Econometric analysis of cross section and panel data woolridge solution

Thisxfilexandxseveralxaccompanyingxfilesxcontainxthexsolutionsxtoxthexodd-

xnumberedxproblemsxinxthexbookxEconometricxAnalysisxofxCrossxSectionxandxPanelxData,

xbyxJeffreyxM.xWooldridge,xMITxPress,x2002.xThexempiricalxexamplesxarexsolvedxusingx

variousxversionsxofxStata,xwithxsomexdatingxbackxtoxStatax4.0.

Partlyxoutxofxlaziness,xbutxalsoxbecausexitxisxusefulxforxstudentsxtoxseexcomputerxou

tput,xIxhavexincludedxStataxoutputxinxmostxcasesxratherxthanxtypextables.x Inxsomexca

ses,xIxdoxmorexhandxcalculationsxthanxarexneededxinxcurrentxversionsxofxStata.

Currently,xtherexarexsomexmissingxsolutions.x Ixwillxupdatexthexsolutionsxoccas

ionallyxtoxfillxinxthexmissingxsolutions,xandxtoxmakexcorrections.xForxsomexproblemsx

IxhavexgivenxanswersxbeyondxwhatxIxoriginallyxasked.xPleasexreportxanyxmistakesxorxdi

.



CHAPTERx2


dE(y|x1,x2) dE(y|x1,x2)
2.1.x a.x -----------------------------------------------------x =x b 1 +xb4x2 andx -----------------------------------------------------xx=x b 2 +x2b3x2 +xb4x1.
d x1 d x2 2x
b.xByxdefinition,xE(u|xx,xx)x=x0. andxxxxx arexjustxfunctions
1x 2 Because xx
2 1x 2

ofx(x1,x2),xitxdoesxnotxmatterxwhetherxwexalsoxconditionxonxthem:
2
E(u| x1,x2,x2,x1x2)x=x0.

c.xAllxwexcanxsayxaboutxVar(u|x1,x2)xisxthatxitxisxnonnegativexforxallxx1

andxx2:x E(u|x1,x2)x=x0xinxnoxwayxrestrictsxVar(u|x1,x2).



2.3.xa.xyx=xb0 +xb1x1 +xb2x2 +xb3x1x2 +xu,xwherexuxhasxaxzeroxmeanxgivenxx1

andxx2:x E(u|x1,x2)x=x0.x Wexcanxsayxnothingxfurtherxaboutxu.

b. dE(y|x1,x2)/dx1 =xb1 +xb3x2.x BecausexE(x2)x=x0,xb1x =




1

,E[dE(y|x1,x2)/dx1].x Similarly,xb2x=xE[dE(y|x1,x2)/dx2].

c. Ifxx1 andxx2 arexindependentxwithxzeroxmeanxthenxE(x1x2)x=xE(x1)E(x2)
2
=x0.x Further,xthexcovariancexbetweenxxxxx andxxx isxE(xxxxW
1x 2 1 x
1x 2 1 ) x= xE(x 1 2)x=
x
2
E(x1)E(x2)x(byxindependence)x=x0.x Axsimilarxargumentxshowsxthatxthe

covariancexbetweenxx1x2 andxx2 isxzero.x Butxthenxthexlinearxprojectionxof

x1x2x ontox(1,x1,x2)xisxidenticallyxzero.x Nowxjustxusexthexlawxofxiterated

projectionsx(PropertyxLP.5xinxAppendixx2A):

L(y|1,x1,x2)x=xL(b0 +xb1x1 +xb2x2 +xb3x1x2|1,x1,x2)

=xb0 +xb1x1 +xb2x2 +x b3L(x1x2|1,x1,x2)

=xb0 +xb1x1 +xb2x2.

