B y t = γ 0 + Γ 1 y t + ε t B(L) y t = γ 0 + ε t ε t iid (0, D) D is diagonal

Transcription

1 Structural VAR Modeling for I(1) Data that is Not Cointegrated Assume y t =(y 1t,y 2t ) 0 be I(1) and not cointegrated. That is, y 1t and y 2t are both I(1) and there is no linear combination of y 1t and y 2t that is I(0). In this case, y t =( y 1t, y 2t ) 0 is I(0) and is assumed to have the SVAR representation B y t = γ 0 + Γ 1 y t + ε t B(L) y t = γ 0 + ε t ε t iid (0, D) D is diagonal

2 The reduced form VAR for y t is where y t = a 0 + A 1 y t 1 + u t A(L) y t = a 0 + u t α 0 = B 1 γ 0, A 1 = B 1 Γ 1, u t = B 1 ε t, E[u t u 0 t] = B 1 DB 10, A(L) =I 2 A 1 L TheWoldMArepresentationis y t = μ + Ψ(L)u t, μ = A(1) 1 a 0, Ψ(L) =A(L) 1 and the SMA representation is y t = μ + Θ(L)ε t, Θ(L) = Ψ(L)B 1

3 Impulse Response Functions Consider the SMA representation at time t + s " y1t+s y 2t+s # = " μ1 μ 2 + # + θ(0) 11 θ (0) " 12 θ (0) 21 θ (0) ε1t+s ε 2t+s 22 " # ε1t ε 2t θ(s) 11 θ (s) 12 θ (s) 21 θ (s) 22 The structural dynamic multipliers are y 1t+s # +. = θ (s) ε 11, y 1t+s = θ (s) 1t ε 12 2t y 2t+s = θ (s) ε 21, y 2t+s = θ (s) 1t ε 22 2t which give the impact of the structural shocks on the first difference of y at horizon t + s. +

4 Using the fact that y it+s = y it 1 + y it + y it y it+s, i =1, 2 the impacts of the structural shocks on the level of y are y it+s = y it ε jt ε jt = θ (0) = ij + θ (1) ij sx k=0 + y it+1 ε jt + + θ (s) ij θ (k) ij, i,j =1, y it+s ε jt The long-run impact of a shock to ε j on the level of y i is then lim s y it+s ε jt = θ ij (1), i,j =1, 2. For stationary y this long-run impact is always zero but for nonstationary y this impact may or may not be zero for some combination of i and j.

5 Beveridge-Nelson Decomposition Using the Wold representation for y t,themultivariate BN decomposition of y t is y t = y 0 + μ t + Ψ(1) ũ t = Ψ(L)u t Ψ(1) = Ψ(L) = X k=0 X k=0 tx k=1 Ψ k =(I 2 A 1 ) 1 Ψ k L k, Ψ k = u k + ũ t ũ 0, X Ψ j j=k+1 The BN decomposition gives the multivariate stochastic trends in y t in terms of the reduced form error terms u t = TS t = Ψ(1) Ã ψ11 (1) ψ 12 (1) ψ 21 (1) ψ 22 (1) tx u k k=1!ã Ptk=1! u P 1k tk=1 u 2k

6 Using u t = B 1 ε t, Θ(1) = Ψ(1)B 1 the multivariate stochastic trends in y t may also be represented in terms of the structural errors ε t = TS t = Θ(1) Ã θ11 (1) θ 12 (1) θ 21 (1) θ 22 (1) tx ε k k=1!ã P! tk=1 ε P 1k tk=1 ε 2k Remarks: 1. TS t is invariant to the use of u t or ε t 2.The bivariate BN decomposition uses different information set than univariate BN decomposition: they may differ substantially! This is an open area of research.

7 Testing Long-run Neutrality King and Watson (1997) survey the use of bivariate SVAR models to test some simple long-run neutrality propositions in macroeconomics. The key feature of long-run neutrality propositions is that changes in nominal variables have no effect on real economic variables in the long-run. Some examples of long-run neutrality propositions are: 1. A permanent change in the nominal money stock has no long-run effect on the level of real output 2. A permanent change in the rate of inflation has no long-run effect on unemployment (a vertical Phillips curve) 3. A permanent change in the rate of inflation has no long-run effect on real interest rates (the longrun Fisher relationship).

8 KW show that testing long-run neutrality within a SVAR framework requires the data to be I(1). They characterize long-run neutrality of money using the SMA representation for y t written as output: y t = μ y + θ yy (L)ε yt + θ ym (L)ε mt money : m t = μ m + θ my (L)ε yt + θ mm (L)ε mt where ε yt represents exogenous shocks to output that are uncorrelated with exogenous shocks to nominal money ε mt. Long-run neutrality of money involves the answer to the question: Does an unexpected and exogenous permanent change in the level of money (m) leadtoapermanent change in the level of output (y)? If the answer is no, then money is long-run neutral towards output.

