Consider the structural VAR (SVAR) model

Transcription

1 Structural VARs Structural Representation Consider the structural VAR (SVAR) model y 1t = γ 10 b 12 y 2t + γ 11 y 1t 1 + γ 12 y 2t 1 + ε 1t y 2t = γ 20 b 21 y 1t + γ 21 y 1t 1 + γ 22 y 2t 1 + ε 2t where à ε1t ε 2t! iid Ãà 0 0!, à σ σ 2 2!!. Remarks: ε 1t and ε 2t are called structural errors In general, cov(y 2t,ε 1t ) 6= 0and cov(y 1t,ε 2t ) 6= 0 All variables are endogenous - OLS is not appropriate!

2 In matrix form, the model becomes 1 b12 b 21 1 y1t or = γ10 γ 20 y 2t + γ11 γ 12 γ 21 γ 22 y1t 1 y 2t 1 + ε1t ε 2t By t = γ 0 + Γ 1 y t 1 + ε Ã! t E[ε t ε 0 σ 2 t] = D = σ 2 2 In lag operator notation, the SVAR is B(L)y t = γ 0 + ε t, B(L) = B Γ 1 L.

3 Reduced Form Representation Solve for y t in terms of y t 1 and ε t : or 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 a 0 = B 1 γ 0, A 1 = B 1 Γ 1, u t = B 1 ε t Note that B 1 = 1 A(L)y t = a 0 + u t A(L) = I 2 A 1 L 1 b 12 b 21 1, =det(b) =1 b 12 b 21 The reduced form errors u t are linear combinations of the structural errors ε t and have covariance matrix E[u t u 0 t] = B 1 E[ε t ε 0 t]b 10 = B 1 DB 10 = Ω. Remark: Parameters of RF may be estimated by OLS equation by equation

4 Identification Issues Without some restrictions, the parameters in the SVAR are not identified. That is, given values of the reduced form parameters a 0, A 1 and Ω, it is not possible to uniquely solve for the structural parameters B, γ 0, Γ 1 and D. 10 structural parameters and 9 reduced form parameters Order condition requires at least 1 restriction on the SVAR parameters Typical identifying restrictions include Zero (exclusion) restrictions on the elements of B; e.g., b 12 =0. Linear restrictions on the elements of B; e.g., b 12 + b 21 =1.

5 MA Representations Wold representation Multiplying both sides of reduced form by A(L) 1 = (I 2 A 1 L) 1 to give y t = μ + Ψ(L)u t Ψ(L) = (I 2 A 1 L) 1 = X k=0 μ = A(1) 1 a 0 E[u t u 0 t] = Ω Ψ k L k, Ψ 0 = I 2, Ψ k = A k 1 Remark: Wold representation may be estimated using RF VAR estimates

6 Structural moving average (SMA) representation SMA of y t isbasedonaninfinite moving average of the structural innovations ε t. Using u t = B 1 ε t in the Wold form gives That is, y t = μ + Ψ(L)B 1 ε t Θ(L) = = μ + Θ(L)ε t X k=0 Θ k L k = Ψ(L)B 1 = B 1 + Ψ 1 B 1 L + Θ k = Ψ k B 1 = A k 1 B 1, k =0, 1,... Θ 0 = B 1 6= I 2

7 Example: SMA for bivariate system Notes y1t y 2t = μ1 + + μ 2 θ(1) 11 θ (1) 12 θ (1) 21 θ (1) 22 θ(0) 11 θ (0) 12 θ (0) 21 θ (0) 22 ε1t 1 ε 2t 1 ε1t ε 2t + Θ 0 = B 1 6= I 2. Θ 0 captures initial impacts of structural shocks, and determines the contemporaneous correlation between y 1t and y 2t. Elements of the Θ k matrices, θ (k) ij, give the dynamic multipliers or impulse responses of y 1t and y 2t to changes in the structural errors ε 1t and ε 2t.

