E 4101/5101 Lecture 8: Exogeneity

E 4101/5101 Lecture 8: Exogeneity Ragnar Nymoen 17 March 2011

Introduction I Main references: Davidson and MacKinnon, Ch 8.1-8,7, since tests of (weak) exogeneity build on the theory of IV-estimation Ch 8.7 in particular (Durbin-Wu-Hausman test) Ch 15.3 on the relationship between tests of exogeneity and encompassing tests (for non-nested models) Some references to the literature along the way.

The VAR, models and exogeneity I In lecture 7 we developed the VAR: ( ) ( ) ( yt a11 a = 12 yt 1 x t a 21 a 22 x t 1 ) ( ɛy,t + ɛ x,t ), (1) and two econometric models of this VAR: An identified systems of equations model (SEM) Simultaneous equations model, and Recursive model (identification by Cholesky factorization in VAR terminology) are two special cases of identified SEM The conditional model (conditional ARDL for y t equation and a marginal equation for x t ).

The VAR, models and exogeneity II These three models are different parameterization of the VAR (1). The parameters of the conditional ARDL and the recursive SEM can be estimated consistently by OLS. For the parameters of the simultaneous SEM, OLS gives inconsistent estimates. We sem to be in the situation that a variable x t can be exogenous in one econometric model but not exogenous in another econometric model.

The VAR, models and exogeneity III In order to clarify this, at the conceptual level, modern econometrics distinguishes between different concepts of exogeneity: Weak exogeneity (WE) Strong exogeneity (StE) Super exogeneity (SuE) Strict exogeneity or pre-determinedness.

Weak exogeneity I From Lecture 7 we have that (1) can be re-parameterized as: y t = φ 1 y t 1 + β 0 x t + β 1 x t 1 + ε t. (2) x t = a 21 y t 1 + a 22 x t 1 + ɛ x,t (3) where (φ 1, β 0, β 1 ) depend on the parameters of the joint distribution of y t and x t as shown, and ε t depend on the VAR disturbances ɛ x,t and ɛ y,t. This representation corresponds to the factorization of the joint density: f (x t, y t ; θ) = f (y t x t ; λ 1 ) f x (x t ; λ 2 ) (4) where the conditioning on x t 1 and y t 1 is suppressed.

Weak exogeneity II Let θ denote the parameters of the joint density. θ 1 and θ 2 are the parameters of conditional and marginal densities. θ = [a 11,..., a 22, σ 2 y, σ 2 x, ω xy ] (5) θ 1 = [ φ 1, β 0, β 1, σ 2 ] (6) θ 2 = [a 21,a 22, σ 2 x ]. (7) Weak exogeneity (WE) is the case where statistically efficient estimation and inference can be achieved by only considering the conditional model and not taking the rest of the system into account. With WE there is no loss of information by abstracting from the marginal model.

Weak exogeneity III WE is defined relative to the parameters of interest. The parameter of interest can be θ or a sub-set. Let ψ denote the vector with parameters of interest. x t in the conditional model is weakly exogenous if 1. ψ = g(θ 1 ), ψ depends functionally on θ 1 and not on θ 2. 2. θ 1 and θ 2 are variation free. The second condition secures the validity of the factorization in (4) in more general situations. Heuristically we will think of 1. as the condition that secures that there is no direct dependence of ψ on θ 2 and 2. and a condition that secures that there are no indirect (e.g. cross-restrictions) dependence between θ 2 and ψ.

Weak exogeneity IV Example Set ψ = β 0. x t is WE because both 1 and 2 is fulfilled. (In fact, x t in (2) is WE with respect to the whole vector ψ = θ 1 = [ φ 1, β 0, β 1, σ 2 ].) Example If ψ = (λ 1, λ 2 ), the eigenvalues of the companion matrix, then x t is not WE, since ψ is a function of a 12 and a 22 which belongs to θ 2.

Strong exogeneity and Granger non-causality I Earlier we defined dynamic multipliers and impulse responses with respect to disturbances of ARMA and VAR time series models. The purpose of an econometric study is often to find the dynamic effects on one economic variable (y t ) of a change in a variable (x t ) elsewhere in the economy. These effects can be found as y t+s x t from the particular solution of (2) for period t + s: y t+s = β 0 x t+s + (β 1 + φ 1 β 0 )x t+s 1 + φ 1 (β 1 + φ 1 β 0 )x t+s 2 + φ 2 1(β 1 + φ 1 β 0 )x t+s 3 +... + φ s 1y t (8)

Strong exogeneity and Granger non-causality II s = 0, y t x t = β 0 s = 1, y t+1 x t = (β 1 + φ 1 β 0 ) s = 2, y t+2 x t = φ 1 (β 1 + φ 1 β 0 ) s = j, y t+j x t If y t is not Granger-causing x t, meaning = φ j 1 1 (β 1 + φ 1 β 0 ) y t 1 x t a 21 = 0 in (1) the multipliers give the correct effect on y t+s of an independent change in x t.

