_quarterly.csv");
y = loadd(fname, "gdp_growth + cpi_inflation + fed_funds");

result = varFit(y, 4);

// Define sign restrictions
ctl = svarControlCreate();

// [variable, shock, horizon, sign]
// Monetary shock (shock 3): FFR up, GDP down, CPI down at impact
ctl.sign_restr = { 3 3 0  1,       // FFR positive
                   1 3 0 -1,       // GDP negative
                   2 3 0 -1 };     // CPI negative

sr = svarIdentify(result, ctl);

print "Structural impact matrix P:";
print sr.p;
print "Rotations tried:" sr.n_tries;

Remarks#

Algorithm: Draws random orthogonal matrices Q from the Haar measure on O(m) via QR decomposition of N(0,1) matrices with sign correction (Mezzadri 2007). For each candidate Q, forms \(P = \text{chol}(\Sigma) \cdot Q\) and checks all sign restrictions on the implied IRFs. Returns the first accepted rotation.

This function finds a single rotation, which is useful for point estimation (e.g., from an OLS VAR). For posterior inference with credible bands, use svarIrfCompute() with a bvarResult or bvarSvResult.

Zero restrictions are not currently supported. Setting ctl.zero_restr raises an error. Zero restrictions require the ARW2018 null-space algorithm, which is planned for a future release.

Model#

Sign-restricted SVAR decomposes the error covariance as \(\Sigma = PP'\) where \(P\) is not unique. The set of valid decompositions is \(\{PQ : Q \in O(m),\; PQ \text{ satisfies sign restrictions}\}\) where \(O(m)\) is the group of orthogonal matrices.

Given a set of sign restrictions \(\Theta_h[i,j] \gtrless 0\) (the IRF of variable \(i\) to shock \(j\) at horizon \(h\) is positive or negative), the function finds a \(Q^*\) such that \(P^* = \text{chol}(\Sigma)' \cdot Q^*\) produces IRFs satisfying all restrictions.

Algorithm#

Compute \(L = \text{chol}(\Sigma)'\).
Draw \(Z \sim N(0, I_{m \times m})\) and compute \(Q, R = \text{QR}(Z)\) with sign correction (Mezzadri 2007 algorithm for Haar-uniform orthogonal matrices).
Form candidate \(P = L \cdot Q\).
Compute IRFs \(\Theta_h = J F^h J' P\) at all restricted horizons.
Check all sign restrictions. If satisfied, return \(P\). Otherwise, go to step 2.
Repeat up to ctl.max_tries times.

Complexity: \(O(\text{max\_tries} \cdot h_{\max} \cdot m^2 p^2)\) worst case. Acceptance rates depend on how restrictive the sign constraints are.

Troubleshooting#

No valid rotation found (max_tries exceeded): The sign restrictions may be too numerous, contradictory, or implausible for this data. Relax some restrictions or increase ctl.max_tries.

Low acceptance rate (< 1%): Many restrictions at long horizons are hard to satisfy. Start with impact-only restrictions and add horizons incrementally.

Verification#

Sign restriction algorithm verified against the Rubio-Ramirez, Waggoner & Zha (2010) analytical examples for 2-variable and 3-variable systems.

References#

Mezzadri, F. (2007). “How to generate random matrices from the classical compact groups.” Notices of the AMS, 54(5), 592-604.
Rubio-Ramirez, J.F., D.F. Waggoner, and T. Zha (2010). “Structural vector autoregressions: Theory of identification and algorithms for inference.” Review of Economic Studies, 77(2), 665-696.
Uhlig, H. (2005). “What are the effects of monetary policy on output? Results from an agnostic identification procedure.” Journal of Monetary Economics, 52(2), 381-419.

Library#

timeseries

Source#

svar.src