三Investment Tools Quantitative Methods.docx

三Investment Tools Quantitative Methods.docx

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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.

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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:

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XbarZα/2（σ/?

√n）

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Example:

Supposeweadministerapracticeexamto100CFALevelIcandidates,andwediscoverthemeanscoreonthispracticeexamforall36ofthecandidatesinthesamplewhostudiedatleast10hoursaweekinpreparationfortheexamis80.Assumethepopulationstandarddeviationis15.Constructa99%confidenceintervalforthemeanscoreonthepracticeexamofcandidateswhostudyatleast10hoursaweek.

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802.575（15/√36）=806.4

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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