E 4101/5101 Lecture 8: Exogeneity - Universitetet i oslo · I The Lucas-critique states that (9) never holds: Policy analysis should never be based on a conditional model|it gives

E 4101/5101Lecture 8: Exogeneity

Ragnar Nymoen

Department of Economics, University of Oslo

17 March 2011

E 4101/5101 Lecture 8: Exogeneity Department of Economics, University of Oslo

Introduction I

I Main references:

I Davidson and MacKinnon,

I Ch 8.1-8,7, since tests of (weak) exogeneity build on thetheory of IV-estimation

I Ch 8.7 in particular (Durbin-Wu-Hausman test)I Ch 15.3 on the relationship between tests of exogeneity and

encompassing tests (for non-nested models)

I Some references to the literature along the way.


The VAR, models and exogeneity I

In lecture 7 we developed the VAR:(ytxt

)=

(a11 a12a21 a22

)(yt−1xt−1

)+

(εy ,tεx ,t

), (1)

and two econometric models of this VAR:

I An identified systems of equations model (SEM)

I Simultaneous equations model, andI Recursive model (identification by Cholesky factorization in

VAR terminology)are two special cases of identified SEM

I The conditional model (conditional ARDL for yt equation anda marginal equation for xt).


The VAR, models and exogeneity IIThese three models are different parameterization of the VAR (1).

I The parameters of the conditional ARDL and the recursiveSEM can be estimated consistently by OLS.

I For the parameters of the simultaneous SEM, OLS givesinconsistent estimates.

We sem to be in the situation that a variable xt can be“exogenous” in one econometric model but “not exogenous” inanother econometric model.


The VAR, models and exogeneity III

In order to clarify this, at the conceptual level, moderneconometrics distinguishes between different concepts ofexogeneity:

I Weak exogeneity (WE)

I Strong exogeneity (StE)

I Super exogeneity (SuE)

I Strict exogeneity or pre-determinedness.


Weak exogeneity I

From Lecture 7 we have that (1) can be re-parameterized as:

yt = φ1yt−1 + β0xt + β1xt−1 + εt . (2)

xt = a21yt−1 + a22xt−1 + εx ,t (3)

where (φ1, β0, β1) depend on the parameters of the jointdistribution of yt and xt as shown, and εt depend on the VARdisturbances εx ,t and εy ,t .This representation corresponds to the factorization of the jointdensity:

f (xt , yt ; θ) = f (yt | xt ; λ1) · fx (xt ; λ2) (4)

where the conditioning on xt−1 and yt−1is suppressed.


Weak exogeneity II

Let θ denote the parameters of the joint density. θ1and θ2 are theparameters of conditional and marginal densities.

θ = [a11,...,a22, σ2y , σ2

x , ωxy ]′ (5)

θ1 = [ φ1, β0, β1, σ2]′

(6)

θ2 = [a21,a22, σ2x ]′. (7)

Weak exogeneity (WE) is the case where statistically efficientestimation and inference can be achieved by only considering theconditional model and not taking the rest of the system intoaccount.With WE there is no loss of information by abstracting from themarginal 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. xt in theconditional model is weakly exogenous if

1. ψ = g(θ1), ψ depends functionally on θ1 and not on θ2.

2. θ1 and θ2 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 thatthere is no direct dependence of ψ on θ2 and 2. and a conditionthat secures that there are no indirect (e.g. cross-restrictions)dependence between θ2 and ψ.


Weak exogeneity IV

Example

Set ψ = β0. xt is WE because both 1 and 2 is fulfilled.

(In fact, xt in (2) is WE with respect to the whole vectorψ = θ1 = [ φ1, β0, β1, σ2]

′.)

Example

If ψ = (λ1, λ2), the eigenvalues of the companion matrix, then xtis not WE, since ψ is a function of a12 and a22 which belongs to θ2.


Strong exogeneity and Granger non-causality I

Earlier we defined dynamic multipliers and impulse responses withrespect to disturbances of ARMA and VAR time series models.The purpose of an econometric study is often to find the dynamiceffects on one economic variable (yt) of a change in a variable (xt)“elsewhere in the economy”.These effects can be found as

∂yt+s

∂xt

from the particular solution of (2) for period t + s:

yt+s = β0xt+s + (β1 + φ1β0)xt+s−1 + φ1(β1 + φ1β0)xt+s−2

+ φ21(β1 + φ1β0)xt+s−3 + . . . + φs

1yt (8)


Strong exogeneity and Granger non-causality II

s = 0,∂yt∂xt

= β0

s = 1,∂yt+1

∂xt= (β1 + φ1β0)

s = 2,∂yt+2

∂xt= φ1(β1 + φ1β0)

s = j ,∂yt+j

∂xt= φj−1

1 (β1 + φ1β0)

If yt is not Granger-causing xt , meaning

yt−1 9 xt ⇐⇒ a21 = 0 in (1)

the multipliers give the correct effect on yt+s of an independentchange in xt .


