Exogeneity and Causality

Transcription

1 Università di Pavia Exogeneity and Causality Eduardo Rossi University of Pavia

2 Factorization of the density DGP: D t (x t χ t 1, d t ; Ψ) x t represent all the variables in the economy. The econometric analysis will focus on explaining a subset of variables in terms of the history of the system and of a contemporaneous subset z t, treated as given. Treating z t as given is motivated by the assumption that z t causes y t. What is causality in econometrics? Factorization of the density y t,z t. y t subset of x t ; z t subset of x t.w t variables in x t that / y t,z t. D = D w,y,z = D w y,z D y z D z (1) ( ) presence (absence) of contemporaneous casuality ( ) presence (absence) of simultaneous relations Eduardo Rossi - Macroeconometria 2

3 Factorization of the density If z t y t w t y t The factor represents D y z completely represents the stochastic mechanism generating y t. Note that y t w t is allowed, and no restriction on the relationships between w t and z t is required, for this to be true. Eduardo Rossi - Macroeconometria 3

4 Factorization of the density Denote by W t 1 = σ (w t 1,w t 2,...) Y t 1 = σ (y t 1,y t 2,...) Z t 1 = σ (z t 1,z t 2,...) Assume there exists a partition of θ into two subvectors θ 1 Θ 1 and θ 2 Θ 2, such that Θ = Θ 1 Θ 2 and D w y,z = D w y,z (w t y t,z t, W t 1, Y t 1, Z t 1 ;d t, θ 2 ) (2) D y z = D y z (y t z t, Y t 1, Z t 1 ;d t, θ 1 ) (3) D z = D z (z t W t 1, Y t 1, Z t 1 ;d t, θ 2 ) (4) D w y,z and D z must not depend on θ 1. D y z must not depend on w t j for j > 0, in the sense that either conditioning or not conditioning on these variables has the same effect. Eduardo Rossi - Macroeconometria 4

5 Factorization of the density Under the condition Θ = Θ 1 Θ 2 the admissible values of θ 1 may not depend on θ 2, so that knowledge of the latter cannot improve influences about the former. In this case θ 1 and θ 2 are said to be variation free. Under those conditions nothing need be known about the forms of D w y,z and D z to analyze D y z since these do not depend on θ 1. The analysis is conducted conditioning on z t and marginalizing on w t. Sequential cut: The separation of θ into two sets. θ 1 parameters of interest for the investigation θ 2 parameters that are not of interest. Eduardo Rossi - Macroeconometria 5

6 Weak exogeneity Suppose that D y z depends on a vector φ of parameter of interest, whose values are the focus of investigation. To make the desired factorization of the DGP, it is only necessary that there exists some parameterization θ such that (2) holds, with θ 1 and θ 2 variation free, and φ = g (θ 1 ). In this analysis, the y t are called endogenous, z t are called weakly exogenous for φ. Weak exogeneity is a relationship between parameters and variables, and is not a property of variables as such. Without the required cut of the parameters, the factorization (1) is not relevant to the investigation. Eduardo Rossi - Macroeconometria 6

7 Granger Causality y t is said to cause z t in Granger s sense if knowledge of y t aids the prediction of z t+j for some j > 0. Since χ t 1 appears as the conditioning variables in D z, y t may Granger-cause z t without violating the weak exogeneity of z t. Suppose there exists the following factorization D z (z 1t,z 2t χ t 1 ) = D z1 z 2 (z 1t z 2t, χ t 1 ) D z2 (z 2t W t 1, Z t 1 ) in this case y t does not Granger-cause z 2t. Granger non-causality is a property of the DGP. It is a relationship between variables. If z 2t is both not Granger-caused by y t and weakly exogenous for φ these variables are said to be strongly exogenous for φ. Eduardo Rossi - Macroeconometria 7

8 Granger Causality The weak endogeneity does not rule out the fact that y t 1 z t whereas the existence of lagged feedback is compatible with making efficient inferences about φ, it rules out treating z t as given in multistep forecasting and policy simulation exercises involving y t. Strongly exogenous variables can be regarded as fixed to all intents an purposes, and from a statistical viewpoint can be treated like the deterministic variables d t. Eduardo Rossi - Macroeconometria 8

