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BES Tutorial Sample Solutions, S2, 201010
WEEK 11 TUTORIAL EXERCISES (To be discussed in the week
starting
October 4) 1. Useacalculator tocomputethesample
leastsquaresregression line for
themodel ,giventhefollowingsixobservations.y 2 8 6 12 9 11x 1 4
3 10 10 8 1 4 3 10 10 86 6;
2 8 6 12 9 116 8
1 62 8 8 611 8 62 1 6 8 6 74
6274 0.8378
8 0.8378 6 2.9732Thusthesampleregressionlineis 2.9732 0.8378 2.
Suppose the relationship between the dependent variable weekly
household consumption expenditure in dollars (y) and the
independentvariable weekly household income in dollars (x) is
represented by
thesimpleregressionmodel(ireferstotheithobservationorhousehold):
Supposeasampleofobservationsyieldsleastsquaresestimatesof
b0=32andb1=0.82.
(a) Whatdoesrepresentinthemodel?
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It is therandomdisturbance term. It includesanypurelyrandom
factorsorerrorsandfactorsthathavebeenleftoutofthemodelbutwhoseinfluenceisconsideredminor.
(b) State the basic (classical) assumptions made about the s in
thismodel.Explaininwordswhattheassumptionsmean.
(i) | 0forallobservations.Theconditionalmeanofthedisturbance
doesnotdependonxandisnormalizedtozero.NotethisisdifferentfromKellerwhoonlymentionsthenormalizationtozero.Thattheconditionalmeanof
thedisturbancesdoesnotdependon
xensuresunbiasednessoftheOLSestimatorandso isthemuchmore
importantcomponentofthisassumption.Relatingback to
thepreviouspartof thequestion it impliesthat omitted factors that
might affect expenditure but appear in
thedisturbanceareassumedtobeuncorrelatedwithx.
(ii) , aredrawnbysimplerandomsamplingandhenceiid.(iii)
Thestandarddeviationofisconstantforallobservations.Itisdenoted
byandwe say thedisturbancesarehomoskedastic.Here that impliesthe
variability in food expenditure doesnot depend on incomewhich
ispossiblyproblematicinpractice.
(iv) The disturbances for any two observations are independent.
Thiswillimply, in particular that there is no correlation between
disturbancesassociatedwithdifferentobservations. In thisexample the
factors in
thedisturbanceforhouseholdiarenotcorrelatedwiththoseforhouseholdj.
(v)
isnormallydistributedforallobservations.Doestheestimateofb0=32makesense?Ifnot,doesthisnecessarilyinvalidatethemodel?Explainyouranswer.This
indicatesthat ifahouseholdhadazeroweekly incomethenonaveragesuch a
householdwould have negative consumption,which does
notmakesense.However,thisdoesnotnecessarilyinvalidatethemodel.Itmaybethatthe
linear model is only a reasonable approximation for some range
ofhousehold incomes, not including incomes near zero. In
particular, therelationshipmaybenonlinear forvaluesof xnearzero.
The conclusion isthatweshouldbecareful in interpretingthe
interceptterm,as itmaynotbeverymeaningfulinsomecases.
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(c)
Interpretboth1andb1.Whatdoesthemodelpredictwouldbethechangeinyfollowinga$10increaseinxfromsomeinitiallevel?
1isthe(unknown)populationchangeinthevalueofyresultingfromaoneunit
increase in x,whereas b1=0.82 isan estimateof1. In
thisparticularexamplethisisthemarginalpropensitytoconsumethatwouldbediscussedineconomics
courses.The predicted change in y following a $10 increase in
xwouldbe10 10 0.82 $8.20.
(d) Supposewemeasured y and x in cents rather than dollars.
Whateffectwouldthishaveontheestimatedcoefficientofx?Whateffectwouldithaveontheestimatedintercept?
Inthiscase:$xbecomes100xcentsand$ybecomes100ycents.Theestimatedcoefficientof
ix whenthevariablesaremeasuredindollarsisgivenby
Ifweletbetheestimatedslopecoefficientwhenthevariablesaremeasuredincents,wehave
100 100100 100
100 100 100
100 Also,denotebytheestimatedinterceptinthiscasethenwehave
100 100 100
100Thusestimationofthismodel(withthesame,butrescaleddata)would
leadtoanunchangedb1,whilsttheintercepttermwouldbecome100 3200.
(e)
Supposeyweremeasuredindollarsbutxweremeasuredincents.Whateffectswouldthishaveontheestimatedcoefficientofx?
