svarIdentify#

Purpose#

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

Format#

sr = svarIdentify(result, sign_restr)#
sr = svarIdentify(result, sign_restr, n_ahead=20, max_tries=10000, seed=42, quiet=0)
Parameters:

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

  • sign_restr (Nx4 matrix) –

    Nx4 matrix of sign restrictions on impulse responses. Each row specifies one restriction:

    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.).

    4

    Sign: 1 for positive response, -1 for negative response.

  • n_ahead (scalar) – Optional keyword, number of IRF horizons to compute. Default = 20.

  • max_tries (scalar) – Optional keyword, maximum number of rotation attempts before giving up. Default = 10000.

  • seed (scalar) – Optional keyword, RNG seed for reproducible rotation draws. Default = 42.

  • quiet (scalar) – Optional keyword, 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);

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

sr = svarIdentify(result, sign_restr);

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 in svarIdentify(). Zero restrictions require the ARW2018 null-space algorithm; use svarIrfCompute() with an svarControl struct for zero restrictions.

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#

  1. Compute \(L = \text{chol}(\Sigma)'\).

  2. 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).

  3. Form candidate \(P = L \cdot Q\).

  4. Compute IRFs \(\Theta_h = J F^h J' P\) at all restricted horizons.

  5. Check all sign restrictions. If satisfied, return \(P\). Otherwise, go to step 2.

  6. Repeat up to 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 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

See also

Functions svarIrfCompute(), svarControlCreate(), irfCompute()

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