TOPICS IN ADVANCED ECONOMETRICS: SVAR MODELS PART I LESSON 2. Luca Fanelli. University of Bologna.

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1 TOPICS IN ADVANCED ECONOMETRICS: SVAR MODELS PART I LESSON 2 Luca Fanelli University of Bologna luca.fanelli@unibo.it

2 The material in these slides is based on the following books and papers: Amisano, G., Giannini, C. (1997), Topics in Structural VAR Econometrics, Springer. Hamilton, J.D. (1994), Time Series Analysis, Princeton University Press. Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Berlin: Springer-Verlag, Pesaran, M. H., Shin, Y. (1998), Generalized Impulse Response Analysis in Linear Multivariate Models, Economic Letters 58, Personal processing.

3 SIMPLE SVAR: CHOLESKY Given w t = Ã r L t r P t!, long term interest rate policy rate µ = 0 υ = 0, C 0 = I M Ã ε P t ε L t! = Ã r P t r L t " p11 0! p 12 p 22 = C(L) Ã ε P t ε L t #Ã u P t u L t!! policy shock monetary market shock ε t = Pu t, E(u t u 0 t)=i M P Cholesky factor of Λ unique P = Λ = PP 0 " p11 0 p 12 p 22 #

4 Accordingly à r P t r L t! = C(L) à ε P t ε L t! = ε t +C 1 ε t 1 +C 2 ε t PP 1 ε t + C 1 PP 1 ε t 1 + C 2 PP 1 ε t = Pu t + C 1 Pu t 1 + C 2 Pu t In summary, the VMA representation of this model based on the maping ε t = Pu t is: w t = Φ 0 u t + Φ 1 u t 1 + Φ 2 u t function of orthogonalized shocks: E(u t u 0 t)=i M Φ 0 = P Φ 1 = C 1 P,...,Φ h = C h P

5 The original idea of Sims (1980, Ecta) was to move from the VMA representation with non orthogonal disturbances to orthgonalized VMA representations (i.e. with disturbances having a diagonal covariance matrix), using the Choleski factorization of Λ. To sum up: A(L) w t = PP 1 ε t A(L) w t = Pu t w t = A(L) 1 Pu t w t = C(L)P u t w t = Φ(L) u t VMA with orthgonalized shocks

6 Alternative - Equivalent Interpretation Pre-multiply the VAR(p) A(L) w t = ε t, ε t WN(0, Λ) by the inverse of the Choleski factor of Λ, obtaining P 1 A(L) w t = P 1 ε t or A (L) w t = u t, u t WN(0, I M ) where A (L) =P 1 A(L) =P 1 P 1 A 1 L P 1 A p L p

7 Example (Causal chain): P 1 w t = P 1 A 1 w t P 1 A p w t p + u t The structure " #Ã! P 1 ϕ11 0 r P w t = t ϕ 12 ϕ 22 rt L =... implies a (simultaneous) system of equations of the form rt P = 1 (lags of all variables) + u 1t ϕ 11 r L t = ϕ 12 ϕ 22 r P t + 1 ϕ 22 (lags of all variables) + u 2t

8 Observation: in Λ = PP 0 by construction, P is triangular inferior, hence it contains free elements. M M(M 1) = 1 M(M +1) 2 Λ is symmetric with 1 2M(M +1)free elements.

9 Example: " σ 2 1 σ 12 σ 12 σ 2 2 # " σ 2 1 σ 12 σ 12 σ 2 2 = # " p11 0 = p 12 p 22 " #" p11 p 12 0 p 22 p 2 11 p 11 p 12 p 12 p 11 p p2 22 # # σ 2 1 = p2 11 σ 12 = p 11 p 12 σ 12 = p 12 p 11 σ 2 2 = p p2 22

10 p 11 = σ 1 p 12 = σ 12 /σ 1 p 22 = (σ 2 2 (σ 12 σ 1 ) 2 ) 1/2 Once we get a consistent estimate of Λ from the reduced-form VAR, we can estimate a unique (consistent) estimate of the Choleskt factor P. We have no identification issues here, meaning that once we get the reduced for estimates of the VAR, we have a one-to-one mapping which allows to recover the structural parameters from the former. EXACT IDENTIFICATION. The Choleski factorization of Λ is unique, however, if we permute the elements in w t (ε t ), we get a different factorization (M! possible permutations). Thus, the SVAR based on the Choleski factorization of Λ depends on the ordering of the variables.