8 Impulse Response Functions Consider the SMA representation at time t + s y1t+s y 2t+s = μ1 + + μ 2 θ(s) 11 θ (s) 12 θ(0) 11 θ (0) 12 θ (0) 21 θ (0) ε1t+s ε 2t+s 22 ε1t +. ε 2t θ (s) 21 θ (s) 22 The structural dynamic multipliers are + y 1t+s = θ (s) ε 11, y 1t+s 1t y 2t+s ε 2t = θ (s) 12 = θ (s) ε 21, y 2t+s = θ (s) 1t ε 22 2t The structural impulse response functions (IRFs) are the plots of θ (s) ij vs. s for i, j =1, 2. These plots summarize how unit impulses of the structural shocks at time t impact the level of y at time t + s for different values of s. Stationarity of y t implies lim s θ(s) ij =0, i,j =1, 2

9 The long-run cumulative impact of the structural shocks is captured by P θ11 (1) θ Θ(1) = 12 (1) s=0 = θ (s) 11 θ 21 (1) θ 22 (1) P s=0 θ (s) 21 θ11 (L) θ Θ(L) = 12 (L) θ 21 (L) θ 22 (L) = P s=0 θ (s) 11 Ls P s=0 θ (s) 12 Ls P s=0 θ (s) 21 Ls P s=0 θ (s) 22 Ls P s=0 θ (s) 12 P s=0 θ (s) 22

10 Digression: Dynamic Regression Models In the SVAR every variable is engodenous. Suppose, for example, y 2t is strictly exogenous which implies b 21 =0 and γ 21 =0. Then, the first equation is an ADL(1,1) y 1t = α + φy 1t 1 + β 0 y 2t + β 1 y 2t 1 + ε 1t cov(y 2t,ε 1t ) = 0 In lag operator notation the equation becomes φ(l)y 1t = α + β(l)y 2t + ε 1t φ(l) = 1 φl, β(l) =β 0 + β 1 L The second equation is an AR(1) model for y 2t y 2t = c + ρy 2t 1 + ε 2t Stationarity now only requires φ < 1 and ρ < 1.

11 The first equation may then be solved for y 1t as a function of y 2t and ε 1t y 1t = α φ(1) + φ(l) 1 β(l)y 2t + φ(l) 1 ε 1t = μ + ψ β (L)y 2t + ψ(l)ε t μ = α φ(1) ψ β (L) = φ(l) 1 β(l), ψ(l) =φ(l) 1 Since y 2t is exogenous, we have two sources of shocks. Note: there can be four types of dynamic multipliers : y 1t+s y 2t, y 1t+s ε 2t, y 1t+s, y 2t+s ε 1t ε 2t

12 The short-run dyamic multipliers with respect to y 2t and ε 1t are y 1t+s = y 1t = ψ y 2t y β,s 2t s y 1t+s = y 1t = ψ ε 1t ε s 1t s In the steady state or long-run equilibrium all variables are constant y1 = μ + ψ β(l)y2 = μ + ψ β(1)y2 y2 = c 1 ρ ψ β (1) = φ(1) 1 β(1) = β 0 + β 1 1 φ The long-run impact of a change in y 2 on y 1 is then y 1 y 2 = ψ β (1) = β 0 + β 1 1 φ = X s=0 y 1t+s y 2t

13 Identification issues In some applications, identification of the parameters of the SVAR is achieved through restrictions on the parameters of the SMA representation. Identification through contemporaneous restrictions Suppose that ε 2t has no contemporaneous impact on y 1t. Then θ (0) 12 =0and Θ 0 = Since Θ 0 = B 1 then θ(0) 11 0 θ (0) 21 θ (0) 22 θ(0) 11 0 = 1 θ (0) 21 θ (0) 22 b 12 =0. 1 b 12 b 21 1 Hence, assuming θ (0) 12 =0in the SMA representation is equivalent to assuming b 12 =0in the SVAR representation.

14 Identification through long-run restrictions Suppose ε 2t has no long-run cumulative impact on y 1t. Then θ 12 (1) = Θ(1) = X θ (s) 12 =0 s=0 θ11 (1) 0 θ 21 (1) θ 22 (1) This type of long-run restriction places nonlinear restrictions on the coefficients of the SVAR since Θ(1) = Ψ(1)B 1 = A(1) 1 B 1 = (I 2 B 1 Γ 1 ) 1 B 1.

