RDP 2008-08: The Role of International Shocks in Australia's Business Cycle Appendix A: Sign Restriction Algorithm

Define an (n × n) orthonormal rotation matrix Q such that:

where Inline Equation. This provides a way of systematically exploring the space of all VMA representations by searching over the range of values of θi,j. While Canova and De Nicolo (2002) propose setting up a grid over the range of values for θi,j, the following algorithm generates the Qs randomly from a uniform distribution:

  1. Estimate the VAR in Equation (10) using OLS to obtain the reduced form variance covariance matrix V and compute Inline Equation.
  2. Compute the Choleski decomposition of Inline Equation11 and Inline Equation22, where H11 = chol(Inline Equation11) and H22 = chol(Inline Equation22).
  3. For both the foreign and domestic block, draw a vector of θi,j from a uniform [0,π] distribution.
  4. Calculate Inline Equation.
  5. Use the candidate rotation matrix Q to compute εt = HQet and its corresponding structural IRFs C(L) for domestic and foreign shocks.
  6. Check whether the IRFs satisfy all the sign restrictions described in Table 2. If so keep the draw, if not, drop the draw.
  7. Repeat (3)–(6) until 2,000 draws that satisfy the restrictions are found.