Wu-Hausman Test: But if X and ε are independent, βˆ. ECON 324 Page 1

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1 Wu-Hausma Test: Detectg Falure of E( ε X ) Caot drectly test ths assumpto because lack ubased estmator of ε ad the OLS resduals wll be orthogoal to X, by costructo as ca be see from the momet codto X' ( y Xβ But f X ad ε are depedet, βˆ OLS ad βˆ IV (wth Z the strumet) should be the same asymptotcally (both are cosstet). If E( ε X ) does ot hold, they wll dffer: OLS s cosstet ad IV cosstet Geeral approach of comparg a estmator that s cosstet uder both the ull ad alteratve hypothess wth oe that s cosstet oly uder the ull hypothess ca also be used for pael data (fxed vs radom effects) ad Multomal Logt ˆ) ECON 34 Page

2 If qˆ = βˆ IV βˆ OLS s the dfferece the coeffcet vectors betwee the two estmators the (for smple regresso) uder the ull hypothess var( βˆ IV qˆ ) var( βˆ OLS ) a ~ χ () Multple regresso verso depeds o a quadratc form, for the dfferece betwee the two coeffcet vectors Ths s ot a test of edogeety (or measuremet error or omtted varable bas) per se, but a test of whether edogeety (or measuremet error or omtted varable bas) has a sgfcat effect o the cosstecy of βˆ ECON 34 Page

3 Added Varable Iterpretato of Hausma Test A smple alteratve to formg the vector of cotrasts s to add the resduals from the frst stage regresso(s) to the model (for each potetally edogeous varable) ad test f coeffcet wth t or F True model y = β x * + ε but we estmate y β x + ε st Stage regresso uses some strumetal varable, z x = γ z + w = xˆ + wˆ where wˆ s = (.e. x ms-measured) frst stageresduals OLS regresses y o x, whle IV/SLS regresses y o xˆ, so f the two estmators gve same aswer, lkely that x ad ε are depedet Base the test o the followg artfcal regresso y = β = β xˆ x + δ wˆ + ( δ + ε β) wˆ + ε otg xˆ = x wˆ f coeffcet o added resduals, the δ=β, ad OLS usg x ad IV usg xˆ gve the same aswer (mplctly, the effect o y of both the systematc ad radom parts of x s the same) ECON 34 Page 3

4 Over-Idetfcato Tests Whe there are more strumets tha potetally edogeous RHS varables, t s possble to test the valdty of the strumets test smply dcates whether oe or more of the strumets should play a drect role explag y; does t show whch oe ca be cosdered a test of the assumpto that corr(z, ε) 0 as sample sze gets large ˆβ SLS regress the SLS resduals, y X o the matrx of strumets Z (cludg exogeous varables X that serve as ther ow strumet) the R from ths auxlary regresso, multpled by sample sze, has a χ dstrbuto, wth d.f. = # of strumets f the model s exactly detfed ths test ca t work because the resduals wll be orthogoal to Z by costructo ECON 34 Page 4

5 SLS as a GMM Estmator Geeralzed Method of Momets estmators clude may commo estmators (OLS, IV, SLS, NLSLS, MLE) as specal cases, whch makes t easy to compare ad evaluate them (e.g. testg E( ε X ) ) GMM s a large sample estmator, so t s cosstet, but usually ot effcet fte samples Recall the steps dervg a GMM estmator. start wth a orthogoalty codto e.g. E ( X' e) or E( Z' e) whch s based o theory. replace populato momets wth sample momets 3. solve k smultaeous equatos (from the sample momets) for the k ukows ths gves a Method of Momets (MoM) estmator what f there are more equatos tha ukows (over-detfed) 4. weght the smultaeous momet equatos accordg to how precsely (measured by ther varace) each s estmated use of the optmal weghtg matrx gves the Geeralzed MM ECON 34 Page 5

6 SLS ad GMM Estmators whe the model s over-detfed (.e. rak(z)>rak(x) MoM estmator has a problem of more momet codtos tha parameters to solve for e.g. SLS where ß = [ α β ] X = [ x ] Z = [ z z ] momet codto s E( Z'e) Z' (y X߈) ad whe replace wth sample verso get.. (3 ) ( ) = 3 smultaeous equatosto solve Gve these 3 eqs for oly ukows could ether (a) drop a equato (b) weght each equato equally ad mmse a sum of squared devatos, or (c) weght equatos accordg to how precsely they are measured Optmal estmator ths class uses (c) ad mmses the quadratc form.. m βˆ [ Z'(y X߈) ] W [ Z'(y X߈) ] where W - s cosstet estmate of ( var[ ( )( Z'e) ]) so amogst the three momet codtos, those that are estmated less precsely get less weght the mmsato problem ECON 34 Page 6

7 the frst order codto of the mmsato problem s: ( X' Z) W ( X' Z) W ( Z' y Z' X߈) Z' y ( X' Z) W Z' X߈ ( X' ZW Z' X) ( X'Z) W Z' y = ߈ GMM If the dsturbaces are homoscedastc ad serally depedet [ Z' ee'z] E[ Z' e] E[ e' Z] = E[ Z' ( σ I) Z] = σ ( Z' ) W = var( Z' e) = E Z usg ( b' Ab) b = Ab Substtutg for W, ad droppg σ whch s scalar ad should t affect the mmsato problem.. ( X' Z(Z' Z) Z' X) = = ߈ ( X' P X) GMM Z = ߈ X' P SLS Z y X' Z(Z' Z) Z' y I cases where dsturbaces are heteroscedastc ad/or autocorrelated, use the Whte or Newey-West varace estmator for W the GMM formula ECON 34 Page 7