15 Estimation Issues In order to compute the structural IRFs, the parameters of the SMA representation need to be estimated. Since Θ(L) = Ψ(L)B 1 Ψ(L) = A(L) 1 =(I 2 A 1 L) 1 the estimation of the elements in Θ(L) can often be broken down into steps: A 1 is estimated from the reduced form VAR. Given c A 1, the matrices in Ψ(L) can be estimated using c Ψ k = c A k 1. B is estimated from the identified SVAR. Given ˆB and ˆΨ k, the estimates of Θ k, k =0, 1,..., are given by ˆΘ k = c Ψ k ˆB 1.

16 Forecast Error Variance Decompositions Idea: determine the proportion of the variability of the errors in forecasting y 1 and y 2 at time t + s based on information available at time t that is due to variability in the structural shocks ε 1 and ε 2 between times t and t + s. To derive the FEVD, start with the Wold representation for y t+s y t+s = μ + u t+s + Ψ 1 u t+s 1 + +Ψ s 1 u t+1 + Ψ s u t + Ψ s+1 u t 1 +. The best linear forecast of y t+s based on information available at time t is y t+s t = μ + Ψ s u t + Ψ s+1 u t 1 + and the forecast error is y t+s y t+s t = u t+s + Ψ 1 u t+s Ψ s 1 u t+1.

17 Using ε t = B 1 u t, Θ k = Ψ k B 1 The forecast error in terms of the structural shocks is y t+s y t+s t = B 1 ε t+s + Ψ 1 B 1 ε t+s Ψ s 1 B 1 ε t+1 = Θ 0 ε t+s + Θ 1 ε t+s Θ s 1 ε t+1 The forecast errors equation by equation are y1t+s y 1t+s t y 2t+s y 2t+s t + = θ(s 1) 11 θ (s 1) 12 θ(0) 11 θ (0) 12 θ (0) 21 θ (0) ε1t+s ε 2t+s 22 ε1t+1 ε 2t+1 θ (s 1) 21 θ (s 1) 22 +

18 For the first equation y 1t+s y 1t+s t = θ (0) 11 ε 1t+s + + θ (s 1) 11 ε 1t+1 +θ (0) 12 ε 2t+s + + θ (s 1) 12 ε 2t+1 Since it is assumed that ε t i.i.d. (0, D) where D is diagonal, the variance of the forecast error in may be decomposed as var(y 1t+s y 1t+s t )=σ 2 1 (s) à µ = σ 2 1 θ (0) 2 µ + + +σ à µ θ (0) 2 µ θ (s 1) 11 θ (s 1) 12 2! 2! The proportion of σ 2 1 (s) due to shocks in ε 1 is then à µ σ 2 1 θ (0) 2 µ θ (s 1)! 2 11 ρ 1,1 (s) = σ 2 1 (s).

19 the proportion of σ 2 1 (s) due to shocks in ε 2 is à µ σ 2 2 θ (0) 2 µ θ (s 1)! 2 12 ρ 1,2 (s) = σ 2 1 (s).

20 The forecast error variance decompositions (FEVDs) for y 2t+s are à µ σ 2 1 θ (0) 2 µ θ (s 1)! 2 21 ρ 2,1 (s) = where ρ 2,2 (s) = σ 2 2 à µ θ (0) 22 σ 2 2 (s), 2 µ! θ (s 1) 22 σ 2 2 (s), var(y 2t+s y 2t+s t )=σ 2 2 (s) à µ = σ 2 1 θ (0) 2 µ + + +σ à µ θ (0) 2 µ θ (s 1) 21 θ (s 1) 22 2! 2!.

21 Identification Using Recursive Causal Orderings Consider the bivariate SVAR. We need at least one restriction on the parameters for identification. Suppose b 12 =0so that B is lower triangular. That is, 1 0 B = b 21 1 B 1 = Θ 0 = 1 0 b 21 1 The SVAR model becomes the recursive model y 1t = γ 10 + γ 11 y 1t 1 + γ 12 y 2t 1 + ε 1t y 2t = γ 20 b 21 y 1t + γ 21 y 1t 1 + γ 22 y 2t 1 + ε 2t The recursive model imposes the restriction that the value y 2t does not have a contemporaneous effect on y 1t. Since b 21 6=0a priori we allow for the possibility that y 1t has a contemporaneous effect on y 2t.

22 The reduced form VAR errors u t = B 1 ε t become u t = = u1t u 2t = ε 1t ε 2t b 21 ε 1t 1 0 b ε1t ε 2t Claim: The restriction b 12 =0is sufficient to just identify b 21 and, hence, just identify B.

