Structural VAR Models and Applications

Transcription

1 Structural VAR Models and Applications Laurent Ferrara 1 1 University of Paris Nanterre M2 Oct. 2018

2 SVAR: Objectives Whereas the VAR model is able to capture efficiently the interactions between the different variables, it does not allow to reveal the underlying causal mecanisms since two different causal schemes can correspond to the same reduced forms. By taking into account certain economic relationships, a Structural VAR model (SVAR) makes it possible to identify structural shocks while letting play the interactions between the different variables

3 Overview of the presentation 1 Structural Vector Auto-Regressions definitions Motivations and Definitions Causal mechanisms 2 Identification issues Cholesky Others 3 Applications 1 Dedola and Lippi 2 Kilian (AER, 2009)

4 SVAR definition Motivations and definitions Motivations Assume 2 stationary processes (x t ) and (z t ). We treat each variable symetrically within the contemporaneous bivariate system: { xt = b 10 b 12 z t + γ 11 x t 1 + γ 12 z t 1 + u xt z t = b 20 b 21 x t + γ 21 x t 1 + γ 22 z t 1 + u zt (1) where u xt and u zt are pure innovation shocks, no cross-correlated at all leads and lags k, ie: u xt WN(σ 2 x) u zt WN(σ 2 z ) ρ(u xt, u zt ) = 0 ρ(u xt, u z,t k ) = 0

5 SVAR definition Motivations and definitions Motivations Definition: This equation is a 1st order VAR process in its primitive form: { xt = b 10 b 12 z t + γ 11 x t 1 + γ 12 z t 1 + u xt z t = b 20 b 21 x t + γ 21 x t 1 + γ 22 z t 1 + u zt Role of coefficients: b 12 = contemporaneous effect of unit change in z t on x t γ 12 = lagged effect of unit change in z t 1 on x t if b 21 0 then a shock u xt has an indirect contemporaneous effect on z t

6 SVAR definition Motivations and definitions Motivations This system is not a reduced form standard VAR equation as z t has a contemporaneous effect on x t and conversely. Let s rewrite equation (1): { xt + b 12 z t = b 10 + γ 11 x t 1 + γ 12 z t 1 + u xt b 21 x t + z t = b 20 + γ 21 x t 1 + γ 22 z t 1 + u zt (2)

7 SVAR definition Motivations and definitions Motivations Then in a matricial form: B 1 y t = Γ 0 + Γ 1 y t 1 + u t, (3) with : y t = (x t, z t ), Γ 0 = (b 10, b 20 ) and u t = (u xt, u zt ) and : ( ) B 1 1 b12 = b 21 1 ( ) γ11 γ Γ 1 = 12 γ 21 γ 22

8 SVAR definition Motivations and definitions Recall If we assume M is a 2 2 matrix such as : ( ) a b M = c d If ad bc 0 then M can be inverted and ( ) M 1 1 d b = ad bc c a

9 SVAR definition Motivations and definitions Motivations Assuming B 1 is inversible, by multiplying by B the primitive form can be rewritten in its reduced form: y t = A 0 + A 1 y t 1 + ε t (4) with A 0 = BΓ 0, A 1 = BΓ 1, ε t = Bu t

10 SVAR definition Motivations and definitions Motivations Thus the reduced form can be rewritten as : { xt = a 10 + a 11 x t 1 + a 12 z t 1 + ε xt z t = a 20 + a 21 x t 1 + a 22 z t 1 + ε zt (5) where residuals ε t are linear combinations of the 2 structural shocks such as: 1 ε xt = (u xt b 12 u zt ) 1 b 21 b 12 1 ε zt = (u zt b 12 u xt ) 1 b 21 b 12 (6)

11 SVAR definition Motivations and definitions Properties of ε t We can show that 1 E(ε xt ) = E(ε zt ) = For all j 0 : E(ε 2 xt) = E(ε 2 zt) = 1 (1 b 12 b 21 ) 2 (σ2 x + b 2 12σ 2 z ) 1 (1 b 12 b 21 ) 2 (σ2 z + b 2 21σ 2 x) E(ε xt ε x,t j ) = 0 E(ε zt ε z,t j ) = 0

12 SVAR definition Motivations and definitions Properties of ε t What about covariance? We can show that: 1 E(ε xt ε zt ) = (1 b 12 b 21 ) 2 (b 21σx 2 + b 12 σz 2 ) In general, non-null covariance except if b 12 = b 21 = 0, that is no contemporaneous effect. Covariance matrix Ω such that: ( ) E(ε 2 Ω = xt ) E(ε xt ε zt ) E(ε xt ε zt ) E(ε 2 zt)

