November 6 Eco 59A WEAK INSTRUMENTS III Testig for Weak Istrumets From the results discussed i Weak Istrumets II we kow that at least i the case of a sigle edogeous regressor there are weak-idetificatio-robust iferece procedures that are as powerful as usual t- ad Wald tests whe istrumets are strog. Thus whe the coefficiet o a sigle edogeous regressor is the mai object of iterest the ecoometricia ca alwas use efficiet robust procedures for example procedures based o Kleiberge s Ecoometrica K-statistic or Moreira s 3 Ecoometrica CLR-statistic istead of SLS estimates ad their stadard errors ad there is o eed to test whether istrumets are strog or weak. I other situatios it might be desirable to have a test for weak istrumets. This ca happe for example whe there are multiple edogeous regressors ad the ecoometricia is iterested i testig a hpothesis about ol oe of the coefficiets. Aother example is whe the ecoometricia is primaril iterested i the coefficiets o exogeous regressors. I such situatios usig projectio-tpe weak-idetificatio-robust procedures ca result i power loss or less iformative cofidece sets relativel to usual tests ad cofidece itervals based o SLS estimates ad their stadard errors. The discussio below is based o Stock ad Yogo 5 Testig for Weak Istrumets i Liear IV Regressio. However to simplif the presetatio we will ol cosider the case of a sigle edogeous regressor ad o exogeous regressors: γ + u Z π + v where γ is a scalar parameter ad π is l. While as was metioed above there is o eed to test for weak istrumets i this case the simplified framework is more coveiet for illustratig the idea behid Stock ad Yogo tests. Their paper hadles the geeral case of a model with multiple edogeous ad exogeous regressors usig the same priciple. Asmptotic distributio of the first-stage F-statistic uder weak IVs Usuall to evaluate the stregth of istrumets oe looks at the first-stage F -statistic. Let ˆπ deote the OLS estimator of π : ˆπ π d N σuq. Hece the usual F -statistic is give b: F ˆπ Z Z ˆ ˆπ /l. Note however whe IVs are weak π C/ Z Z Z Z /l ˆσ v ˆπ Z π + Z Z v d C + Q Ψ where Ψ N σvq.
Hece whe IVs are weak C + Q F Ψ Q C + Q Ψ /l d σv d Q / C + Q / Ψ /l d λ + Z v /l d χ l λ /l where λ Q / C Z v N I l. Here λ is the so-called cocetratio parameter which measures the stregth of IVs: λ C Q C σ v π Q π σv. The cocetratio parameter is a ratio of the sigal cotaied i IVs relativel to the oise of the errors. Whe IVs are strog π is fixed λ as. We are iterested i testig the ull hpothesis of weak IVs agaist the alterative of strog IVs: H : π C/ vs H : π fixed. Ufortuatel such hpotheses caot be tested eve asmptoticall because the asmptotic distributio of F uder H depeds o ukow parameter λ ad therefore oe caot select critical values for the test. A more practical defiitio of weak istrumets is eeded. Note also that usual F -critical values caot be used because the are determied for λ. Asmptotic distributio of the Wald statistic uder weak IVs To formulate a practicall useful defiitio of weak IVs Stock ad Yogo propose to defie it b cosiderig actual sizes of Wald tests for differet values of the cocetratio parameter λ. IVs will be defied as weak if their cocetratio parameter results i substatial size distortios for Wald tests. Recall that ˆγ γ Z Z Z Z u Z Z Z Z ad uder strog IVs ˆγ γ d N σu/π Q π. Thus the usual Wald statistic for testig H : γ γ is uder H : W H ˆσ u ˆγ γ Z Z Z Z ˆγ γ ˆσ Z Z Z Z u u Z Z Z Z. We would like to decide that IVs are strog whe there is sufficiet evidece i data agaist the ull hpothesis of weak IVs. Suppose istead oe decides first to test the ull hpothesis of strog IVs agaist the alterative of weak IVs. The if the ull is accepted i the secod step to use the usual SLS-based o-robust iferece approach. Sice the power of a test is ofte substatiall below oe whe IVs are i fact weak oe is likel to accept the ull of strog IVs with o-egligible error i the first step. Cosequetl i the secod step oe would mistakel use a o-robust approach which ca result i substatial size distortios. Such a situatio ofte arises whe oe uses pre-tests with preferred specificatios as a ull hpothesis.
