Structural-VAR II. Dean Fantazzini. Dipartimento di Economia Politica e Metodi Quantitativi. University of Pavia

Transcription

1 Structural-VAR II Dipartimento di Economia Politica e Metodi Quantitativi University of Pavia

2 Overview of the Lecture 1 st Structural VARs with Combinations of I(1) and I(0) Data: The Blanchard and Quah (1989) model July

3 Overview of the Lecture 1 st Structural VARs with Combinations of I(1) and I(0) Data: The Blanchard and Quah (1989) model 2 nd Identifying the SVAR Using Long-Run Restrictions July a

4 Overview of the Lecture 1 st Structural VARs with Combinations of I(1) and I(0) Data: The Blanchard and Quah (1989) model 2 nd Identifying the SVAR Using Long-Run Restrictions 3 rd Estimating the SVAR in the Presence of Long-Run Restrictions July b

5 Overview of the Lecture 1 st Structural VARs with Combinations of I(1) and I(0) Data: The Blanchard and Quah (1989) model 2 nd Identifying the SVAR Using Long-Run Restrictions 3 rd Estimating the SVAR in the Presence of Long-Run Restrictions 4 rd An empirical example with the BQ model (I) July c

6 Overview of the Lecture 1 st Structural VARs with Combinations of I(1) and I(0) Data: The Blanchard and Quah (1989) model 2 nd Identifying the SVAR Using Long-Run Restrictions 3 rd Estimating the SVAR in the Presence of Long-Run Restrictions 4 rd An empirical example with the BQ model (I) 5 th An empirical example: Impulse responses with the BQ model (II) July d

7 Overview of the Lecture 1 st Structural VARs with Combinations of I(1) and I(0) Data: The Blanchard and Quah (1989) model 2 nd Identifying the SVAR Using Long-Run Restrictions 3 rd Estimating the SVAR in the Presence of Long-Run Restrictions 4 rd An empirical example with the BQ model (I) 5 th An empirical example: Impulse responses with the BQ model (II) 6 th A review empirical example: Cholesky factorization as structural factorization July e

8 Structural VARs with Combinations of I(1) and I(0) Data: The Blanchard and Quah (1989) model Consider two observed series y 1t and y 2t such that y 1t is I(1) and y 2t is I(0). This is the case, for example, in the famous analysis in Blanchard and Quah (1989): y 1 = log of real GDP y 2 = unemployment rate Define y t = ( y 1t, y 2t ) = y t I(0) Suppose y t has the structural representations By t = γ 0 + Γ 1 y t 1 + ε t y t = µ + Θ(L)ε t Θ(L) = Ψ(L)B 1 E[ε t ε t] = D (= diagonal) July

9 Structural VARs with Combinations of I(1) and I(0) Data: The Blanchard and Quah (1989) model The reduced form representations are: y t = B 1 γ 0 + B 1 Γ 1 y t 1 + B 1 ε t = a 0 + A 1 y t 1 + u t = µ + Ψ(L)u t, Ψ(L) = (I 2 A 1 L) 1, E[u t u t] = B 1 E[ε t ε t]b 1 = B 1 DB 1 = Ω Note: BQ loosely interpret ε 1t as a permanent (supply) shock since it is the innovation to the I(1) real output series y t and interpret ε 2t as a transitory (demand) shock since it is the innovation to the I(0) unemployment series. The structural IRFs are given by y 1t+s = θ (s) 11 ε, y 1t+s 1t ε 2t y 2t+s = θ (s) 21 ε, y 2t+s 1t ε 2t = θ (s) 12 = θ (s) July

10 Structural VARs with Combinations of I(1) and I(0) Data: The Blanchard and Quah (1989) model Since y 1t (output) is I(1), the long-run impacts on the level of y 1 of shocks to ε 1 and ε 1 are y 1t+s lim = θ 11 (1) = θ (s) 11 s ε 1t lim s y 1t+s ε 2t = θ 12 (1) = s=0 s=0 Since y 2t (unemployment) is I(0), the long-run impacts on the level of y 2 of shocks to ε 1 and ε 2 are zero: For y 2, lim s θ 21 (1) = θ 22 (1) = y 2t+s ε 1t = lim s θ(s) 2j = 0 s=0 s=0 θ (s) 21 θ (s) 22 θ (s) 12 represent the cumulative impact of shocks to ε 1 and ε 2 on the level of y July