Strong exogeneity and Granger non-causality IIIDefinition (Strong exogeneity)

xt is strongly exogenous, (StE) if xt is WE in (2) and yt is notGranger-causing xt .


Strong exogeneity and Granger non-causality IV

“Scenario forecasts”

I Often forecasting institutions interpret their forecast as being“not necessarily unbiased”.

I Instead it is meant as scenario or a “main alternative”forecast.

I Such forecasts are from open systems, with exogenousvariables, and are conditional forecasts in another sense than aforecast that conditions on initial conditions of allvariables—they condition on a specific hypothetical path for(xt+s ; s = 1, 2, . . .) in (8)

I Scenario forecasts seem to assume (or know) strongexogeneity of xt .


Super exogeneity (autonomy and invariance) I

I If a change in θ2 does not affect θ1 we say that θ1 is invariantor autonomous (Haavelmo (1944)) with respect to the (classof) interventions that caused the change in θ2.

I If θ1 and θ2 are two scalars, autonomy implies:

θ1 = atθ2t

meaning that the parameter θ1 of the conditional modelremains a constant also when the parameter of the marginalmodel is a non-constant function of time.


Super exogeneity (autonomy and invariance) II

I For example θ2t can be constant over one time period,corresponding to one “regime”, and then change to a newlevel, temporarily, or more permanently. The change can befast or slow. In such cases we speak structural breaks in themarginal model. The term intervention is also common.

Definitionxt is super exogenous (SuE) in (2) if xt is WE and the parameters(φ1, β0, β1,σ2) are invariant with respect to structural breaks inthe marginal model (3).


Super exogeneity (autonomy and invariance) IIII If we look back at the expression for the bivariate normal case

we have that SuE of xt requires

ωxy = β0σ2x , (9)

since only then can β0 be unaffected by changes in σ2x , for

example an intervention in the marginal model.

I Note that super-exogeneity does not require strong exogeneity.

Further remarks:

I While there is nothing hindering that (9) may hold, there alsonothing that “makes it hold”.

I Invariance is a relative concept: A conditional model can havesuper exogeneity with respect to certain interventionsstructural breaks, but not all.


Super exogeneity (autonomy and invariance) IV

I All models break down sooner or later!

I It it not obvious that all structural breaks (in the marginalmodel) affect β0 or other “derivative coefficients”. Might be astrong incidence of structural breaks that mainly affectconditional mean, i.e., the constant term (which we haveabstracted from for simplicity her)—Return to that when wediscuss forecasting.

I The Lucas-critique states that (9) never holds: Policy analysisshould never be based on a conditional model—it gives thewrong answer to the question“what happens to yt when xt ischanged?”

I See Lecture note on Lucas-critique


Super exogeneity (autonomy and invariance) V

I If the conditional model does not have super exogenousvariables, it may well be that another parameterization, i.e.,another econometric model of the VAR has parameters thatare invariant. This is the constructive part of the Lucas’critique:

I Estimate models where the parameters of interest arecoefficients of variables that are subject to rationalexpectations

I These coefficients will be deep structural parameters with highdegree of invariance (certainly, or possibly, can it be tested?)


Strict exogeneity and pre-determinedness I

I Even though the concepts of WE, StE and SuE goes back toEngle, Hendry and Richard (1983),they are still regarded as“new” and perhaps adding little to the “usual concepts” of

I strict exogeneity (disturbances uncorrelated with anyrandomness in the DGP that generated xt)

I and the pre-determinedness secured by sequential conditioning(the work-horse of time series econometrics)

I One reconciliation of views may be that in several situations itpays off to be clear about parameters of interest—as theLucas critique shows:

I If the parameters of interest is given by the rationalexpectations model then xt cannot be weakly exogenous

I Even if xt is predetermined in the condition model.


Testing exogeneity I

I Weak exogeneity.

I In the stationary case (that we have developed here) one couldsay that WE is implied by model specification:

I If the parameters of interest are “in” the conditional model(ARDL for example), then the variables of the model are WE

I That said, the usual exogeneity tests, like the Wu-Hausmantest can be interpreted as a test of WE (see first part of theLecture note from 4160)

I We will see later that WE has relevance and is testable in thecointegrated I(1)-model

I Strong exogeneity

I Granger non-causality is testable in a well specified VAR

I Super exogeneity


Testing exogeneity II

I Lack of invariance with respect to structural breaks(interventions) that have occurred in the sample is a testablehypothesis. We will see specific examples later. (Autometericsis a useful tool)

I When the model “under test” is a conditional model, theseinvariance tests are tests of super exogeneity.

I But invariance tests are also relevant for the parameters in anequations in a simultaneous equation model, and other deepstructural parameters (Euler equations for consumption , NPCfor inflation).

I Given overidentification—testing is possible andI the statistics have power.


References

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


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E 4101/5101 Lecture 8: Exogeneity - Universitetet i oslo · I The Lucas-critique states that (9) never holds: Policy analysis should never be based on a conditional model|it gives