11 In small systems (2 variables) this is not a major problem... With a tri-variate VAR it becomes difficult to establish which is the shock that takes the first place.

12 Forecast Error Variance Decomposion (FEVD) VAR forecast: w f t+h = E(w t+h F t ) Forecast error: η t+h = w t+h w f t+h = ε t+h + C 1 ε t+h C h 1 ε t+1. By definition: MSE(w f t+h )=E(η t+hη 0 t+h ) = E[(ε t+h + C 1 ε t+h C h 1 ε t+1 ) (ε t+h + C 1 ε t+h C h 1 ε t+1 ) 0 ] = Λ + C 1 ΛC C h 1ΛC 0 h 1. We want to investigate how the ortogonalized disturbances in u t (u t = P 1 ε t ) contributes to MSE(w f t+h ).

13 More precisely, we are interested in measuring the contribution of u qt to the MSE(w f i,t+h ) (variance of the h-step ahead error forecast). Of course, MSE(w f i,t+h )isthei-th diagonal element of MSE(w f t+h ). Moreover, MSE(w f h 1 t+h )= X MSE(w f h 1 t+h ) = X = = j=0 Λ = PP 0 j=0 h 1 X j=0 h 1 X j=0 C j ΛC 0 j C j ΛC 0 j C j PP 1 ΛP 10 P 0 C 0 j Φ j Φ 0 j

14 MSE(w f i,t+h )=e0 i h 1 X j=0 Φ j Φ 0 j e i. Now, let s come back on w t+h w f t+h = ε t+h + C 1 ε t+h C h 1 ε t+1 = PP 1 ε t+h +C 1 PP 1 ε t+h C h 1 PP 1 ε t+1 = Φ 0 u t+h + Φ 1 u t+h Φ h 1 u t+1 which implies, considering the i-th equation: where w i,t+h w f h 1 i,t+h = X MX j=0 q=1 φ j i,q u q,t+h j φ j i,q impact of u q,t on w i,t+j.

15 Using simple algebra: w i,t+h w f h 1 i,t+h = X MX j=0 q=1 φ j i,q u q,t+h j = MX q=1 h 1 X j=0 φ j i,q u q,t+h j = MX q=1 (φ 0 i,q u q,t+h + φ 1 i,q u q,t+h φ h 1 Thus: E h (w i,t+h w f i,t+h )2i E(η i,t+h η 0 i,t+h ) i,q u q,t+1). = MX q=1 h (φ 0 i,q ) 2 +(φ 1 i,q ) (φ h 1 i,q )2i

16 MSE h w f i,t+h i = M X q=1 h (φ 0 i,q ) 2 +(φ 1 i,q ) (φ h 1 i,q )2i Note that each component in the summation above (φ 0 i,q )2 +(φ 1 i,q ) (φ h 1 i,q )2 = h 1 X j=0 ³ e 0 i Φ j e q 2 can be interpreted as the contribution of u qt to the MSE(w f i,t+h ). Therefore, the ratio: ³ e 0 i Φ j e q 2 P h 1 j=0 MSE(w f i,t+h ) = e 0 i P h 1 j=0 ³ e 0 i Φ j e q 2 ³ Ph 1 j=0 Φ j Φ 0 j ei measures the proportion of the h-step forecast error variance of variable i accounted for by innovations in variable q.