9 In terms of the SMA representation, ε mt represents exogenous unexpected changes in money. θ mm (1)ε mt = permanent effect of ε mt on the m θ ym (1)ε mt = permanent effect of ε mt on the y With the data in logs, the long-run elasticity of output with respect to permanent changes in money is γ ym = θ ym(1) θ mm (1) Result: money is neutral in the long-run when θ ym (1) = 0 or γ ym =0 That is, money is neutral in the long-run when the exogenous shocks that permanently alter money, ε mt, have no permanent effect on output.

10 The restriction that money is long-run neutral for output imposes the restriction that the long-run impact matrix Θ(1) is lower triangular. The lower triangularity of Θ(1) implies that the multivariate stochastic trend for y t has the form " TSyt TS mt # = " θyy (1) 0 θ my (1) θ mm (1) #" P # tk=1 ε P yk tk=1. ε mk Hence, the stochastic trend in y t,ts yt, only involves shocks to ε y.

11 SVAR(1) Model y t = c y + λ ym m t +γ 1 yy y t 1 + γ 1 ym m t 1 + ε yt m t = c m + λ my y t cov(ε yt,ε mt ) = 0 where B = +γ my y t 1 + γ 1 mm m t 1 + ε mt à 1 λ ym λ my 1 λ ym = impact elasticity of y wrt m λ my = impact elasticity of m wrt y!

12 To test the long-run neutrality proposition, the SVAR model for y t must be identified and estimated and then the long-run impact coefficients θ ym (1) and θ mm (1) can be estimated from the derived SMA model. From the previous discussion of identification, at least one restriction on the parameters of SVAR is need for identification. KW consider the following identifying assumptions: the impact elasticity of y with respect to m, λ ym, is known, the impact elasticity of m with respect to y,λ my, is known, the long-run elasticity of y with respect to m, γ ym, is known, the long-run elasticity of m with respect to y, γ my, is known.

13 Estimating the SVAR assuming λ ym or λ my is known Suppose λ ym is known. Given that λ ym is known the SVAR(1) may be rewritten as y t + λ ym y 2t = c y + γ 1 yy y t 1 + γ 1 ym m t 1 + ε yt y 2t = γ 20 + λ my y t + γ my y t 1 + γ 1 mm m t 1 + ε mt The first equation may be estimated by OLS since only lagged values of y and m are on the right-hand-side. However, the second equation cannot be estimated by OLS because y t will be correlated with ε mt unless λ my =0. Need an instrument for y t in the second equation

14 Result: If λ ym 6= 0, the second equation may be estimated by IV/GMM using the residual from the estimated first equation, ˆε yt, together with y t 1 and m t 1 as instruments. The residual ε yt is a valid instrument because p lim T 1 T since E[ε yt y t ] 6= 0and TX t=1 ˆε yt y t 6=0 p lim T since E[ε yt ε mt ]=0. 1 T TX t=1 ˆε yt ε mt =0 Remark: Hausman, Newey and Taylor (1987) show that this procedure is asymptotically equivalent to maximum likelihood estimation assuming normal errors. However, the OLS standard errors for the parameters in the second equation must be adjusted because ε yt is used instead of ε yt. See the appendix of King and Watson for details.

15 GMM estimation of SVAR (preferred method) Ignoring the variances and with λ ym known, the SVAR has 7 parameters: c y,γ 1 yy,γ 1 ym,c m,λ my,γ 1 my,γ 1 mm There are 7 population moment conditions E[ε yt ] = E[ε mt ]=E[ε yt ε mt ]=0 E[ y t 1 ε yt ] = E[ y t 1 ε mt ]=0 E[ m t 1 ε yt ] = E[ m t 1 ε mt ]=0 SVAR with λ ym known is exactly identified, and GMM estimation may proceed using the identity weight matrix. Do not have to adjust standard errors for 2nd equation estimates

16 Estimating long-run elasticities and extracting structural shocks Given the estimates ˆB, ˆγ 0 and ˆΓ of the SVAR parameters obtained using the above IV procedure, the SMA representation may be obtained directly. The process is Solve for the reduced form VAR y t = ˆB 1 ĉ 0 + ˆB 1ˆΓ y t 1 + ˆB 1ˆε t = â 0 + Â 1 Y t 1 + ˆB 1ˆε t Define A(L) =I Â 1 L so that Â(1) = I Â 1. Solve for the SMA representation by inverting the reduced form VAR y t = ˆμ + ˆΘ 0ˆε t + ˆΘ 1ˆε t 1 + ˆμ = Â(1) 1 â 0, ˆΘ 0 = ˆB 1 ˆΘ k = Â k 1 ˆB 1 =(ˆB 1ˆΓ) k ˆB 1

