Time Series Analysis for Macroeconomics and Finance Bernd Süssmuth IEW Institute for Empirical Research in Economics University of Leipzig December 12, 2011 Bernd Süssmuth (University of Leipzig) Time Series Analysis December 12, 2011 1 / 15
Ambiguity of IR functions Contents I 1 Orthogonalizing VARs Ambiguity of IR functions 2 Structural VAR: Example(s) Bernd Süssmuth (University of Leipzig) Time Series Analysis December 12, 2011 2 / 15
Ambiguity of IR functions Any time series can be represented with arbitrary linear combinations of any set of IRs Orthogonalization = process of selecting the most interesting IR Transforming VARs to systems w/ orthogonal errors = convenient Orthogonalization assumptions Starting point: estimated VAR A (L) x t = ɛ t, A (0) = I (lhs of basic regression), E ɛ t ɛ 0 t = (1) or alternatively, in MA notation x t = B (L) ɛ t, B (0) = I (lhs of basic regression), E ɛ t ɛ 0 t =, (2) where B (L) = A 1 (L), i.e. an inverted linear lter Bernd Süssmuth (University of Leipzig) Time Series Analysis December 12, 2011 3 / 15
Ambiguity of IR functions Instead of analyzing IRs of x t with regard to new shocks, we could have looked at x t s responses to linear combinations of old shocks For example: reactions to ɛ yt and ɛ zt + 0.5ɛ yt Call new shocks η t so that η 1t = ɛ yt ; η 2t = ɛ zt + 0.5ɛ yt : η t = Qɛ t, Q = 1 0 0.5 1 MA representation in terms of new shocks: x t = B (L) Q 1 Qɛ t {z } C(L) ) x t = C (L) η t C (L) is a linear combination of original IRs B (L) Bernd Süssmuth (University of Leipzig) Time Series Analysis December 12, 2011 4 / 15
Contents I 1 Orthogonalizing VARs Ambiguity of IR functions 2 Structural VAR: Example(s) Bernd Süssmuth (University of Leipzig) Time Series Analysis December 12, 2011 5 / 15
We would like to think about IRs in causal terms For example, the e ect of money on GNP Convenience = centering shocks to unit variance Thus, we want to choose Q from η t = Qɛ t so that E (η t ηt 0 ) = I Therefore, we need a Q s.t. Q 1 Q 1 0 =, b/c then E η t η 0 t = E Qɛ t ɛ 0 tq 0 = I One way to do this: Choleski decomposition Bernd Süssmuth (University of Leipzig) Time Series Analysis December 12, 2011 6 / 15
In a nutshell I Why do we do this? IRs are the instrument to track the impact of any variable of a dynamic system on other variables in the system Hence, IRs = essential tools in causal analysis and policy e ectiveness For example, the e ect" of money on GNP Starting point (k-dimensional vectors) x t = A 1 x t 1 +... + A p x t p + ɛ t (3) x t = B (L) ɛ t = θ i ɛ t i, (4) i=0 where MA coe cient θ jk,i represents the response of variable j to a unit impulse in variable k occuring i periods ago Bernd Süssmuth (University of Leipzig) Time Series Analysis December 12, 2011 7 / 15
In a nutshell II As usually is non-diagonal, it is impossible to shock one variable, holding other variables xed ) Some kind of transformation is needed The Choleski decomposition is one such transformation (the most popular one) ) Let Q 1 = P be a lower triangular matrix such that = PP0 ) Allows us to rewrite eq. (2) as x t = eθ i η t i, i =0 where eθ i = θ i P, η t = P 1 ɛ t = Qɛ t, and E (η t η 0 t ) = I Bernd Süssmuth (University of Leipzig) Time Series Analysis December 12, 2011 8 / 15
Choleski decomposition Note, B shares its diagonals with Q and Recall that C (L) = Q 1 B (L) = PB (L) x t = D (0) x t + D (1) x t 1 +... + D (p) x t p + v t, where D (0) = I k C (L) 1, D (i) = C (L) 1 A i As D (0) is lower triangular with 0 diagonals, Choleski imposes a recursive causal structure from the top variables to the bottom variables but not the other way around Problem: Makes it sensitive to variable ordering! Unfortunately, there are many Qs that act as square root matrices for s.t.q 1 Q 1 0 =, E (ηt η 0 t ) = E Qɛ tɛ 0 tq 0 = I Bernd Süssmuth (University of Leipzig) Time Series Analysis December 12, 2011 9 / 15
Contents I 1 Orthogonalizing VARs Ambiguity of IR functions 2 Structural VAR: Example(s) Bernd Süssmuth (University of Leipzig) Time Series Analysis December 12, 2011 10 / 15
Sims orthogonalization: Specifying C(0) The C (0) or D (0) speci cation shown above is one such choice of a square root matrix Q, namely the Sims orthogonalization The Sims orthogonalization is numerically equivalent to estimating the system by OLS with contemporaneous y t in the z t eq, but not vice versa, and then scaling each equation so that the error variance is one To see this, recall that OLS produces residuals that are by construction uncorrelated with rhs variables: y t = a 1yy y t 1 +... + a 1yz z t 1 + η yt z t = a 0zy y t +a 1zy y t 1 +... + a 1zz z t 1 + η zt η yt = y t E (y t j y t 1,..., z t 1 ): linear combination of fy t, y t 1,..., z t 1 g Bernd Süssmuth (University of Leipzig) Time Series Analysis December 12, 2011 11 / 15
As OLS residuals are orthogonal to rhs variables, η zt is orthogonal to any linear combination of fy t, y t 1,..., z t 1 g by construction Hence, η zt and η yt are uncorrelated with each other a 0zy captures all the contemporaneous correlation of news in y t and news in z t Choleski produces already a lower triangular Q s.t. C (0) = B (0) Q 1 = Q 1 Hence, Choleski decomp already produces Sims ortho! Some rule of thumb for ordering your VAR s variables: Bernd Süssmuth (University of Leipzig) Time Series Analysis December 12, 2011 12 / 15
Ideally, try to use economic theory to decide on the order of your VAR s variables For example, it is plausible that a Central Bank cannot see GDP Y until the end of the quarter ) money cannot respond within the quarter to a GDP shock In our notation, a monetary shock η Mt does not a ect the rst variable, i.e. money M Mt C0MM 0 ηmt = + C 1 η t 1 +... Y t C 0YM C 0YY η Yt It makes sense to let M be the 1 st and Y be the 2 nd VAR variable Bernd Süssmuth (University of Leipzig) Time Series Analysis December 12, 2011 13 / 15
Blanchard-Quah orthogonalization: restrictions on C(1) Rather than restricting the immediate response of one variable to another shock, BQ s idea is to examine shocks that are de ned so that the long-run response of one variable to another shock is zero If a system is speci ed in changes x t = C (L) η t, then C (1) is the long-run response of the levels of x t to η shocks For example, BQ argue that demand shocks have no long-run e ect on GDP Thus, the BQ logics requires, for example, C (1) to be lower diagonal in a VAR with GDP in the rst eq Bernd Süssmuth (University of Leipzig) Time Series Analysis December 12, 2011 14 / 15