OLS bias for econometric models with errors-in-variables. The Lucas-critique Supplementary note to Lecture 17

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1 OLS bias for ecoomeric models wih errors-i-variables. The Lucas-criique Supplemeary oe o Lecure 7 RNy May 6, 03 Properies of OLS i RE models I Lecure 7 we discussed he followig example of a raioal expecaios (RE) model: Y = β E(X + I ) + ε () X = λx + ɛ x, < λ < () ε IID(0, σ ) (3) ɛ x IID(0, σ x) (4) Cov(ε, ɛ xs ) = 0 for all ad s (5) () is he srucural equaio ad β is he parameer of ieres of his model: β shows by how much Y is adjused i period whe he expecaio abou X i period + is chaged based o iformaio ha he ages have available i period. Hece he ages are forward-lookig i his model, bu like oher forecasers hey mus build o hisory i order o formulae heir expecaios abou he fuure X. The iformaio se used by he ages o form expecaios is deoed by I i (). I his oe, we focus o he salie implicaios of his ype of expecaios for ecoomeric modellig. Algebraic delails aside, hose cosequeces are he same if we have E(X + I ) i (), allowig ages o form expecaios o he basis of period iformaio, ad also if E(X I ) which would oly remove he forward-lookig aspec of he model, o he ecoomeric cosequeces of raioal expecaios formaio. The disurbaces ε ad ɛ x are idepede from each oher, as saed i assumpio (5) ad hey have classical properies codiioal o he ages iformaio se I. Wihou loss of geeraliy, his is coveyed by he assumpios ha each disurbace is IID wih expecaio zero ad cosa variace. The model is specified wihou a iercep, wihou loss of geeraliy, ad because i simplifies he algebra.

2 is As show i he lecures, give he model specificaio, he soluio for X + wih codiioal mahemaical expecaio X + = λ X + λɛ x + ɛ x+ (6) E(X + I ) = λ X. (7) (The ucodiioal expecaio of X + is zero, i does o play a role i he derivaios below.) The ucodiioal variace of X + does o deped o ad i is herefore he same as V ar(x ). From he lecure o dyamic regressio we foud i o be V ar(x ) = σ x λ. The expecaio error is X + E(X + I ) = λɛ x + ɛ x+ (8) We kow ha OLS gives (a leas) cosise esimaors of he codiioal expecaio of Y give X +. Bu is β a parameer i he codiioal expecaio? To aswer ha quesio we ca ivesigae he probabiliy limi of he OLS esimaor ˆβ = (X +)Y (9) X + for Noe ha he disurbace u is as a resul of usig (7) o replace Y = β X + + u (0) u = ε β ɛ x+ β λɛ x E(X + I ) i he srucural equaio by he righ had side of E(X + I ) = X + ɛ x+ λɛ x.

3 The deailed algebra for fidig plim( ˆβ ): plim( ˆβ ) = plim( (X +)Y ) X + = β + plim( (X +)u ) = β + = β + X + V ar(x ) plim( T V ar(x ) plim( T = β + β λ σ x β σ x V ar(x ) (X + )(ε β ɛ x+ β λɛ x )) (λ X + λɛ x + ɛ x+ )(ε β ɛ x+ β λɛ x ) = β + β λ σ x β σ x σ x λ = β ( + λ σx σx ) = β σx ( (λ + )( λ ) = β λ [ = β λ ( λ ) + λ ] = β λ 4 [ λ ( λ ) ( λ ) ] This meas ha he OLS esimaor of β i (0) is icosise whe he rue model is ()-(5). The asympoic bias is: plim( ˆβ ) β = β (λ 4 ) < 0 () Noe ha i Lecure 7 we foud ha i he case where () is replaced by Y = β E(X I ) + ε ad he model is oherwise kep uchaged, he bias is plim( ˆβ ) β = β (λ ) < 0. Errors-i-variables bias The source of he OLS bias is ha we coamiae he disurbace erm by he forecas error for he X + (or X, depedig o he specificaio of ()). Therefore, he bias is a special case of he errors-i-varaibles bias. As oed i Lecure 7, aoher famous example is he measureme-error bias. The measureme-error bias is ofe preseed for he cross-secio daa model: Y i = β 0 + β X i + ε i i =,,..., () where Cov(ε i, X i ) = 0, ad ε i has he oher classical properies as well We assume ha Xi is a uobservable radom variable which is replaced by he observable X i i he esimaio of (). The differece bewee X i ad Xi is radom: e i = X i Xi 3

4 Eve if all e i ad ε i are idepede, OLS o Y i = β 0 + β X i + ε i i =,,..., (3) will produce a icosise esimaor of β, because Cov(ε i, X i ) 0 as a cosequece of ε i = ε i β e i As usual, he probabiliy limi of ˆβ β is plim( ˆβ i= β ) = plim (X i X)ε i i= (X i X) (4) The deomiaor of (4) is equal o he heoreical variace of X, which is he sum of he variace of X, σ X, ad he measureme-error variace,σ e: The umeraor is [ ] [ plim (X i X)ε i = plim i= V ar(x) = σ X + σ e (5) ] (Xi X ) + (e i ē))(ε i β e i ) i= = β V ar(e) = β σ e sice he probabiliy limi of all he oher erms are zero by he assumpios of he model. Collecig resuls, we have he compac expressios plim( ˆβ β ) = β σ e V ar(x) = β σe σx + σ e (6) for he asympoic bias of he OLS esimaor i he measureme-error model. To see he errors-i-variables ierpreaio of he RE bias of he OLS esimaor () i he model ()-(5), se V ar(x ) ad iser i (6): σ e V ar(u ) = V ar(ɛ x+ + λɛ x ) = σ x + λ σ x σ x λ plim( ˆβ β ) = β σe V ar(x) = β (σx + λ σx) σx λ = β ( + λ )( λ ) = β ( λ 4 ) which is he same expressio as i () above. 4

5 3 The Lucas-criique The Lucas-criique aacks he idea ha if here is a chage i he expecaios abou X +, he effec of his chage o Y ca be prediced by usig he OLS esimae ˆβ from a regressio model where Y is regressed o X +. By chage i he expecaios he criique meas a chage i a parameer of he equaios ha are used o form he mahemaical expecaio of X +. By lookig a () we see ha he releva parameer mus be λ. We see ha we if λ chage we call his a srucural break i he X process he probabiliy limi of ˆβ. mus also chage. This meas ha he esimaed effec of a chage i X o Y will be wrogly esimaed by he use of OLS esimaio. The soluio, as we have discussed i Lecures 8 ad 9, is o use IV esimaio. More geerally, he criique implies ha policy aalysis cao be based o OLS esimaed codiioal expecaios (regressio models). Empirically, he relevace of he criique ca be cofirmed by fidig proof of a srucural break i he codiioal model ha occurs a he same ime as a srucural break i model for X. However, he relevace ca also be refued if he srucural break i he codiioal expecaio for X does o lead o a srucural break i he codiioal model. As oed above, ad as show i Lecure 7, he same applies if he sucural equaio is specified wih E(X I ) as he explaaory varaible for Y.. 5