RDP 2010-07: Monetary Policy and the Exchange Rate: Evaluation of VAR Models Appendix A: Sign Restriction Algorithm
September 2010
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Consider a general VAR(p) model with n variables Y_{t}:
where: A(L) = A_{1}L + ... + A_{p}L^{p} is a p^{th} order matrix polynomial; B is a (n × n) matrix of coefficients that reflect the contemporaneous relationships among Y_{t}; and ε_{t} is a set of (n × 1) normally distributed structural disturbances with mean zero and variance covariance matrix Σ, Σ_{i,j} = 0∀i ≠ j. The structural representation has the following reduced form:
where π(L) = B^{−1}A(L) and e_{t} is a set of (n × 1) normally distributed reduced-form errors with mean zero and variance covariance matrix V, V_{i,j} ≠ 0∀i,j. The aim is to map the statistical relationships summarised by the reduced-form errors e_{t} back into economic relationships described by ε_{t}. Let P = B^{−1}. The reduced-form errors are related to the structural disturbances in the following manner:
for some matrix H such that HH^{′} = p∑p^{′}. An identification problem arises if there are not enough restrictions to uniquely pin down H from the matrix V.
The central idea behind SVAR analysis is to decompose the set of reduced-form shocks, characterised by V, into a set of orthogonal structural disturbances characterised by ∑. However, there are an infinite number of ways in which this orthogonality condition can be achieved. Let H be an orthogonal decomposition of V = HH′. The multiplicity arises from the fact that for any orthonormal matrix Q (where QQ′ = I), such that is also an admissible decomposition of V, where . This decomposition produces a new set of uncorrelated shocks , without imposing zero-type restrictions on the model.
Define an (n × n) orthonormal rotation matrix Q such that:
where θ_{i,j} ∈ [0, π]. This provides a way of systematically exploring the space of all VMA representations by searching over the range of values θ_{i,j}. We generate the Qs randomly from a uniform distribution using the following algorithm:
- Estimate the VAR to obtain the reduced form variance covariance matrix V.
- For both the foreign and domestic block, draw a vector θ_{i,j} from a uniform [0, π] distribution.
- Calculate Q, as in Equation (A4).
- Use the candidate rotation matrix Q to compute ε_{t} = HQe_{t} and its corresponding structural IRFs for domestic and foreign shocks.
- Check whether the IRFs satisfy all the sign restrictions described in Table 2. If so, keep the draw, if not, drop the draw.
- Repeat (2)–(5) until 1,000 draws that satisfy the restrictions are found.