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12
JAKUBROK
isapplied,developedbyRiversandVuong(1988).Inthefirststageitrunsan
OrdinaryLeastSquares(OLS)regressionfortheendogenouscomponent,us-
inginstrumentalvariablesaspredictors.Theresidualsarethenusedalongthe
potentiallyendogenousvariableinthesecond-stageorderedmodel.The2SCML
approachallowstoobtainconsistentandefficientestimatesofthecoefficientsas
wellastoconductasimpleandrobustexogeneitytestforselectedvariables.It
wasprovedtooutperformalternativeapproachessuchasinstrumentalvariable
probitand2-stageleastsquares(AlvarezandGlasgow1999);itispreferredfor
smalldatasamples(Arendt2004),andwassuccessfullyappliedtoinvestigate
theendogeneityofperceivedAPinthelifesatisfactionfunctionbyGoetzkeand
Rave(2015).Twostepsofthe2SCLMarespecifiedasfollows:
(5)Step1:
y
1
i
=
X
11
α
+
Z
11
β
+
v
1
(6)Step2:
y
*
2
i
=
X
2
β
2
+
y
1
i
γ
+
u
1
with:
y
*
2
i
={
[
|
|
|
|
[
2
10
if
if
3
0
µ
if
ify
1
<
µ
<
2
y
2
y
i
<
2
2
i
≤
i
≤
y
≤
0
2
µ
i
µ
1
2
wherey
1iisthehypothesisedendogenouseffect,i.e.theAQ_todayforrespon-
denti,y*
2iistheorderedLS_tr/Happyscoreforani-threspondent;y
2iisthei-th
respondent’sleveloflatentSWBfunction;µ
jdepictsSWBthresholdvaluesfor
1≤j≤3;Z
irepresentsavectorforobjectiveAP;X
irepresentsavectorofother
explanatoryvariables;α,β,γareregressioncoefficients,whilev
1andu
1areun-
observederrorterms.
FollowingWooldridge(2002)andGoetzkeandRave(2015),severalassump-
tionsaremade.First,itisassumedthaterrortermsv
1andu
1areidenticallydis-
tributed,withameanofzero.Second,furthernormalisationisperformedtoallow
identificationoftheparametersinequation(6),assumingvar(v)≡1andsetting
u
1≡v
1λ+η
1whereλ=cov(v,u)var(u).Theequation(6)mightberewrittenas:
(7)
y
2
*
i
=
X
2
β
2
+
y
1
i
γ
+
v
i
λη
+
1
Thestartingpointforthe2SCMLestimationmethodistheestimationofthe
unobservedu
1byrunningtheOLSregression(Equation5)ofthehypothesised
endogenousvariabley
1i,usingtheobservedvariablesX
1andZ
i.TheOLSre-
sidualsaretheninsertedasapredictorvariableinthelogitmodelofy*
2i,along
observedvariablesX
2andhypothesisedendogenousvariabley
1i.Theobtained
estimatorsofthescaledcoefficientsaresufficienttoverifythesignificanceand
thesignoftheoriginalcoefficients.The2SCMLhasabuilt-intestforexogeneity
-iftheestimateforresidualsisnon-significantasapredictorinthefinalmodel,
thehypothesisofendogeneitycanberejected.