9 Other notions of exogeneity Exogeneity is sometimes defined in terms of the independence of the variables in question from the disturbances in a model. In the regression model y t = x tβ + ε t t = 1,...,T x t is independent of ε t+j, j 0, E[x t ε t ] = 0. x t is said to be predetermined. If the independence holds for all j, x t is said to be strictly exogenous. Eduardo Rossi - Macroeconometria 9

10 Gaussian VAR(1). No marginalized variables, x t = (y t, z t ). Structural equations y t + α 12 z t = γ 1 + β 11 y t 1 + β 12 z t 1 + u 1t α 21 y t + z t = γ 2 + β 21 y t 1 + β 22 z t 1 + u 2t Structural form Γx t = Cd t +Bx t 1 + u t the vector d t consists of dummy variables, one of which is equal to 1 in all periods, representing the intercept of the linear relations, and the others might include trends and seasonals. Eduardo Rossi - Macroeconometria 10

11 E (u t χ t 1 ) = 0 E (u t u t χ t 1 ) = Σ u t = u 1t u 2t NID 0 0, σ 11 σ 12 σ 21 σ 22 Eduardo Rossi - Macroeconometria 11

12 Parameters of interest φ = (α 12, γ 1, β 11, β 12 ). The joint density of the variables is D (y t, z t y t 1, z t 1 ) = = 1 ω 11 ω 12 2π ω 21 ω 22 1/2 exp 1 2 [ ] ε 1t ε 2t ω 11 ω 12 ω 21 ω 22 1 ε 1t ε 2t ε t = Γ 1 u t E (ε t χ t 1 ) = 0 E (ε t ε t χ t 1 ) = Ω Ω = Γ 1 Σ (Γ ) 1 Eduardo Rossi - Macroeconometria 12

13 To factorize the joint density into conditional and marginal components, remember that for x N (µ,σ), where x = x 1 (n 1 1) x 2 (n 2 1) µ = µ 1 Σ = µ 2 (n 1 1) (n 2 1) Σ 11 Σ 12 Σ 21 Σ 22 Eduardo Rossi - Macroeconometria 13

14 where f (x 1,x 2 ) = (2π) n/2 Σ 11 Σ 12 Σ 1 { exp Σ 21 1/2 Σ 22 1/2 [ (x1 µ 1 ) E (x 1 µ 1 ) (x 1 µ 1 ) EΣ 12 Σ 1 22 (x 2 µ 2 ) (x 2 µ 2 ) Σ 1 22 Σ 21E (x 1 µ 1 ) + (x 2 µ 2 ) ( Σ Σ 1 22 Σ 21EΣ 12 Σ 1 22 E = ( Σ 11 Σ 12 Σ 1 22 Σ 21 ) 1 ) (x2 µ 2 ) ]} Eduardo Rossi - Macroeconometria 14

15 The terms in the exponent can be rearranged as [ x1 µ 1 Σ 12 Σ 1 22 (x 2 µ 2 ) ] E [ x1 µ 1 Σ 12 Σ 1 22 (x 2 µ 2 ) ] + (x 2 µ 2 ) Σ 1 22 (x 2 µ 2 ) so we have the factorization f (x 1,x 2 ) = f (x 1 x 2 )f (x 2 ): f (x 1 x 2 ) = (2π) n 1/2 Σ 11 Σ 12 Σ 1 22 Σ 1/2 21 { exp 1 2 [ x1 µ 1 Σ 12 Σ 1 22 (x 2 µ 2 ) ] E [ x1 µ 1 Σ 12 Σ 1 22 (x 2 µ 2 ) ]} f (x 2 ) = (2π) n 2/2 Σ 22 1/2 exp { 1 } 2 (x 2 µ 2 ) Σ 1 22 (x 2 µ 2 ) Eduardo Rossi - Macroeconometria 15