23 To establish this result, we show how b 21 can be uniquely identified from the elements of the reduced form covariance matrix Ω. Note ω 2 1 ω 12 ω 12 ω 2 2 = 1 0 b 21 1 σ σ 2 2 σ = 2 1 b 21 σ 2 1 b 21 σ 2 1 σ2 2 + b2 21 σ2 1 Then,wecansolveforb 21 via b 21 = ω 12 ω 2 1 = ρ ω 2 ω 1, 1 b where ρ = ω 12 /ω 1 ω 2 is the correlation between u 1 and u 2. Notice that b 21 6=0provided ρ 6= 0.

24 Estimation Procedure 1. 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 2. Estimate b 21 and B from b Ω : b ω 21 = b 12, bb = bω b Estimate SMA from estimates of a 0, A 1 and B: y t = bμ + Θ(L)bε c t bμ = ba 0 (I 2 A c 1 ) 1 cθ k = A ck 1B b 1,k =0, 1,... cd = b B b Ω b B 0.

25 Remark: Above procedure is numerically equivalent to estimating the triangular system by OLS equation by equation: y 1t = γ 10 + γ 11 y 1t 1 + γ 12 y 2t 1 + ε 1t y 2t = γ 20 b 21 y 1t + γ 21 y 1t 1 + γ 22 y 2t 1 + ε 2t Why? Since cov(ε 1t,ε 2 )=0by assumption, cov(y 1t,ε 2t )= 0

26 Recovering the SMA representation using the Choleski Factorization of Ω. Claim: The SVAR representation based on a recursive causal ordering may be computed using the Choleski factorization of the reduced form covariance matrix Ω. Recall, the Choleski factorization of the positive semidefinite matrix Ω is given by Ω = PP 0 P = p11 0 p 21 p 22 A closely related factorization obtained from the Choleski factorization is the triangular factorization T = Ω = TΛT λ1 0, Λ = t λ 2 λ i 0,i=1, 2.,

27 Consider the reduced form VAR y t = a 0 + A 1 y t 1 + u t, Ω = E[u t u 0 t] Ω = TΛT 0 Construct a pseudo SVAR model by premultiplying by T 1 : or T 1 y t = T 1 a 0 + T 1 A 1 y t 1 + T 1 u t By t = γ 0 + Γ 1 y t 1 + ε t where B = T 1, γ 0 = T 1 a 0, Γ 1 = T 1 A 1, ε t = T 1 u t.

28 The pseudo structural errors ε t have a diagonal covariance matrix Λ E[ε t ε 0 t] = T 1 E[u t u 0 t]t 10 = T 1 ΩT 10 = T 1 TΛT 0 T 10 = Λ. In the pseudo SVAR, 1 0 B = b 21 1 = T 1 = 1 0 t 21 1 b 12 = 0, b 21 = t 21

29 Ordering of Variables The identification of the SVAR using the triangular factorization depends on the ordering of the variables in y t. In the above analysis, it is assumed that y t =(y 1t,y 2t ) 0 so that y 1t comes first in the ordering of the variables. When the triangular factorization is conducted and the pseudo SVAR is computed the structural B matrix is B = T = b 21 1 b 12 =0 If the ordering of the variables is reversed, y t =(y 2t,y 1t ) 0, then the recursive causal ordering of the SVAR is reversed and the structural B matrix becomes B = T = b 12 1 b 21 =0

30 Sensitivity Analysis Ordering of the variables in y t determines the recursive causal structure of the SVAR, This identification assumption is not testable Sensitivity analysis is often performed to determine how the structural analysis based on the IRFs and FEVDs are influenced by the assumed causal ordering. This sensitivity analysis is based on estimating the SVAR for different orderings of the variables. If the IRFs and FEVDs change considerably for different orderings of the variables in y t then it is clear that the assumed recursive causal structure heavily influences the structural inference.

31 Residual Analysis One way to determine if the assumed causal ordering influences the structural inferences is to look at the residual covariance matrix ˆΩ from the estimated reduced form VAR. If this covariance matrix is close to being diagonal then the estimated value of B will be close to diagonal and so the ordering of the variables will not influence the structural inference.