三Investment Tools Quantitative Methods.docx

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三Investment Tools Quantitative Methods.docx

三InvestmentToolsQuantitativeMethods

三、InvestmentTools:

QuantitativeMethods

1.A:

SamplingandEstimation

a:

Definesimplerandomsampling.

Simplerandomsamplingisamethodofselectingasampleinsuchawaythateachitemorpersoninthepopulationbeginstudiedhasthesame(non-zero)likelihoodofbeingincludedinthesample.Thisisthestandardsamplingdesign.

b:

Defineandinterpretsamplingerror.

Samplingerroristhedifferencebetweenasamplestatistic(themean,variance,orstandarddeviationofthesample)anditscorrespondingpopulationparameter(themean,varianceorstandarddeviationofthepopulation). 

Thesamplingerrorofthemean=samplemean-populationmean=Xbar-µ.

c:

Defineasamplingdistribution

Thesamplestatisticitselfisarandomvariable,soitalsohasaprobabilitydistribution.Thesamplingdistributionofthesamplestatisticisaprobabilitydistributionmadeupofallpossiblesamplestatisticscomputedfromsamplesofthesamesizerandomlydrawnfromthesamepopulation,alongwiththeirassociatedprobabilities.

d:

Distinguishbetweensimplerandomandstratifiedrandomsampling.

Simplerandomsamplingiswheretheobservationsaredrawnrandomlyfromthepopulation.Inarandomsampleeachobservationmusthavethesamechanceofbeingdrawnfromthepopulation.Thisisthestandardsamplingdesign.

?

Stratifiedrandomsamplingfirstdividesthepopulationintosubgroups,calledstrata,andthenasampleisrandomlyselectedfromeachstratum.Thesampledrawncanbeeitheraproportionaloranon-proportionalsample.Aproportionalsamplerequiresthatthenumberofitemsdrawnfromeachstratumbeinthesameproportionasthatfoundinthepopulation.

e:

Distinguishbetweentime-seriesandcross-sectionaldata.

Atime-seriesisasampleofobservationstakenataspecificandequallyspacedpointsintime.ThemonthlyreturnsonMicrosoftstockfromJanuary1990toJanuary2000areanexampleoftime-seriesdata.

 

Cross-sectionaldataisasampleofobservationstakenatasinglepointintime.ThesampleofreportedearningspershareofallNasdaqcompaniesasofDecember31,2000isanexampleofcross-sectionaldata.

f:

Statethecentrallimittheoremanddescribeitsimportance.

Thecentrallimittheoremtellsusthatforapopulationwithameanµandafinitevariance σ2,thesamplingdistributionofthesamplemeansofallpossiblesamplesofsizenwillbeapproximatelynormallydistributedwithameanequaltoµandavarianceequaltoσ2/n.

Thecentrallimittheoremisextremelyusefulbecausethenormaldistributionisrelativelyeasytoworkwithwhendoinghypothesistestingandformingconfidenceintervals.Wecanmakeveryspecificinferencesaboutthepopulationmean,usingthesamplemean,nomatterthedistributionofthepopulation,aslongasthesamplesizeis"large."

 

Whatyouneedtoknowfortheexam:

1.Ifthesamplesizenissufficientlylarge(greaterthan30),thesamplingdistributionofthesamplemeanswillbeapproximatelynormal. 

2.Themeanofthepopulation,µ,andthemeanofallpossiblesamplemeans,µx,areequal.

3.Thevarianceofthedistributionofsamplemeansisσ2/n.

g:

Calculateandinterpretthestandarderrorofthesamplemean.

Standarderrorofthesamplemeansisthestandarddeviationofthesamplingdistributionofthesamplemeans.Thestandarderrorofthesamplemeanswhenthestandarddeviationofthepopulationisknowniscalculatedby:

σx=σ/√n,where:

σx=thestandarderrorofthesamplemeans,σ=thestandarddeviationofthepopulation,andn=thesizeofthesample.

Example:

ThemeanhourlywageforIowafarmworkersis$13.50withastandarddeviationof$2.90.LetxbethemeanwageperhourforarandomsampleofIowafarmworkers.Findthemeanandstandarderrorofthesamplemeans,x,forasamplesizeof30.

Themeanμxofthesamplingdistributionofxisμx=μ=$13.50.Sinceσisknown,thestandarderrorofthesamplemeansis:

σx=σ/√n=2.90/√30=$.53.Inconclusion,ifyouweretotakeallpossiblesamplesofsize30fromtheIowafarmworkerpopulationandprepareasamplingdistributionofthesamplemeansyouwillgetameanof$13.50andstandarderrorof$.53.

h:

Distinguishbetweenapointestimateandaconfidenceintervalestimateofapopulationparameter.

Pointestimatesaresingle(sample)valuesusedtoestimatepopulationparameters.Theformulaweusetocomputethepointestimateiscalledtheestimator.Forexample,thesamplemeanXbarisanestimatorofthepopulationmeanµ,andiscomputedusingthefollowingformula:

Xbar=(Σx/n)

Thevalueweobtainfromthiscalculationforaspecificsampleiscalledthepointestimateofthemean.

