悉尼大學(xué)計量經(jīng)濟(jì)學(xué)原理課件lec2-ECMT5001-sem1-

ECMT5001

PrinciplesofEconometricsLecture2Semester1,2010InThisLecture:

RandomvariablesDiscreterandomvariablesContinuousrandomvariablesProbabilitiesfornormalrandomvariablesRandomVariablesRandomVariablesArandomvariableisavariablewhosevalueisunknownuntilitisobserved

TheoutcomeofanexperimentisarandomvariablebeforetheexperimenttakesplaceWeuserandomvariablestoassignnumericalvaluestotheoutcomesofanexperimentRandomVariablesArandomvariablehasasetofpossiblevaluesitcantakee.g.,numberofheadsoccurringwhentossingtwocoinsisarandomvariableSetofpossiblevalues:0,1,2Distinguishtwotypes:Discrete&ContinuousrandomvariableAdiscreterandomvariablecantakeonlyalimitedorcountablenumberofvalues

(e.g.no.ofchildren,TV’sowned)Acontinuousrandomvariablecantakeanyvalueinaninterval.Ithasanuncountablenumberofpossiblevalues

(GDPin2008,income,Stockpriceindex)RandomVariablesNotationWhenwerefertoarandomvariablewhoseoutcomeisunknownwegenerallydenoteitwithacapitalletter(uppercase)e.g.XWhenwerefertoonespecificoutcomeofanexperimentoftherandomvariablewedenoteitthelowercaseversionofthevariablenamee.g.xWhenwehavenoutcomesofanexperimentweuseasubscripttodistinguishthem,e.g.x1,x2,x3,…,xn.ProbabilityDensityFunctionTheprobabilitydensityfunction(pdf)summarisestheprobabilitiesattachedtopossibleoutcomesofarandomvariableApdfcanbepresentedasatable,agraph,oraformula,f(x)Fordiscreterandomvariablesthepdfisalsocalledtheprobabilitymassfunction

(pmf)WeshallusethetermprobabilitydensityfunctionforbothcontinuousanddiscreterandomvariablesCumulativeDistributionFunctionAnotherwaytorepresentprobabilitiesisthecumulativedistributionfunction(cdf)Thecdf

forarandomvariableXgivestheprobabilitythatXislessthanorequaltoaspecificvaluex

F(x)=P(X≤x)DiscreteRandomVariablesDiscreteRandomVariablesForadiscrete

randomvariabletheprobabilitythatarandomvariableXtakesthevaluexis

f(x)=P(X=x)f(x)isaprobabilityso0≤f(x)≤1forallvaluesofxIfXcanonlytakenvaluesx1,…,xn,theprobabilitiesmustsumto1:

f(x1)+f(x2)+…+f(xn)=1, i.e.Inthislectureweshallconsiderafewcommonlyuseddiscretedistributions:BernoulliBinomialPoisson

DiscreteRandomVariablesBernoulliRandomVariableRandomvariableXisaBernoullivariableifitonlytakesvalues0or1Usedforexperimentswithonly2possibleoutcomes(labelledsuccess(1)orfail(0))Examples:Heads,TailsRent,OwnEmployed,unemployedPass,Fail,Purchase,Don’tPurchaseMale,FemalepdfforBernoulliVariable

Probabilitydensityfunctioninequationform:f(x;p)=px(1–p)1-xx=0,10p1f(0;p)=P(x=0)=1-pf(1;p)=P(x=1)=pHerepisaparametergivingtheprobabilityofsuccessE.g.Ifp=0.6wehave:

f(0)=P(X=0)=0.4

f(1)=P(X=1)=0.6GraphicalRepresentationofBernoullipdfBernoullipdfinaTableEmploymentStatusxf(x)Unemployed00.4Employed10.6BinomialRandomVariablesABinomialrandomvariableisobtainedifwerepeattheBernoulliexperimentanumber(n)oftimesLetX=numberofsuccessesinnindependenttrialsThepdf

forthenewbinomialrandomvariableXis

f(x;p,n)=px(1–p)n-xn!/{x!(n-x)!}x=0,1,2,….,n

0p1BinomialDistributionNotethatthebinomialdistributionhastwoparameters:p–theprobabilityofasuccessineachtrial,andn–thenumberoftrialsAlso,thesymbol“!”means“factorial”n!=n*(n-1)*(n-2)*…*2*1,and

