ECON 4160, Spring term Lecture 12

Transcription

1 ECON 4160, Spring term Lecture 12 Cointegrated VAR, SVAR and recursive models. Ragnar Nymoen Department of Economics 14 Nov / 18

2 From I(1 to I(0 systems I Assume that Y 1t I (1 and Y 2t (1, and that after testing, we accept that they are cointegrated Let ˆβ 1 denote the estimated cointegration parameter. (to save notation we have abstracted from a constant term in the cointegrating equation To formulate the corresponding I (0 system (for stationary variables, we regard ˆβ 0 and ˆβ 1 as knows and define the I (0 variable: ( ( 1 Y1t EC t = = Y β 1t ˆβ 1 Y 2t Y 2t 2 / 18

3 From I(1 to I(0 systems II ( Y1t Y 2t = ( α11 α 21 } {{ } α = α ˆβ ( Y t 1 X t 1 EC t 1 + ε t (1 + ε t, (2 which the same form as equation (11 in the slides about Granger s representation theorem. The system (1 has I (0 variables on both sides of the equation sign. We call it the cointegrated VAR (CI-VAR 3 / 18

4 Cointegrated VAR I Even if this particular cointegrated VAR: ( ( Y1t α11 = EC t 1 + ε t Y 2t is very stylized, thereis a couple of things that we can can use it for α Robustness test of cointegration: Internal logical consistency requires that H 0 : α 11 = α 21 = 0 can be rejected against H 1 (how specify? when the cointegrated VAR is estimated (by OLS, assume ε t normally distributed 2. Granger causation: If either (α 11 < 0, α 21 = 0, or (α 11 = 0, α 21 > 0 we have Granger-causality in one direction. Sometimes that has economic theoretical implications. Like testing the consumption Euler-equation. 4 / 18

5 SEM representation of cointegrated VAR I We can interpret the cointegrated VAR as the reduce from of a SEM. Define: ( α11 α 21 ε t = B 1 ɛ t ( = B 1 a11 a 21 ( 1 b12 B = b 21 1 Y 1t + b 12 Y 2t = a 11 EC t 1 + ɛ 1t b 21 Y 1t + Y 2t = a 21 EC t 1 + ɛ 2t (3 This SEM is not identified if all the structural parameter are = 0. 5 / 18

6 SEM representation of cointegrated VAR II But if for example a 21 = 0, and b 12 = 0, we have identification of each equation of the SEM. In practice, the CI-VAR often contains exogenous I (0 variables, for example Z 1t and Z 2t, which will be on the right hand side of (3. Increased the scope for identification. Main point: all the concepts and estimation methods that we have worked with, are relevant to use for the CI-VAR! 6 / 18

7 VAR systems (unrestricted reduced forms I For simplicity of notation, we now go back to the stationary VAR, so do not have to include cointegration relationships. Consider the general open-var (also known as VARX, that we specify when we do Multiple-Equation Modelling: y t = p i=1 Π i y t i + q i=0 Γ i z t i + ε t (4 ε t = IN(0,Σ (5 t = 1, 2,..., T (6 In the program, this is called the Unrestricted System also called Unrestricted Reduced Form (URF. 7 / 18

8 VAR systems (unrestricted reduced forms II In modern macro, main parameters of interest are the impulse responses (dynamic multipliers of random shocks. We have produced some of these in seminars ourselves. 8 / 18

9 Are VAR impulse responses identified parameters? I Consider the open VAR with 2 endogenous variables: ( ( ( Y1t π11 π = 12 Y1t 1 + Y 2t π 21 π 22 Y 2t 1 }{{}}{{}}{{} y t Π 1 y t 1 ( ( γ11 γ 12 Z1t + γ 21 γ 22 Z 2t }{{}}{{} Γ 0 z t ( ε1t ε 2t } {{ } ε t (7 After estimation, we can calculate and plot the partial derivatives of Y 1t+j and Y 2t+j with respect to shocks in ε 1t and ε 2t. 9 / 18

10 Are VAR impulse responses identified parameters? II But we cannot identify them as effects of structural shocks, for example supply schock or demand shocks. (One of Paul Romers points. The under-identification is easily illustrated: Assume that we have a SEM that we write compactly as: By t = Φ 1 y t 1 + Υ 0 z t + ɛ t (8 where the structural disturbances are ɛ t = (ɛ 1t, ɛ 2t, for example a demand shock ɛ 1t and a supply-shock ɛ 2t, if the first structural equation is a demand function, and the second structural equation is a supply function. Let the covariance between the structural disturbances be ω 12, the variances ω 2 1 and ω / 18