Strong exogeneity and Granger non-causality III Definition (Strong exogeneity) x t is strongly exogenous, (StE) if x t is WE in (2) and y t is not Granger-causing x t.

Strong exogeneity and Granger non-causality IV Scenario forecasts Often forecasting institutions interpret their forecast as being not necessarily unbiased. Instead it is meant as scenario or a main alternative forecast. Such forecasts are from open systems, with exogenous variables, and are conditional forecasts in another sense than a forecast that conditions on initial conditions of all variables they condition on a specific hypothetical path for (x t+s ; s = 1, 2,...) in (8) Scenario forecasts seem to assume (or know) strong exogeneity of x t.

Super exogeneity (autonomy and invariance) I If a change in θ 2 does not affect θ 1 we say that θ 1 is invariant or autonomous (Haavelmo (1944)) with respect to the (class of) interventions that caused the change in θ 2. If θ 1 and θ 2 are two scalars, autonomy implies: θ 1 = a t θ 2t meaning that the parameter θ 1 of the conditional model remains a constant also when the parameter of the marginal model is a non-constant function of time.

Super exogeneity (autonomy and invariance) II For example θ 2t can be constant over one time period, corresponding to one regime, and then change to a new level, temporarily, or more permanently. The change can be fast or slow. In such cases we speak structural breaks in the marginal model. The term intervention is also common. Definition x t is super exogenous (SuE) in (2) if x t is WE and the parameters (φ 1, β 0, β 1,σ 2 ) are invariant with respect to structural breaks in the marginal model (3).

Super exogeneity (autonomy and invariance) III If we look back at the expression for the bivariate normal case we have that SuE of x t requires ω xy = β 0 σ 2 x, (9) since only then can β 0 be unaffected by changes in σ 2 x, for example an intervention in the marginal model. Note that super-exogeneity does not require strong exogeneity. Further remarks: While there is nothing hindering that (9) may hold, there also nothing that makes it hold. Invariance is a relative concept: A conditional model can have super exogeneity with respect to certain interventions structural breaks, but not all.

Super exogeneity (autonomy and invariance) IV All models break down sooner or later! It it not obvious that all structural breaks (in the marginal model) affect β 0 or other derivative coefficients. Might be a strong incidence of structural breaks that mainly affect conditional mean, i.e., the constant term (which we have abstracted from for simplicity her) Return to that when we discuss forecasting. The Lucas-critique states that (9) never holds: Policy analysis should never be based on a conditional model it gives the wrong answer to the question what happens to y t when x t is changed? See Lecture note on Lucas-critique

Super exogeneity (autonomy and invariance) V If the conditional model does not have super exogenous variables, it may well be that another parameterization, i.e., another econometric model of the VAR has parameters that are invariant. This is the constructive part of the Lucas critique: Estimate models where the parameters of interest are coefficients of variables that are subject to rational expectations These coefficients will be deep structural parameters with high degree of invariance (certainly, or possibly, can it be tested?)

Strict exogeneity and pre-determinedness I Even though the concepts of WE, StE and SuE goes back to Engle, Hendry and Richard (1983),they are still regarded as new and perhaps adding little to the usual concepts of strict exogeneity (disturbances uncorrelated with any randomness in the DGP that generated x t ) and the pre-determinedness secured by sequential conditioning (the work-horse of time series econometrics) One reconciliation of views may be that in several situations it pays off to be clear about parameters of interest as the Lucas critique shows: If the parameters of interest is given by the rational expectations model then x t cannot be weakly exogenous Even if x t is predetermined in the condition model.

Testing exogeneity I Weak exogeneity. In the stationary case (that we have developed here) one could say that WE is implied by model specification: If the parameters of interest are in the conditional model (ARDL for example), then the variables of the model are WE That said, the usual exogeneity tests, like the Wu-Hausman test can be interpreted as a test of WE (see first part of the Lecture note from 4160) We will see later that WE has relevance and is testable in the cointegrated I(1)-model Strong exogeneity Granger non-causality is testable in a well specified VAR Super exogeneity

Testing exogeneity II Lack of invariance with respect to structural breaks (interventions) that have occurred in the sample is a testable hypothesis. We will see specific examples later. (Autometerics is a useful tool) When the model under test is a conditional model, these invariance tests are tests of super exogeneity. But invariance tests are also relevant for the parameters in an equations in a simultaneous equation model, and other deep structural parameters (Euler equations for consumption, NPC for inflation). Given overidentification testing is possible and the statistics have power.

References Engle, R.F.,D.F. Hendry and J.F. Richard (1983) Exogeneity, Econometrica, 51, Supplement, 277 304 Haavelmo, T. (1944) The Probability Approach in Econometrics, Econometrica, 12, Supplement, 1 118