0!=1(bydefinition)Calculationforn=10,X=6BinomialExampleLetXbethenumberofhouseholdswith1phoneinarandomsampleofnhouseholdsSupposen=2Iftheprobabilitythatasinglerandomlydrawnhousehas1phoneispthenP(X=0)=f(0;p;2)=(1-p)2

P(X=1)=f(1;p;2)=2p(1-p)

P(X=2)=f(2;p;2)=p2BinomialExampleforn=2&p=0.4Example:SuccessfulTradesAlong-termtradercomparestheannualreturnonstocksinhisportfoliowiththatoftheAllOrdinariesIndexrepresentingthemarketaverage.HebelieveshischosenstocksreturnmorethantheAllOrds70%ofthetime.LetXbearandomvariablecountingthenumberofstocksinhis5-stockportfoliothatbeattheAllOrdsin2008.Ifhisbeliefiscorrectfindtheprobabilitythat4ormoreofhisstocksbeattheAllOrdsindexthisyearExample:SuccessfulTradesPoissonRandomVariablesThePoissondistributionisusedtomodeltheprobabilityof“rareevents”thatoccurrandomlyintime,distanceorspacee.g.NumberofaccidentsonNSWroadsinaweekNumberofarrivalsathetrainstationticketingofficeevery5minutesNumberofwombatsencounteredonatripfromSydneytoCanberraNumberofdaysinagivenyearinwhichtheAllOrdsindexhasamoveofover1%PoissonCharacteristicsAPoissonexperimentconsistsofcountingthenumber,x,oftimesaparticulareventoccursduringagivenunitoftime,oragivenarea,orvolume,(orweight,ordistance,oranyotherunitofmeasurement)Theprobabilitythataneventoccursinagivenunitoftime,area,orvolumeisthesameforallunitsThenumberofeventsthatoccurinoneunitoftimeareaorvolumeisindependentofthenumberthatoccurinotherunitsPoissonpdfThepdfforaPoissondistributionisgivenby

where

x=Numberofrareeventsperunitoftime, distanceorspace

μ

=Meanvalueofx

e=2.71828…PoissonExampleLetXbethenumberofcallsreceivedbyacustomerhelp-deskinaten-minuteperiodIfXisPoissondistributedwiththemeannumberofcallsinaten-minuteperiodbeingequalto5,findthefollowing:ProbabilityofnocallsProbabilityofonly1callProbabilityofatleast2callsPoissonExample

ContinuousRandomVariablesContinuousRandomVariables

ContinuousrandomvariablescantakeanyvalueinanintervalonthenumberlineTheyhave(atleasttheoretically)anuncountable(infinite)numberofvaluesTheprobabilityofanyspecificvalueiszero(1/∞)InsteadwetalkaboutoutcomesbeinginacertainrangeContinuousRandomVariablesConsiderforexamplethepopulationofsalariesforallAustralianresidentsWhatistheprobabilitythatarandomlyselectedindividualhasasalarylessthan$100,000?Toanswerthisquestionweneedtoknowtheproportionofallsalariesthatarebelow$100,000Thisproportionistheareaundertherelativefrequencyhistogramthatliestotheleftof$100,000(seeslide)HypotheticalSalariesDistributionProportionofsalariesbelow$100,000ContinuousRandomVariablesForexample,iftheshadedareaunderthehypotheticalrelativefrequencydistributionis80%ofthetotalareaunderthecurve,thentheprobabilitythatarandomlyselectedindividual’ssalaryislessthan$100,000is0.8However,withoutthefullpopulationwewillnotknowtheexactshapeofthedistributionInthiscase,wepostulateamodel,i.e.selectasmoothcurveasamodelforthepopulationrelativefrequencydistributionContinuousRandomVariablesTofindtheprobabilitythataparticularobservationliesinaparticularintervalweusethemodelandfindtheareaunderthecurvethatfallsoverthatintervalOfcourseweneedtobesurethepopulationrelativefrequencyandourmodelareverysimilarWeshallseeinlaterlectureswhywebelievethatthemodelsweusearegoodapproximationstorealityContinuousRandomVariablesOurmodelsarecontinuousfunctions,definedfortherangeofvaluesourunderlyingrandomvariableofinterestcantakeThemodelistheprobabilitydensityfunction(pdf)oftherandomvariable:f(x)TofindtheprobabilityofobservinganoutcomewithinarangeofvaluesweintegratethepdfacrossthatrangeContinuousrandomvariablesProbabilitiesarefoundbyintegratingthepdfovertherange:IntegrationgivestheareaunderthecurveThetotalareaunderthepdfmustbe1:CumulativeDistributionFunctionAnotherwaytorepresentprobabilitiesisthecumulativedistributionfunction