11 Identifying the SVAR Using Long-Run Restrictions BQ (1989) achieve identification of the SVAR/SMA by assuming Transitory (demand) shocks (shocks to ε 2 ) have no long-run impact on the level of output or unemployment. They allow permanent (supply) shocks (shocks to ε 1 ) to have a long-run impact on the level of output but not on the level of unemployment. The restriction that shocks to ε 2 have no long-run impact on the level of y 1 implies that θ 12 (1) = s=0 θ (s) 12 = July

12 Identifying the SVAR Using Long-Run Restrictions The restriction that shocks to ε 1 and ε 2 have no long-run effect on the level of y 2 is lim s y 2t+s ε jt = lim s θ(s) 2j = 0 which follows automatically since y 2 I(0). BQ (1989) assumptions imply that the long-run impact matrix Θ(1) is lower triangular Θ(1) = θ 11(1) 0 θ 21 (1) θ 22 (1) Claim: The lower triangularity of Θ(1) can be used to identify B July

13 Identifying the SVAR Using Long-Run Restrictions To see why, consider the long-run covariance matrix of y t defined from the Wold representation Since Λ = Ψ(1)ΩΨ(1) = (I 2 A 1 ) 1 Ω(I 2 A 1 ) 1 Ω = B 1 DB 1 Θ(1) = Ψ(1)B 1 = (I 2 A 1 ) 1 B 1, Λ may be re-expressed as, Λ = (I 2 A 1 ) 1 B 1 DB 1 (I 2 A 1 ) 1 = Θ(1)DΘ(1) In order to identify B, BQ (1989) make the additional assumption D = I 2 so that the structural shocks ε 1t and ε 2t have unit variances. Then Λ = Θ(1)Θ(1) July

14 Identifying the SVAR Using Long-Run Restrictions Since Θ(1) is lower triangular, Λ can be obtained uniquely using the Choleski factorization; that is, Θ(1) can be computed as the lower triangular Choleski factor of Λ. The Choleski factorization of Λ is Λ = PP = Θ(1)Θ(1) = Θ(1) = P Given that Θ(1) = P can be computed directly from Λ, B can then be computed using P = Θ(1) = Ψ(1)B 1 = (I 2 A 1 ) 1 B 1 = B = [(I 2 A 1 )P] 1 and the SVAR model is exactly identified July

15 Estimating the SVAR in the Presence of Long-Run Restrictions The estimation of B and Θ(L) using the BQ identification scheme can be accomplished with the following steps: Estimate the reduced form VAR by OLS equation by equation: y t = â 0 + Â 1 y t 1 + û t ˆΩ = 1 T û t û t T t=1 Compute a parametric estimate of the long-run covariance matrix: ˆΛ = (I 2 Â 1 ) 1ˆΩ(I 2 Â 1 ) 1 Compute the Choleski factorization of ˆΛ: ˆΛ = ˆPˆP July

16 Estimating the SVAR in the Presence of Long-Run Restrictions Define the estimate of Θ(1) as the lower triangular Choleski factor of ˆΛ, ˆΘ(1) = ˆP Estimate B using [ ˆB = (I 2 Â 1 )ˆΘ(1) Estimate Θ k using ˆΘ k = ˆΨ kˆb 1 = Â k 1 ] 1 ˆB 1 From the estimated ˆΘ k matrices the structural IRF may then be computed. Also, estimates of the structural shocks ˆε 1t and ˆε 2t may be extracted using ˆε t = ˆBû t July

17 An empirical example with the BQ model (I) The following program replicates the long-run restriction model by Blanchard-Quah (1989) (the data are not the same as those in the original paper so that the results do not match). The Y (output growth) and U (unemployment) series are already demeaned following the procedure by Blanchard-Quah (1989). The program estimates the factorization matrix by two different methods which should all yield the same result July

18 An empirical example with the BQ model (I) Blanchard-Quah long-run restriction (11/5/99) verify estimates using two methods change path to program path %path cd %path create workfile wfcreate blanquah q 1948:1 1987:4 fetch data from database (already demeaned) fetch(d=data svar) dy u estimate VAR (no constant) var var1.ls 1 8 dy July