16 In this case ω 11 ω 12 ω 21 ω 22 = ( ) ω 11 ω2 21 ω 22 ω 22 by simmetry ω 21 = ω 12 ω 11 ω 21 ω 21 ω 22 1 = [( ) ] 1 ω 11 ω2 21 ω 22 ω 22 ω 21 ω 22 ω 21 ω 11 [ ] ε 1t ε 2t ω 11 ω 12 ω 21 ω 22 1 ε 1t ε 2t = ( ε 1t ω 21 ω 22 ε 2t ) 2 ω 11 ω2 21 ω 22 + ε 2 2t ω 22 Eduardo Rossi - Macroeconometria 16

17 The conditional density D y z (y t z t, y t 1, z t 1 ) = 2π ( 1 ( ) exp ω 11 ω ω 22 ) 2 ε 1t ω 21 ω 22 ε 2t ( ) ω 11 ω2 21 ω 22 The marginal density D z (z t y t 1, z t 1 ) = { } 1 exp ε2 2t 2πω22 2ω 22 note that ε 1t = y t E (y t y t 1, z t 1 ) ε 2t = z t E (z t y t 1, z t 1 ) so that ε 2t is treated as conditionally fixed in D y z. Eduardo Rossi - Macroeconometria 17

18 The factorization does not divide the parameters into subsets θ 1 and θ 2, where φ = g (θ 1 ) ε t = Γ 1 u t Γ 1 = ε 1t ε 2t 1 1 α 12 α 21 = = 1 1 α 12 α α 12 α 21 1 α 12 α α 12 α 21 1 u 1t α 12 u 2t u 2t α 21 u 2t u 1t u 2t Eduardo Rossi - Macroeconometria 18

19 Ω = Γ 1 Σ (Γ ) 1 ¾ ω 11 ω 12 ω 21 ω 22 1 = (1 α 12 α 21 ) 2 ¾ σ 11 2α 12 σ 21 + α 2 12σ 22 (1 + α 12 α 21 ) σ 12 α 12 σ 22 α 21 σ 11 σ 22 2α 21 σ 21 + α 2 21σ 11 Eduardo Rossi - Macroeconometria 19

20 All the parameters appear in both factors. In the case that α 21 = 0 and σ 21 = 0, it is easy verify that the conditional and marginal factors become { } 1 D y z (y t z t, y t 1, z t 1 ; θ 1 ) = exp u2 1t 2πσ11 D z (z t y t 1, z t 1, θ 2 ) = { 1 exp u2 2t 2πσ22 2σ 22 2σ 11 where θ 1 = (α 12, γ 1, β 11, β 12, σ 11 ) and θ 2 = (γ 2, β 21, β 22, σ 22 ). These conditions are both necessary and sufficient for the factorization to cut the parameters as required. } Eduardo Rossi - Macroeconometria 20

21 The model has a statistically recursive structure. Causality is one way, from z t to y t. y t depends on both u 1t and u 2t (the latter through z t ), but z t does not depend on u 1t, which is independent of u 2t. The mere observation of the variables without knowing the structure of the DGP is not sufficient to establish this fact, given that y t and z t have the same date. If β 21 = 0, y t does not Granger-cause z t, which is predictable only from its past values. β 21 = α 21 = σ 21 = 0 yields strong exogeneity. z t can be treated as if fixed in repeated samples. Eduardo Rossi - Macroeconometria 21

22 Weak exogeneity conditions α 21 = σ 21 = 0 are needed in order to estimate consistently the parameters of the model of y t z t. This is true in any model in which the conditional expectations are linear functions of the conditioning variables. Gaussianity is sufficient but not necessary for this property to hold. Eduardo Rossi - Macroeconometria 22

23 Exogeneity and regression If a variable is weakly exogenous it is valid to treat it as a regressor in LS estimation of the equation of interest. The equation is y t = α 12 z t + γ 1 + β 11 y t 1 + β 12 z t 1 + u 1t y t E[y t z t, y t 1, z t 1 ] = ǫ 1t ω 12 ω 22 ǫ 2t substituting ε 1t ε 2t = 1 1 α 12 α 21 u 1t α 12 u 2t u 2t α 21 u 2t Eduardo Rossi - Macroeconometria 23