Denotetheestimatedslopeandinterceptinthiscasebyletand.Then
100 100
100 100 100
100 100
100
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Now estimationof thismodelwould lead to the estimated
coefficientof theincome variablebeing0.0082and estimated
interceptwouldbeunchanged.Thismakessensesince: If income ismeasured
in dollars,we predict expenditure (in
dollars)willincreaseby$0.82ifhouseholdincomeincreasesbyonedollar.
If income is measured in cents, we predict expenditure (in
dollars)
willincreaseby$0.0082ifhouseholdincomeincreasesbyonecent.
(f) Distinguish between and (or , the residual associated
withobservationi).Illustrateyouranswerwithadiagram
Wecan thinkof asanestimateof the
truerandomdisturbanceassociatedwithobservationi, . 3.
ComputingExercise#4
Refer to the Computing Work document in Course Documents in
theBlackboardwebsite.Answerthe2questionsassociatedwithsimplelinearregressiononpages21and22.
As indicated below in the Line fit plot produced for the first
part of
thequestion,thereisapositivecorrelationbetweenthereturnsonIntelstockandtheoverallmarketreturn.Howeverthereisconsiderablevariationaroundthesuperimposedlinearrelationship.
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Discussion: i)
Whatisthesampleregressionline?FromtheExcelregressionoutputbelow:
0.022 1.472,ii)
Istheresufficientevidencetoinferatthe5%significancelevelthat
thereisalinearrelationshipbetweenthereturnonIntelCorporationstockandthereturnonthetotalmarket?
Appropriatehypothesistobetestedis:
: 0;:
0whichaccordingtotheExceloutputyieldsapvalueof0.0069andsoforanysignificancelevelgreaterthan0.0069(whichincludes5%)wewouldrejectthenullandconcludethereisevidencetosuggestalinearrelationship.
iii)
Istheresufficientevidencetoinferatthe5%significancelevelthatIntelCorporationstockismoresensitivethantheaveragestock?
Nowtheappropriatehypothesistobetestedis:
: 1;: 1Thestandardizedteststatisticforthishypothesisis:
1.47163 10.52052
0.9061Usingatcriticalvalueand40degreesoffreedom(actually47degreesoffreedombutthisvaluenotintables)yieldsarejectionregionoft>1.684.AlternativelywitharelativelylargesamplesizewecaninvoketheCLTandusethe5%normalcriticalvalueof1.675.Ineithercasethecalculatedteststatisticfallswellshortoftherejectionregionandwecannotrejectthenullhypothesis.
iv)
Discussthesignificanceofthefindings?Whilethereisevidenceofastrongpositiverelationshipbetweenthereturns,theevidenceofwhethertheIntelstockismoreorlesssensitivetothemarketis
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weak.Thepointestimateof1.472indicatesevidenceinfavourofbeingmoresensitivebutwecannotexcludethepossibilitythatitisinfactlesssensitive.The95%CIprovidedbyExcelis(0.424,2.519)andhenceincludesvaluesconsistentwithbothpossibilities.
v) Explainthemeaningoftheregressionandresidualsumofsquares.The
total sums of squares representing the total variation (0.4446) in
thedependent variable (returns on Intel stocks) can be decomposed
into twoparts:aregressionsumofsquares(0.0658)
representingthatpartexplainedbytheregressionmodelandtheresidualsumofsquares(0.3788)representingthatpartleftoverandunexplainedbythemodel.InthiscasethelatterislargerelativetotheformerleadingtoanR2of0.148indicatingthatonly14.8%ofthevariationinIntelstockisbeingexplainedbythemarketmodel.Thisisconsistentwithourinitialobservationfromthescatterplotthattherewasconsiderablevariationaroundthetrendline.Seealsothelinefitplotthatoverlaystheestimatedmarketmodelonthebivariatescatter.
SUMMARY OUTPUT
Regression StatisticsMultiple R 0.3848R Square 0.1480Adjusted R
Square 0.1295Standard Error 0.0907Observations 48
ANOVAdf SS MS F Significance F
Regression 1 0.065822161 0.065822 7.993182 0.0069287Residual 46
0.378800255 0.008235Total 47 0.444622416
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Lower 95.0%Upper 95.0%Intercept 0.02192 0.01508 1.45365 0.15283
-0.00843 0.05228 -0.00843 0.05228INDEX 1.47163 0.52052 2.82722
0.00693 0.42387 2.51938 0.42387 2.51938
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0.200.150.100.050.00
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INTEL
INDEX
INDEXLineFitPlot
INTEL
PredictedINTEL