19 An empirical example with the BQ model (I) method 1: text form long-run restrictions var1.cleartext(svar) var1.append(svar) freeze(tab1) var1.svar(rtype=text,conv=1e-4) show tab1 store estimated A and B matrices from method 1 matrix mata1 = var1.@svaramat A should be identity matrix matb1 = var1.@svarbmat method 2: long-run restrictions in short-run form *not recommended; only for checking* July

20 An empirical example with the BQ model (I) get unit (cumulated) long-run response var1.impulse(imp=u) matrix clr = var1.@lrrsp var1.cleartext(svar) = c(1)*@u1 + c(2)*@u2 = -clr(1,1)*c(1)/clr(1,2)*@u1 + c(4)*@u2 freeze(tab2) var1.svar(rtype=text,conv=1e-5) show tab2 store estimated A and B matrices from method 2 matrix mata2 = var1.@svaramat A should be identity matrix matb2 = var1.@svarbmat July

21 An empirical example with the BQ model (I) You should get these results: Structural VAR Estimates Sample (adjusted): 1950Q2 1987Q4 Convergence achieved after 9 iterations Structural VAR is just-identified Model: Ae = Bu where E[uu ]=I Restriction Type: long-run text form Long-run response pattern: 0 C(2) C(1) C(3) Coefficient Std. Error z-statistic Prob. C(1) C(2) C(3) Log likelihood Estimated A matrix: Estimated B matrix: July

22 An empirical example with the BQ model (I) Response to Nonfactorized One Unit Innovations 1.2 Response of DY to DY 1.2 Response of DY to U Response of U to DY 10 Response of U to U July

23 An empirical example with the BQ model (I) Structural VAR Estimates Sample (adjusted): 1950Q2 1987Q4 Included observations: 151 after adjustments Convergence achieved after 8 iterations Structural VAR is just-identified Model: Ae = Bu where E[uu ]=I Restriction Type: short-run text = C(1)*@U1 + = -CLR(1,1)*C(1)/CLR(1,2)*@U1 + C(4)*@U2 represents DY represents U residuals Coefficient Std. Error z-statistic Prob. C(1) C(2) C(4) Log likelihood Estimated A matrix: Estimated B matrix: July

24 An empirical example: Impulse responses with the BQ model (II) The following program replicates the long-run restriction model by Blanchard-Quah (1989) (the data are not the same as those in the original paper so that the results do not match). The Y (output growth) and U (unemployment) series are already demeaned following the procedure by Blanchard-Quah (1989). To plot the accumulated response and the level response in one graph, we first store the impulse responses in separate matrices, extract and place the relevant columns into a third matrix, and use the line view of that matrix July

25 An empirical example: Impulse responses with the BQ model (II) replicate impulse response graph from long-run restriction (Blanchard-Quah 1989) change path to program path %path cd %path create workfile wfcreate blanquah q 1948:1 1987:4 fetch data from database (already demeaned) fetch(d=data svar) dy u change to percentage dy = 100*dy estimate VAR (no constant) var var1.ls 1 8 dy July

26 An empirical example: Impulse responses with the BQ model (II) impose long-run = aggregate demand = aggregate supply shock var1.cleartext(svar) estimate factorization freeze(tab1) var1.svar(rtype=text,conv=1e-5) set response horizon!hrz = 40 replicate Figures 3-4 (p.662) accumulate to get level response of output freeze(fig3) var1.impulse(!hrz,imp=struct,se=a,a,matbys=rsp acc) 1 freeze(fig4) var1.impulse(!hrz,imp=struct,se=a,a) July

27 An empirical example: Impulse responses with the BQ model (II) freeze(fig34) fig3 fig4 show fig34 replicate Figures 5-6 freeze(fig5) var1.impulse(!hrz,imp=struct,se=a,matbys=rsp lvl) 1 freeze(fig6) var1.impulse(!hrz,imp=struct,se=a) 2 freeze(fig56) fig5 fig6 show fig56 replicate Figure 1 matrix(!hrz,2) mtmp get accumulated output response vector v acc,1) colplace(mtmp,v,1) get level unemployment response July

28 An empirical example: Impulse responses with the BQ model (II) v lvl,2) colplace(mtmp,v,2) freeze(fig1) mtmp.line fig1.draw(line,left) 0 fig1.elem(1) legend(output response to demand) fig1.elem(2) legend(unemployment response to demand) show fig1 replicate Figure 2 get accumulated output response vector v acc,3) colplace(mtmp,v,1) get level unemployment response v lvl,4) July

