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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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0.1 0.05 0 0.05 0.1

INTEL

INDEX

INDEXLineFitPlot

INTEL

PredictedINTEL