13 SVAR definition Motivations and definitions Identification issue The residuals ε t have good properties, meaning that the reduced form can be efficiently estimated. Question: Is it possible to recover all the information present in the primitive form from the estimation of the reduced form? Answer: NO! 9 parameters in the reduced form but 10 parameters in the primitive form : The system is not fully identified

14 SVAR definition Motivations and definitions Identification issue Solution (Sims, 1980): recursive identification scheme, ie : recursive system with restrictions on the primitive form. Ex. Assume z t does not have any contemporanesous effect on x t, ie : b 21 = 0 Thus B 1 is a lower diagonal matrix: ( ) B = b 21 1 Alternative: b 12 = 0, ie: x t has no effect on z t Main issue in SVAR modelling: how to define B 1?

15 SVAR definition Causal mechanisms Causal mechanisms in a stylized economy IRFs from SVAR with Inflation, Unemployment and FFR

16 SVAR definition Causal mechanisms Causal mechanisms in a stylized economy Assume that the GDP growth g t is affected by some real shocks u r,t following: g t = 0.3(i t 1 π t ) + u r,t where i t is the nominal interest rate, π t is the inflation rate. Besides, assume that we have a simple Taylor rule governing short-term interest rates : i t = 0.9i t π t + u mp,t where u mp,t is a monetary-policy shock. The inflation rate is supposed to follow a backward-looking Phillips curve: π t = 0.9π t g t 1 + u n,t where u n,t is a cost-push shock.

17 SVAR definition Causal mechanisms Causal mechanisms in a stylized economy The structural shocks u t = (u r,t, u mp,t, u n,t ) are uncorrelated (i.e., the covariance matrix of u t, denoted with Ω u is diagonal) The structural shocks u t are serially uncorrelated (i.e. Cov(u t k, u t ) = 0 for any t and k > 0).

18 SVAR definition Causal mechanisms Causal mechanisms in a stylized economy The structural model reads g t = 0.3(i t 1 π t ) + u r,t i t = 0.9i t π t + u mp,t π t = 0.9π t g t 1 + u n,t. To get it in a reduced-form, let us substitute π t in the right-hand sides of the first two equations: g t = 0.06g t 1 0.3i t π t u n,t + u r,t i t = 0.9i t π t g t 1 + u mp,t + 1.5u n,t π t = 0.9π t g t 1 + u n,t.

19 SVAR definition Causal mechanisms Causal mechanisms in a stylized economy In matrix form g t i t π t = g t 1 i t 1 π t 1 + ε g,t ε i,t ε π,t ε g,t u r,t u r,t with ε i,t = u mp,t = B u mp,t ε π,t u n,t u n,t With the procedure described above, one only gets an estimate of Ω ε where Ω ε = Var ε g,t ε i,t ε π,t..

20 Identification issues Identification in SVAR We saw that a structural model can be thought in its general form : ε t = y t E(y t I t 1 ), (7) with ε t = Bu t (8) In our case of SVAR(p), equation (7) is: or p y t = A 0 + A i y t i + ε t, (9) i=1 A(L)y t = ε t, (10)

21 Identification issues SVAR: Orthogonalization Estimation of AR parameters via MLE then estimation of structural parameters in B Note that we must have Ω ε = BΩ u B where Ω u is diagonal positive such as: Ω u = I. In addition, given the structural framework, one knows that B is a triangular matrix (lower or upper according to the specification).

22 Identification issues SVAR: Orthogonalization It has been shown previously that a SVAR is a structural model that draws from a theoretical framework. As a starting point, we always have Ω ε = BΩ u B that provides us with n(n + 1)/2 restrictions to recover the B matrix. Consequently, to get the B matrix, one have to impose n(n 1)/2 additional restrictions.

23 Identification issues SVAR: Restrictions There exist several restrictions that can be implemented in a SVAR: a short-run restriction prevents a structural shock from affecting an endogenous variable contemporaneously by setting to zero some entries of B. a long-run restriction prevents a structural shock from affecting an endogenous variable in a cumulative way. a sign restriction imposes negative or positive parameters in matrix B.

24 Identification issues SVAR: Short-Run restrictions Short-run restrictions are simpler to implement than the other ones. There are particular cases in which some well-known matrix decompostion can be used to easily estimate some specific SVAR. Imagine a context in which you can argue that there exists an ordering of the shocks: A first shock structural u 1,t can affect instantaneously (i.e., in t) only one of the endogenous variable (say, y 1,t ) through ε 1,t ; A second shock, u 2,t can affect instantaneously the first two endogenous variables (say, y 1,t and y 2,t ) through ε 1,t and ε 2,t ;...