(cdf)ThecdfforarandomvariableX,F(x),givestheprobabilitythatXislessthanorequaltoaspecificvaluexWeshallconsidertwocontinuousdistributionslecture:UniformdistributionNormaldistributionOtherswe’llcomeacrosslaterStudent’stdistribution,Chi-squaredistribution,FdistributionContinuousRandomVariablesUniformDistributionSupposeyouweretorandomselectanumberxfromtheintervala≤x

≤bxiscalledauniformrandomvariableanditsdistributioniscalledtheuniformdistributionTheshapeofthedistributionisarectangle

(seefollowingslide)Theheightoftherectangleis1/(b-a)

(whichguaranteestheareaundertherectangleisone)UniformDistribution1/(b-a)f(x)xTheUniformDistributionabUniformDistributionProbabilitydensityfunctionf(x) =1/(b-a)a≤x≤b

=

0elsewhereWhatistheprobabilitythatarandomlyselectednumberislessthanc(wherea≤c≤b)?(seenextslide)UniformDistributionP(X≤c)Areaofrectangle=baseheight

=(c-a)1/(b-a)

=(c-a)/(b-a)UniformExampleConsiderrandomlychoosinganumberbetween2and6.Whatistheprobabilitythatthechosennumberisbetween3and5?

f(x)=1/(6-2)=0.25 2≤x≤6=0 elsewhereUniformExampleP(3.5≤X≤5)=0.375NormalDistributionThemostwidely-usedandwell-knowndistributioninstatistics/econometricsistheNormaldistributionTheNormaldistributioniscommonlyreferredtoasthebell-curveduetoitsfamiliarshapeProposedbyGauss(1777-1855),itremarkablyprovidesanadequatedistributionforthemodellingofmanyvariablesasweshallseelaterinthecourseNormalDensityFunctionPropertiesBell-shapedSpreadsoutmoreassgetslargerSymmetricaroundthemean,m

P(X>m)=P(X<m)=0.5Totalareaunderthecurve=1ProbabilitiesforNormalRandomVariablesNormalDistributionThepdfforanormalrandomvariablewithmeanandvariance2isgivenby where-∞<x<∞and>0IfXfollowsanormaldistributionwewriteitasX~N(m,s2)NormalDistributionUnfortunately,integrationofthepdfforthenormaldistributionisintractableThatis,thereisnosimplemathematicalsolutionasthereiswiththeuniformdistributionHowever,probabilitiesforthestandardnormaldistributionhavebeentabulatedforvariousvaluesandwecanusethesetoobtainprobabilitiesforanynormaldistributionStandardNormalDistributionThestandardnormalrandomvariableisusuallydenotedZandanobservedoutcomeisdenotedzThestandardnormaldistributionisanormaldistributionwithmean=0andvariance=1