Aconfidenceintervalisarangeofestimatedvalueswithinwhichtheactualvalueoftheparameterwillliewithagivenprobabilityof1- α.Thetermαisalsocalledthesignificancelevelofthetest.Itisalsoknownastheconfidencelevel.

i:

Identifyanddescribethedesirablepropertiesofanestimate.

Whenwehaveachoiceamongseveralestimators,wewanttoselecttheonewiththemostdesirablestatisticalproperties:

unbiasedness,efficiency,andconsistency.

Anunbiasedestimatorisoneforwhichtheexpectedvalueoftheestimatorisequaltotheparameteryouaretryingtoestimate.

Anunbiasedestimatorisalsoefficientifthevarianceofitssamplingdistributionissmallerthanalltheotherunbiasedestimatorsoftheparameteryouaretryingtoestimate.Thesamplemean,forexample,isanefficientestimatorofthepopulationmean.

Aconsistentestimatorprovidedamoreaccurateestimateoftheparameterasthesamplesizeincreases.Asthesamplesizeincreases,thestandarddeviation(standarderror)ofthesamplemeanfallsandthesamplingdistributionbunchesmorecloselyaroundthepopulationmean.

j:

Calculateandinterpretaconfidenceintervalforapopulationmean,givenanormaldistributionwithaknownpopulationvariance.

Ifthedistributionofthepopulationisnormalandweknowthepopulationvariance,wecanconstructtheconfidenceintervalforthepopulationmeanasfollows:

?

XbarZα/2(σ/?

√n)

?

Example:

Supposeweadministerapracticeexamto100CFALevelIcandidates,andwediscoverthemeanscoreonthispracticeexamforall36ofthecandidatesinthesamplewhostudiedatleast10hoursaweekinpreparationfortheexamis80.Assumethepopulationstandarddeviationis15.Constructa99%confidenceintervalforthemeanscoreonthepracticeexamofcandidateswhostudyatleast10hoursaweek.

?

802.575(15/√36)=806.4

?

The99%confidencehasalowerlimitof73.6andanupperlimitof86.4.

k:

DescribethepropertiesofStudent'st-distribution.

Thestudent'st-distributionissimilar,butnotidenticaltothenormaldistributioninshape.Itisdefinedbyasingleparameter(thedegreesoffreedom),whereasthenormaldistributionisdefinedbytwoparameters(themeanandvariance).

 

Thestudent'st-distributionhasthefollowingproperties:

∙Itissymmetrical.

∙Itisdefinedbyasingleparameter,thedegreesoffreedom(df),wherethedegreesoffreedomareequaltothenumberofsampleobservationsminusone.(n-1).

∙Itislesspeakedthananormaldistribution,withmoreprobabilityinthetails.

∙Asthedegreesoffreedom(thesamplesize)getslarger,theshapeofthet-distributionapproachesastandardnormaldistribution.

l:

Calculateandinterpretaconfidenceintervalforapopulationmean,givenanormaldistributionwithanunknownpopulationvariance.

Example:

Supposeyoutakeasampleofthepast30monthlyreturnsforMcCrearyInc.Themeanreturnis2%,andthesamplestandarddeviationis20%.Thestandarderrorofthesamplewasfoundtobe3.6%.Constructa95%confidenceintervalforthemeanmonthlyreturn.

 

Becausethereare30observations,thedegreesoffreedomare30-1=29.Remember,becausethisisatwo-tailedtest,wewantthetotalprobabilityinthetailstobeα=5%;becausethet-distributionissymmetrical,theprobabilityineachtailwillbe2.5%whendf=29.Fromthet-table,wecandeterminethatthereliabilityfactorfortα/2,ort25,is2.045.Thentheconfidenceintervalis:

 

22.045(20/ √30)=2%7.4%

 

The95%confidenceintervalhasalowerlimitof-5.4%andanupperlimitof9.4%.

m:

Discusstheissuessurroundingselectionoftheappropriatesamplesize.

Whenthedistributionisnon-normal,thesizeofthesampleinfluenceswhetherornotwecanconstructtheappropriateconfidenceintervalforthesamplemean.Ifthedistributionisnon-normal,butthevarianceisknown,wecanstillusetheZ-statisticaslongasthesamplesizeislarge(n>30).Wecandothisbecausethecentrallimittheoremassuresusthatthedistributionofthesamplemeanisapproximatelynormalwhenthesampleislarge.

 

Ifthedistributionisnon-normalandthevarianceisunknown,wecanusethet-statisticaslongasthesamplesizeislarge(n>30).

 

Thismeansthatifwearesamplingfromanon-normaldistribution(whichissometimesthecaseinfinance),wecannotcreateaconfidenceintervalifthesamplesizeislessthan30.So,allelseequal,makesureyouhaveasamplelargerthan30,andthelarger,thebetter.

n:

Defineanddiscussdata-snooping/data-miningbias.

Data-snoopingbiasoccurswhentheresearcherbaseshisresearchonthepreviouslyreportedempiricalevidenceofothers,ratherthanonthetestablepredictionsofwell-developedeconomictheories.

 

Datasnoopingoftenleadstodatamining,whentheanalystcontinuallyusesthesamedatabasetosearchforpatternsortradingrulesuntilhefindsonethat"works."Forexample,someresearchersarguethatthevalueano

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