25 Identification issues SVAR: Short-Run restrictions Within this framework, the structural recursive mechanism of variable ordering allows the creation of the lower-triangular B matrix: ε 1,t = u 1,t ε 2,t = β 21 u 1,t + u 2,t ε 3,t = β 31 u 1,t + β 32 u 2,t + u 3,t... =... ε n,t = β n1 u 1,t β n,n 1 u n 1,t + u n,t where the u t are the structural shocks and are not cross-correlated at all leads and lags.

26 Identification issues SVAR: Short-Run restrictions The triangular structural model imposes the recursive causal ordering: y 1 y 2... y n y 1 causes y 2, y 3,..., y n but y 2, y 3,..., y n do not cause y 1 y 2 causes y 3,..., y n but y 3,..., y n do not cause y 2... Obviously, a different ordering of variables leads to a different result

27 Identification issues Cholesky decomposition/factorization Decomposition of a symmetric positive definite matrix M into a lower-triangular matrix and its conjugate transpose (by André-Louis Cholesky), ie: M = LL with l 11 l 21 l 22 L =..... l n1 l n2... l nn

28 Identification issues Cholesky decomposition/factorization Property: The ML estimate of B is the Cholesky decomposition of ˆΩ ε the sample covariance matrix of VAR residuals. If the model is just-identified, ˆΩ ε (BB ) 1 = I n and the log-likelihood simplifies to: L = Const 0.5 log ˆΩ ε 0.5n

29 Identification issues IRF for SVAR If A(L) is invertible in eq. (10) then the VMA representation is: y t = A(L) 1 ε t = Θ(L) = ε t + Θ 1 ε t (11) thus the structural VMA representation is: Thus, the IRF: y t = Bu t + Θ 1 Bu t = Q 0 u t + Q 1 u t 1... (12) I i,j,h = y i,t+h u j,t (13) = (Q h ) ij (14) Confidence intervals are generally computed by Bootstrap

30 Identification issues Forecast Error Variance Decomposition FEV after h steps: Ω h = h Q k Q k (15) k=0 Hence the variance v i for y i is : h v i = (Ω h ) ii = e i Q k Q k e i = k=0 h n k=0 l=1 (q k i,l )2 where e i is a zero-vector with 1 at the i th element and (q k i,l )2 is the (i, l) element of Q k. Thus the share of variance on y i attributed to the j th shock after h periods is: VD i,j,h = h k=0 (qk i,l )2 h n k=0 l=1 (qk i,l )2

31 Applications Bloom (2009): Macro response to a VIX shock Bloom (Ect,2009) : Macro impact of stock market volatility shock Monthly SVAR from June 1962 to June Identification scheme is based on Cholesky decompositions, the ordering of the endogenous variables being (from the most exogeneous): log(sp500), VIX, FFR, log(earnings), log(cpi), hours, log(employment) and log(ip) This ordering implies that shocks instantaneously influence the stock market (level and volat), then prices and finally quantities

32 Applications VIX index

33 Applications Macro response to a VIX shock

34 Applications Macro response to a VIX shock

35 Applications Kilian (2009): Oil prices Not all shocks are alike: Disentangling demand from supply shocks in the crude oil market AER. Objective: Distinct effects of supply and demand shocks on oil prices Methodology: A structural monthly VAR is used to identify aggregate supply of oil, aggregate demand of oil and precautionary shocks Findings: Precautionary shocks have adverse effects on oil prices.

36 Applications Kilian (2009) Data Global oil production Global demand estimated by monthly industrial commoditites Real price of oil Model: A 0 y t = α + 24 i=1 A i y t 1 + ε t where y t = ( Prod t, rea t, rpo t )

37 Applications Kilian (2009) u Prod,t u Rea,t u Rpo,t = a a 21 a 22 0 a 31 a 32 a 33 ε Oilsupply,t ε Aggregatedemand,t ε Oilspecific,t

38 Applications

39 Applications References Blanchard, O. and Quah, D. (1989). The Dynamic Effects of Aggregate Demand and Supply Disturbances. American Economic Review, vol. 79. Dedola, L. and Lippi, F. (2005). The monetary transmission mechanism: Evidence from the industries of five OECD countries. European Economic Review, vol. 49 (6). Gerlach, S. and F. Smets (1995). The monetary transmission mechanism: evidence from the G7 countries. CEPR Discussion Paper, no Hamilton, J. (1994). Time Series Analysis. Princeton University Press. Kilian, L. (2009. Not all shocks are alike: Diesntangling supply from demand shocks on the crude oil marketamerican Economic Review, vol. Sims, C. (1980). Macroeconomics and Reality. Econometrica, vol. 48. Smets, F. and Tsatsaronis, K. (1997). Why does the yield curve predict economic activity? BIS Working Paper No. 49.