Theconditionaldistributionof SampleMidterm
Name___________________________________
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
1) YgivenX = x ,Pr(Y = y X=x ),is
l
Pr(Y = y )
A) Pr(X=x i, Y=y ).
B)
.
Pr(X=x )
i =1
C)
Pr(X=x , Y=y )
.
Pr(X=x )
D)
1)
Pr(X = x , Y = y )
Pr(Y=y )
^
2) AnestimatorYofthepopulationvalueYisconsistentif
A) Y
p
2)
0.
^
B) Y p .
Y
C) Yisnormallydistributed.
D) itsmeansquareerroristhesmallestpossible.
^
3) Whentheestimatedslopecoefficientinthesimpleregressionmodel, 1 ,iszero,then
A) R2 =0.
B) R2 >( SSR/TSS ).
C) R2 =Y.
3)
D) 0<R2 <1.
4) Toobtaintheslopeestimatorusingtheleastsquaresprinciple,youdividethe
4)
A) samplecovarianceof Xand YbythesamplevarianceofY.
B) samplevarianceof XbythesamplecovarianceofX and Y.
C) samplecovarianceof Xand Ybythesamplevarianceof X.
D) samplevarianceof Xbythesamplevarianceof Y.
5)
5) Theerrortermishomoskedasticif
A) var(ui Xi=x ) isconstantfori=1,, n.
B) therearenooutliers.
C) Xiisnormallydistributed.
D) var(ui Xi = x ) dependson x.
1
SHORTANSWER.Writeyouranswerinthespaceprovidedoronaseparatesheetofpaper.Ifyouaddanothersheetof
paper,itmustbestapledtotheexamtoreceivecredit.
6) Adultmalesaretaller,onaverage,thanadultfemales.Yousuggesttoyourlittlesisterthatshecollectdataon
heightandgenderofchildrenin4 thto6thgradeaspartofherscienceproject.Theaccompanyingtableshows
herfindings.
HeightofYoungBoysandGirls,Grades4 -6,ininches
YBoys
Boys
SBoys
nBoys
57.8
3.9
YGirls
55
Girls
SGirls
nGirls
58.4
4.2
57
(a)Letyournullhypothesisbethatthereisnodifferenceintheheightoffemalesandmalesatthisagelevel.
Specifythealternativehypothesis.
(b)Findthedifferenceinheightandthestandarderrorofthedifference.
(c)Generatea95%confidenceintervalforthedifferenceinheight.
(d)Calculatethe t- statisticforcomparingthetwomeans.Isthedifferencestatisticallysignificantatthe1%
level?Whichcriticalvaluedidyouuse?Whywouldthisnumberbesmallerifyouhadassumedaone - sided
alternativehypothesis?
7) Yourtextbookdefinedthecovariancebetween X and Y asfollows:
n
1
(X X)(YiY)
n1 i
i =1
Provethatthisisidenticaltothefollowingalternativespecification:
n
n
1
XiYi- n-1 XY
n- 1
i =1
8) Youhaveobtainedasub- sampleof1744individualsfromtheCurrentPopulationSurvey(CPS)andare
interestedintherelationshipbetweenweeklyearningsandage.Theregression,using
heteroskedasticity - robuststandarderrors,yieldedthefollowingresult:
Earn=239.16 + 5.20 Age, R2 =0.05, SER=287.21
(12.3)(3.2)
whereEarnand Agearemeasuredindollarsandyearsrespectively.
(a)Interprettheinterceptandslopecoefficients.
(b)Istheeffectofageonearningsstatisticallysignificantatthe5%level?
(c)Provideaneconomicrationaleforwhyageshouldmatterinthedeterminationofearnings.Doyouthink
thattherelationshipbetweenageandearningsislinear?Explain.
(d)Theaverageageinthissampleis37.5years.Whatisaverageannualincomeinthesample?
(e)Interpretthemeasuresoffit.
9) Considerthefollowingmodel:
Yi=1 Xi+ ui.
DerivetheOLSestimatorfor1 .(Note:youwillneedtousecalculus)
2