17 Solve for the long-run elasticity ˆγ ym = ˆθ ym (1) ˆθ mm (1) " ˆθ ˆΘ(1) = Â(1) 1 ˆB 1 = yy (1) ˆθ ym (1) ˆθ my (1) ˆθ mm (1) # and compute standard errors using delta method The estimated structural errors ˆε t = y t ˆγ 0 ˆΓ 1 y t 1 may be used to compute the BN decompostion and extract the stochastic trends " d TS yt dts mt # = " θyy (1) θ ym (1) θ my (1) θ mm (1) #" P # tk=1 ˆε P yk tk=1 ˆε mk One may also use the reduced form errors û t = y t â 0 Â 1 Y t 1 " # " #" TS d P # yt ˆψ = yy (1) ˆψ ym (1) tk=1 û P yk dts mt ˆψ my (1) ˆψ mm (1) tk=1 û mk where ˆΨ(1) = Â(1) 1.

18 Summary of King and Watson Results Use quarterly data from 1949:I :4 Reduced form VAR is estimated with 6 lags of all variables Long-run money neutrality is not rejected at 5% level for values of λ my (initial impact of money to ouput) < 1.40 Long-run money neutrality is not rejected at 5% level for values of λ ym (initial impact of output to money) > 4.61

19 Structural VARs with Combinations of I(1) and I(0) Data Consider two observed series y t and y 2t such that y t is I(1) and y 2t is I(0). For example, in the analysis in Blanchard and Quah (1989) Define y 1 = log of real GDP y 2 = unemployment rate. y t = ( y 1t,y 2t ) 0 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 0 t] = D =diagonal

20 with reduced form representations y 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 0 t] = Ω = B 1 DB 10 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) ε 11, y 1t+s 1t ε 2t y 2t+s = θ (s) ε 21, y 2t+s 1t ε 2t = θ (s) 12 = θ (s) 22

21 Since y t (output) is I(1), the long-run impacts on the level of y 1 of shocks to ε 1 and ε 2 are lim s lim s y 1t+s ε 1t = θ yy (1) = y 1t+s ε 2t = θ 12 (1) = X s=0 X s=0 θ yy, θ (s) 12. 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 y 2t+s ε jt θ 21 (1) = θ 22 (1) = = lim s θ(s) 2j =0. X s=0 X s=0 θ (s) 21 θ (s) 22 represent the cumulative impact of shocks to ε 1 and ε 2 on the level of y 2.

22 Identifying the SVAR Using Long-Run Restrictions BQ 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) = X s=0 θ (s) 12 =0.

23 The restriction that shocks to ε 1 and ε 2 have no longrun 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 assumptions imply that the long-run impact matrix Θ(1) is lower triangular " # θ11 (1) 0 Θ(1) = θ 21 (1) θ 22 (1) Claim: The lower triangularity of Θ(1) can be used to indentify B.

24 To see why, consider the long-run covariance matrix of y t defined from the Wold representation Since Λ = Ψ(1)ΩΨ(1) 0 = (I 2 A 1 ) 1 Ω(I 2 A 1 ) 10. Ω = B 1 DB 10 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 10 (I 2 A 1 ) 10 = Θ(1)DΘ(1) 0 In order to identify B, BQ 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) 0.

25 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 0 = Θ(1)Θ(1) 0 Θ(1) = P Given that Θ(1) = P can be computed directly from Λ, B canthenbecomputedusing 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!

26 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 in two steps. Estimate the reduced form VAR by OLS equation by equation: y t = ba 0 + c A 1 y t 1 +bu t bω = 1 T TX t=1 bu t bu 0 t Compute a parametric estimate of the long-run covariance matrix: ˆΛ =(I 2 Â 1 ) 1 ˆΩ(I 2 Â 1 ) 10 Compute the Choleski factorization of ˆΛ : ˆΛ = b P b P 0

27 Define the estimate of Θ(1) as the lower triangular Choleski factor of ˆΛ : ˆΘ(1) = ˆP Estimate B using ˆB = h (I 2 Â 1 ) ˆΘ(1) i 1 Estimate Θ k using ˆΘ k = ˆΨ k ˆB 1 = Â k 1 ˆB 1. From the estimated Θ k matrices the structural IRF and FEVD may then be computed. Also, estimates of the structural shocks ˆε 1t and ˆε 2t may be extracted using ˆε t = ˆBû t.