17 Reinterpreting impluse response analysis: GENERALIZED IMPULSE RESPONSE ANALY- SIS δ = δ 1. δ M size of the shocks at time t F t information set at time t GI(h, δ,f t 1 )= E(w t+h ε t = δ, F t 1 ) E(w t+h F t 1 ) Generalized impulse response function Koop -Pesaran -Potter (1996, JoE) nonlinear systems, Pesaran and Shin (1998, EL) linear systems.

18 Given w t = υ + ε t + C 1 ε t 1 + C 2 ε t leading w t+h = υ+ε t+h +C 1 ε t+h C h ε t +C h+1 ε t Now E(w t+h ε t = δ, F t 1 ) = υ + C h δ + C h+1 ε t 1 + C h+2 ε t E(w t+h F t 1 )=υ + C h+1 ε t 1 + C h+2 ε t hence, subtracting: GI(h, δ,f t 1 )=C h δ does not depend on F t 1!

19 We know that for h =0, 1, 2,... C h e q is M 1 measures the impact on all variables in w t (w t+h )ofa (non orthogonal) shock in ε q,t h (ε q,t ), i.e. affecting the q-th equation of the system. Hence,thechoiceoftheappropriatevectorofshocks δ is central.

20 GI(h, δ,f t 1 )=C h δ Traditional approach (Sims, 1980, Ecta), δ = Pe q where P is the Cholesky factor of Λ, Λ = PP 0.With this particular choice: GI(h, δ,f t 1 ) OI(h, δ,f t 1 )=C h δ = C h Pe q = Φ h e q. h =0, 1, 2,... We get orthogonalized impulses (based on Cholesky). By construction, here the ordering of the variables matters.

21 The idea of Pesaran and Shin (1998, EL) is to use the following definition: GI(h, δ q,f t 1 ) = E(w t+h ε q,t = δ q, F t 1 ) E(w t+h F t 1 ) i.e. we shock only one element of ε t, the one relative to the q-th equation. If ε t N (0, Λ), then from the properties of the multivariate normal distribution E(ε t ε q,t = δ q )= = Λe q σ 1 q,qδ q. σ 1,q /σ q,q. σ M,q /σ q,q δ q

22 Therefore E(w t+h ε q,t = δ q, F t 1 ) = υ + C h E(ε t ε q,t = δ q, F t 1 )+... = υ + C h E(ε t ε q,t = δ q )+... so that = υ + C h Λe q σ 1 q,qδ q +... GI(h, δ q,f t 1 )=C h Λe q σ 1 q,qδ q = C h Λe q δ q σ q,q h =0, 1, 2,... For h =0, 1, 2,... this is the M 1vectorofunscaled generalized impulse response of the effect of one standard error shock in the q-thequationattime t on (all the variables in) w t+h.

23 Usually written as GI(h, δ q,f t 1 )= C hλe q σ 1/2 q,q so that by setting δ q = σ 1/2 q,q shock) δ q σ 1/2 q,q (one standard error GI(h, δ q,f t 1 )= C hλe q σ 1/2, h =0, 1, 2,... q,q scaled generalized impulse response of the effect of a one standard error shock in the q-thequationattime t on (all the variables in) w t+h.

24 Differences and relations between GIRs and OIRs The GIRs do not depend on the ordering of the variables. The OIRs depend on the ordering of the variables. Shocks are orthogonalized by construction. GI(h, δ q,f t 1 )= C hλe q σ 1/2 q,q OI(h, δ,f t 1 )=C h Pe q = Φ h e q Λ diagonal GIRs = OIRs; (stems from the definitions). Λ non-diagonal GIRs 6= OIRsforq =2, 3,...,M but GIRs = OIRs for q =1.

25 The GIRs do not convey information about economic causation among the variables. In practise with GIRs we talk about system wide shocks. GIR can be thought of as tracing out the implications of the forecast error in equation q : E(w t+h ε q,t = δ q, F t 1 ) E(w t+h F t 1 ) in terms of revisions in forecasts of all variables included in the model. Allows to investigate the interaction of variables in a more neutral way. OIRs requires that we have in mind an ordering of the variables from the less endogenous to the most endogenous.