29 An empirical example: Impulse responses with the BQ model (II) colplace(mtmp,v,2) freeze(fig2) mtmp.line fig2.draw(line,left) 0 fig2.elem(1) legend(output response to supply) fig2.elem(2) legend(unemployment response to supply) show fig July

30 An empirical example: Impulse responses with the BQ model (II) Output response to demand Unemployment response to demand July

31 An empirical example: Impulse responses with the BQ model (II) Output response to supply Unemployment response to supply July

32 An empirical example: Impulse responses with the BQ model (II) 1.4 Accumulated Response of DY to Structural One S.D. Shock Accumulated Response of DY to Structural One S.D. Shock July

33 An empirical example: Impulse responses with the BQ model (II).1 Response of U to Structural One S.D. Shock Response of U to Structural One S.D. Shock July

34 A review empirical example: Cholesky factorization as structural factorization The following program illustrates the two forms of imposing the identifying restrictions in structural decompositions. For illustration purposes and to check that the restrictions are correctly imposed, we impose restrictions that replicate the Cholesky factorization. As you can see the results resulting from the three methods are the same July

35 A review empirical example: Cholesky factorization as structural factorization replicate cholesky factorization using SVAR change path to program path %path cd %path create workfile wfcreate cholsvar q 1948:1 1979:3 fetch data from database fetch(d=data svar) rgnp rinv m1 take log of each series series lgnp = log(rgnp) series linv = log(rinv) series lm1 = log(m1) estimate unrestricted VAR July

36 A review empirical example: Cholesky factorization as structural factorization var var1.ls 1 4 lgnp linv c method 1: short-run restrictions in text form var1.cleartext(svar) = c(1)*@u1 = -c(2)*@e1 + c(3)*@u2 = -c(4)*@e1 - c(5)*@e2 + c(6)*@u3 freeze(tab1) var1.svar(rtype=text,conv=1e-5) show tab1 store estimated A and B matrices from method 1 matrix mata1 = var1.@svaramat matrix matb1 = var1.@svarbmat July

37 A review empirical example: Cholesky factorization as structural factorization compute factorization matrix matrix fact method 2: short-run restrictions in pattern matrix create pattern matrices matrix(3,3) pata fill matrix in row major order pata.fill(by=r) 1,0,0, na,1,0, na,na,1 matrix(3,3) patb patb.fill(by=r) na,0,0, 0,na,0, 0,0,na freeze(tab2) var1.svar(rtype=patsr,conv=1e-5,namea=pata,nameb=patb) show tab2 store estimated A and B matrices from method July

38 A review empirical example: Cholesky factorization as structural factorization matrix mata2 = var1.@svaramat matrix matb2 = var1.@svarbmat compute factorization matrix matrix fact method 3: built-in cholesky do impulse response with default Cholesky ordering var1.impulse(10) store cholesky factor matrix fact3 = var1.@impfact July

39 A review empirical example: Cholesky factorization as structural factorization Model: Ae = Bu where E[uu ]=I Restriction Type: short-run text = = -C(2)*@E1 + = -C(4)*@E1 - C(5)*@E2 + C(6)*@U3 represents LGNP represents LINV represents LM1 residuals Coefficient Std. Error z-statistic Prob. C(2) C(4) C(5) C(1) C(3) C(6) Estimated A matrix: Estimated B matrix: July

40 A review empirical example: Cholesky factorization as structural factorization Model: Ae = Bu where E[uu ]=I Restriction Type: short-run pattern matrix A = C(1) 1 0 C(2) C(3) 1 B = C(4) C(5) C(6) Coefficient Std. Error z-statistic Prob. C(1) C(2) C(3) C(4) C(5) C(6) Estimated A matrix: Estimated B matrix: July

41 A review empirical example: Cholesky factorization as structural factorization Response to Cholesky One S.D. Innovations.015 Response of LGNP to LGNP.015 Response of LGNP to LINV.015 Response of LGNP to LM Response of LINV to LGNP.03 Response of LINV to LINV.03 Response of LINV to LM Response of LM1 to LGNP.012 Response of LM1 to LINV.012 Response of LM1 to LM July

42 A review empirical example: Cholesky factorization as structural factorization The matrices FACT1, FACT2 and FACT3 are all equal to: July