11 Are VAR impulse responses identified parameters? III The VAR disturbances are then (in general just a linear combination of the structural disturbances: ε t = B 1 ɛ t. Therefore the estimated impulse responses from the VAR cannot be interpreted as the dynamic effects of unique demand and supply shocks. The VAR disturbances are correlated even if the structural disturbances are uncorrelated. To obtain identified impulse responses (if we want them, see Paul Romer s grausame salbe we need to go beyond the VAR. For example: An identified SEM 11 / 18

12 Are VAR impulse responses identified parameters? IV A model made up of a system of recursive equations. Since we have already discussed SEMs a good deal, we spend a little time on the recursive model. 12 / 18

13 SVAR and recursive model I For simplicity set Υ 0 = 0 to give the closed SEM ( ( ( ( 1 b12 Y1t φ11 φ = 12 Y1t 1 b 21 1 Y 2t φ 21 φ 22 Y 2t 1 The unrestricted reduced form, URF, i.e. the VAR is: ( Y1t Y 2t = + ( φ21 b 12 b 12 b 21 1 φ 11 b 12 b 21 1 φ 11 b 21 b 12 b 21 1 φ 21 b 12 b 21 1 ( 1 b 12 b 21 1 ɛ 1t + b 12 b 12 b 21 1 ɛ 2t b 21 b 12 b 21 1 ɛ 1 1t b 12 b 21 1 ɛ 2t φ 22 b 12 b 12 b 21 1 φ 12 b 12 b 21 1 φ 12 b 21 b 12 b 21 1 φ 22 b 12 b 21 1 ( ɛ1t + ɛ 2t (9 ( Y1t 1 Y 2t.1 13 / 18

14 SVAR and recursive model II If we impose the recursive structure: b 21 = 0 = φ 21 = 0 we obtain a structural VAR, SVAR: ( ( Y1t φ 11 b = 1 φ φ 12 ( 1 Y1t 1 Y 2t 0 φ 22 Y 1 2t 1 ( ɛ 1t + b 12 1 ɛ 2t 0ɛ 1t 1 1 ɛ 2t ( Y1t Y 2t ( ( φ11 φ = 12 φ 22 b 12 Y1t 1 0 φ 22 Y 2t 1 + ( ε 1t What is the difference between this SVAR and the VAR? First: We have one-way Granger causality ε 2t (10 14 / 18

15 SVAR and recursive model III Second: The SVAR disturbances ε 1t and ε 2t : ε 1t = ɛ 1t b 12 ɛ 2t ε 2t = ɛ 2t are uncorrelated if b 12 ω 12 /ω 2 2. This means that if b 12 ω 12 /ω2 2, the impulse-responses with respect to ε 1t and ε 1t are identified. Now, go back to the SEM in (9 and see what b 21 = φ 21 = 0 implies there: ( ( ( ( ( 1 b12 Y1t φ11 φ = 12 Y1t 1 ɛ + 1t 0 1 Y 2t 0 φ 22 Y 2t 1 ɛ2t (11 15 / 18

16 SVAR and recursive model IV We have one-way causal ordering both contemporaneously and in terms of Granger-Causality: There are two lower-triangular matrices with coeffi cients In addition we see that ɛ 2t ε 1t = ɛ 2t since the second line in the recursive model (11 is identical to the second row of the structural VAR Moreover, if b 12 ω 12 /ω2 2, the equation in the first row in the recursive model is a regression model of Y 1t given Y 2t and therefore ɛ1t ε 1t = ɛ 1t ω 12 /ω2ɛ 2 2t, so that Cov(ɛ 1t, ɛ 2t = 0 in the recursive model. 16 / 18

17 SVAR and recursive model V In the recursive model: The first equation (11 is the conditional model of Y 1t given Y 2t, and the second row is a marginal model for Y 2t, given only Y 2t 1.The disturbances of the recursive model are the same as the SVAR disturbances. OLS estimation, equation by equation, gives effi cient ML estimation of both the dynamic recursive model (11 and of the SVAR (10. It is the triangular form of the contemporaneous matrix that is necessary for the identification of the two impact multipliers of the SVAR. It is known as the Cholesky decomposition in the literature. The triangulaization of the matrix with autoregressive coeffi cient represents a kind of overidentfication of the impulse responses. 17 / 18

18 SVAR and recursive model VI Class questions: Why is the recursive model (11 exactly identified, even though it fails on the order and rank conditions? Use orthogonality of disturbances as condition for a recursive model. How would you specify the recursive model if we start from: By t = Φ 1 y t 1 + Υ 0 z t + ɛ t where y t is 3 1 and z t 2 1? 18 / 18