d. Equationx(2.47)xisxmorexusefulxbecausexitxallowsxusxtoxcomputexthe

partialxeffectsxofxx1 andxx2 atxanyxvaluesxofxx1 andxx2.x Underxthe

assumptionsxwexhavexmade,xthexlinearxprojectionxinx(2.48)xdoesxhavexasxits

slopexcoefficientsxonxx1 andxx2 thexpartialxeffectsxatxthexpopulationxaverage

valuesxofxx1 andxx2 --xzeroxinxbothxcasesx--xbutxitxdoesxnotxallowxusxto

obtainxthexpartialxeffectsxatxanyxotherxvaluesxofxx1xandxx2.x Incidentally,

thexmainxconclusionsxofxthisxproblemxgoxthroughxifxwexallowxx1 andxx2 toxhave

anyxpopulationxmeans.



2.5.x Byx definition,x Var(u1|x,z)x =x Var(y|x,z)x andx Var(u2|x)x =x Var(y|x).x By
2x 2x
assumption,x thesex arex constantx andx necessarilyx equalx tox s _x Var(ux)x andx s _
1 1 2
2 2
Var(ux2),xrespectively.x ButxthenxPropertyxCV.4ximpliesxthatxs 2 >xs1. This

simplexconclusionxmeansxthat,xwhenxerrorxvariancesxarexconstant,xthexerrorxvaria

ncexfallsxasxmorexexplanatoryxvariablesxarexconditionedxon.



2.7.xWritexthexequationxinxerrorxformxas

2

, yx=xg(x)x+xzBx +xu,xE(u|x,z)x=x0.

Takexthexexpectedxvaluexofxthisxequationxconditionalxonlyxonxx:

E(y|x)x =x g(x)x +x [E(z|x)]B,xandx

subtractxthisxfromxthexfirstxequationxtoxget

yx -x E(y|x)x =x [zx -x E(z|x)]Bx +x u
~ ~ ~ ~
y zB z or z) =
+xu.xxBecause isxaxfunctionxofx(x,z),xE(u| =x0x(sincexE(u|x,z)x=
~x~ ~
E(y
0),xand |z)
xso zB =

.xxThisxbasicxresultxisxfundamentalxinxthexliteraturexonxxxxxxxxestimatingx part

ialx linearx models.x First,x onex estimatesx E(y|x)x andx E(z|x)xusingxveryxflexiblexm

ethods,xtypically,xso-calledxnonparametricxmethods.
~ ^ ~
Then,xafterxobtainingxresidualsxofxthexformxyix _x yix -xE(yi|xi)xandxzix _x zix -x-
^ ~ ~
E(zi|xi),xBxisxestimatedxfromxanxOLSxregressionxyixonxzi,xix=x1,...,N.x Under

generalxconditions,xthisxkindxofxnonparametricxpartialling-
-----
outxprocedurexleadsxtoxaxrN-
consistent,xasymptoticallyxnormalxestimatorxofxB.xxSeexRobinsonx(1988)

andxPowellx(1994).



CHAPTERx3



3.1.xToxprovexLemmax3.1,xwexmustxshowxthatxforxallxex >x0,xtherexexistsxbex <x8

andxanxintegerxNex suchxthatxP[|xN|x >x be]x<xe,xallxNx>x Ne.x Wexusexthe
p
followingxfact:x sincexxNx L a,xforxanyxex >x0xtherexexistsxanxintegerxNe such

thatxP[|xN - a|x >x1]x<xex forxallxNx>x Ne.x [ThexexistencexofxNe isximpliedxby

Definitionx 3.3(1).]x Butx |xN|x =x |xN - ax+xa|x <x |xN - a|x +x|a|x (byxthextriangle

inequality),xandxsox|xN|x -x|a|x <x |xNx -xa|.x ItxfollowsxthatxP[|xN|x -x|a|x >x1]

<xP[|xN - a|x >x1].x Therefore,xinxDefinitionx3.3(3)xwexcanxtakexbe _x|a|x+x1

3

, (irrespectivexofxthexvaluexofxe)xandxthenxthexexistencexofxNe followsxfrom

Definitionx3.3(1).




4