Next Z Z Z Z u Z Z Z Z d Z Z Z Z Z Z Z u Z Q C + Ψ Q C + Q Ψ Φ Q C + Q Ψ σvσ u Q / C + Q / Ψ σ u Q / Φ σv Q / C + Q / Ψ d σu λ + Zv Z u λ + Z v where Note also that Z u ad Z v are joitl ormal: [ [ Zu N Z v Z u d σ u Q / Φ N I l. ρ σ uv σ u. [ ρ ρ I l where 3 To derive the limitig distributio of ˆσ u we eed the limitig distributio of ˆγ γ which ca be derived similarl to. Thus assumig that IVs are weak: ˆγ γ d Z Z Z Z u Z Z Z Z Z Z Z Z Z Z Z u Z σ u λ + Z v Z u σ v λ + Z v σ u λ + Z v Z u λ + Z v. 4 3
Now ˆσ u ˆγ ˆγ u ˆγ γ u ˆγ γ u u + ˆγ γ ˆγ γu u u + ˆγ Z C/ + v Z C/ + v γ d σ u + σ u λ + Z v Z u λ + Z v σ v σ u λ + Z v Z u λ + Z v σ uv ˆγ γ u Z C/ + v σ u λ + Z v 4 + σ u λ + Zv Z u σu σ uv λ + Z v λ + Z v Z u λ + Z v 4 5 where covergece i the lie before the last follows b 4. Lastl b ad 5 we obtai that the ull asmptotic distributio of the Wald statistic is give b: W σ uσ H v λ + Zv Z u λ + Zv d σuσ v λ + Z v 4 + σuσ v λ + Zv Z u σu σ uv λ + Z v λ + Z v Z u λ + Zv Z u λ + Zv λ + Z v 4 + λ + Z v Z u ρ λ + Zv λ + Z v Z u W λ ρ. The asmptotic ull distributio of W is ostadard however as we show below it is completel determied b two scalar parameters: λ which measures the stregth of istrumets Z ad ρ which measures the degree of edogeeit of. Note that sice W λ ρ is a fuctio of ρ λ + Z v ad λ + Z v Z u it suffices to show that the joit distributio of λ + Z v ad λ + Z v Z u deped ol o λ ad ρ. Lemma. Let X u X u... X ul ad X v X v... X vl be joitl ormal radom vectors such that [ [ [ Xu ρ N I ρ l. X v The λ + Z v ca be represeted as λ + X v + l j X vj ad λ + Z v Z u ca be represeted as λ + X v X u + l j X vjx uj. Proof. Let b... b l be orthoormal l -vectors b i b i ad b i b j for i j such that b λ/ λ. Defie a l l matrix λ λ b B. ad ote that BB I l B B I l ad Bλ b l λ.. 4
Defie We have [ Xu X v X u BZ u ad X v BZ v. N [ [ N [ [ Zu B [ ρ ρ [ ρ ρ Z v BB I l where the result i the secod lie follows b 3. Now λ + Z v λ + Z v B λ + Z v B B B λ + Z v Bλ + X v Bλ + X v λ + X v + l Xvj. j Z u λ + Z v BZ u B B B λ + Z v X u Bλ + X v λ + X v X u + l j X vj X uj. b The asmptotic probabilit of rejectio uder H of a test with omial sigificace level α is thus give P W > χ H α P W λ ρ > χ α. The maximum rejectio probabilit for a give value of the cocetratio parameter λ is: Rα max λ max W P λ ρ ρ > χ α where fuctio Rα max ca be computed b simulatios sice the probabilities i its defiitio are completel determied give α λ ad ρ. Stock ad Yogo s 5 quatitative characterizatio of weak IVs ad the test The differece Rα max λ α represets the maximum size distortio of a Wald test with asmptotic omial size of α. Stock ad Yogo propose the followig quatitative defiitio of weak istrumets. Defiitio. Istrumetal variables with cocetratio parameter λ are weak if Rα max λ r for some chose < α r <. For example with α.5 ad r. we sa that istrumets are strog if the maximal size of a 5% Wald test does ot exceed %. With such a defiitio a ull hpothesis of weak IVs ad the correspodig alterative of strog IVs ca be stated as: H : R max α λ r vs.t H : R max α λ < r. 6 5
The above testig problem ivolves oe ukow parameter: λ. If λ was kow we could alwas determie whether IVs are weak or strog accordig to the above defiitio sice fuctio Rα max r is determiistic ad α ad r are chose b the ecoometricia. It is therefore coveiet to reformulate the testig problem i terms of the cocetratio parameter λ. Let L α deote the iverse fuctio of Rα max : L α r Rα max r. Thus for a give value r L α r is the smallest value of the cocetratio parameter λ eeded so that a size-α Wald test would have the maximum size smaller tha r. Whe the cocetratio parameter is λ the testig problem i 6 ca be alterativel stated as H : λ L α r vs. H : λ > L α r. 7 Recall that the first-stage F -statistic is distributed as χ l λ /l. Let χ l τ d be the τ-th quatile of a ocetral χ l distributio with a ocetralit parameter d. A asmptotic level-τ test of weak IVs as defied i 7 ca be costructed as follows:. Select α the omial asmptotic size of a Wald test ad r the maximum allowed asmptotic size of a Wald test.. Compute L α r the smallest ocetralit parameter eeded so that size-α Wald test would have the maximum size smaller tha r. 3. Compute F the first-stage F -statistic. 4. Select τ the desired sigificace level for the test of weak IVs ad compute χ l τ L αr/l the critical value. 5. Reject H if weak IVs whe F > χ l τ L αr/l. Oe ca easil see that the above procedure is asmptoticall valid: P F > χ l τ L α r/l Weak IVs P F > χ l τ L α r/l λ L α r P F > χ l τ λ /l τ where the result i the last lie holds because F d χ l λ /l. Critical values χ l τ L αr/l are tabulated ad reported i Table 5. i Stock ad Yogo 5 for τ.5 ad α.5. For example whe l ad r. χ.95l.5. 6.38. This ca be used as follows. Suppose the ecoometricia is iterested i testig a hpothesis about γ at sigificace level. i a model with a sigle edogeous regressor ad a sigle IV. Whe the first-stage F-statistic exceeds 6.38 it is safe to proceed usig a usual SLS-based Wald statistic however oe should compare it to 5% critical values istead of % critical values.. 6