i.e.Z~N(0,1)Probabilitiesforthestandardnormalcumulativedistributionaretabulated(seefollowingslide)StandardNormalTable*-page1z0123456789-30.00130.00130.00130.00120.00120.00110.00110.00110.0010.001-2.90.00190.00180.00180.00170.00160.00160.00150.00150.00140.0014-2.80.00260.00250.00240.00230.00230.00220.00210.00210.0020.0019-2.70.00350.00340.00330.00320.00310.0030.00290.00280.00270.0026-2.60.00470.00450.00440.00430.00410.0040.00390.00380.00370.0036-2.50.00620.0060.00590.00570.00550.00540.00520.00510.00490.0048-2.40.00820.0080.00780.00750.00730.00710.00690.00680.00660.0064-2.30.01070.01040.01020.00990.00960.00940.00910.00890.00870.0084-2.20.01390.01360.01320.01290.01250.01220.01190.01160.01130.011-2.10.01790.01740.0170.01660.01620.01580.01540.0150.01460.0143-20.02280.02220.02170.02120.02070.02020.01970.01920.01880.0183-1.90.02870.02810.02740.02680.02620.02560.0250.02440.02390.0233-1.80.03590.03510.03440.03360.03290.03220.03140.03070.03010.0294-1.70.04460.04360.04270.04180.04090.04010.03920.03840.03750.0367-1.60.05480.05370.05260.05160.05050.04950.04850.04750.04650.0455-1.50.06680.06550.06430.0630.06180.06060.05940.05820.05710.0559-1.40.08080.07930.07780.07640.07490.07350.07210.07080.06940.0681-1.30.09680.09510.09340.09180.09010.08850.08690.08530.08380.0823-1.20.11510.11310.11120.10930.10750.10560.10380.1020.10030.0985-1.10.13570.13350.13140.12920.12710.12510.1230.1210.1190.117-10.15870.15620.15390.15150.14920.14690.14460.14230.14010.1379-0.90.18410.18140.17880.17620.17360.17110.16850.1660.16350.1611-0.80.21190.2090.20610.20330.20050.19770.19490.19220.18940.1867-0.70.2420.23890.23580.23270.22960.22660.22360.22060.21770.2148-0.60.27430.27090.26760.26430.26110.25780.25460.25140.24830.2451-0.50.30850.3050.30150.29810.29460.29120.28770.28430.2810.2776-0.40.34460.34090.33720.33360.330.32640.32280.31920.31560.3121-0.30.38210.37830.37450.37070.36690.36320.35940.35570.3520.3483-0.20.42070.41680.41290.4090.40520.40130.39740.39360.38970.3859-0.10.46020.45620.45220.44830.44430.44040.43640.43250.42860.4247-00.50.4960.4920.4880.4840.48010.47610.47210.46810.4641*AppendixG.1,Wooldridge(2009)StandardNormalTable*-page2z012345678900.50.5040.5080.5120.5160.51990.52390.52790.53190.53590.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.57530.20.57930.58320.58710.5910.59480.59870.60260.60640.61030.61410.30.61790.62170.62550.62930.63310.63680.64060.64430.6480.65170.40.65540.65910.66280.66640.670.67360.67720.68080.68440.68790.50.69150.6950.69850.70190.70540.70880.71230.71570.7190.72240.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.75490.70.7580.76110.76420.76730.77040.77340.77640.77940.78230.78520.80.78810.7910.79390.79670.79950.80230.80510.80780.81060.81330.90.81590.81860.82120.82380.82640.82890.83150.8340.83650.838910.84130.84380.84610.84850.85080.85310.85540.85770.85990.86211.10.86430.86650.86860.87080.87290.87490.8770.8790.8810.8831.20.88490.88690.88880.89070.89250.89440.89620.8980.89970.90151.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.91771.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.93191.50.93320.93450.93570.9370.93820.93940.94060.94180.94290.94411.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.95451.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.96331.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.97061.90.97130.97190.97260.97320.97380.97440.9750.97560.97610.976720.97720.97780.97830.97880.97930.97980.98030.98080.98120.98172.10.98210.98260.9830.98340.98380.98420.98460.9850.98540.98572.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9892.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.99162.40.99180.9920.99220.99250.99270.99290.99310.99320.99340.99362.50.99380.9940.99410.99430.99450.99460.99480.99490.99510.99522.60.99530.99550.99560.99570.99590.9960.99610.99620.99630.99642.70.99650.99660.99670.99680.99690.9970.99710.99720.99730.99742.80.99740.99750.99760.99770.99770.99780.99790.99790.9980.99812.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.998630.99870.99870.99870.99880.99880.99890.99890.99890.9990.999*AppendixG.1,Wooldridge(2009)UsingStandardNormalTablesThestandardnormaltablesweusegivethecumulativedistributionfunctionIfyoulookupanumbersayz*,thetableprovides

P(-∞<Z<z*)Inthelefthandcolumnofthetablelookupthenumberz*tothefirstdecimalplaceThenlookuptheseconddecimalplaceacrossthetopofthetableExercisesIfZ~N(0,1),findthefollowing:P(Z≤-1.8)P(Z>1.32)P(0.08<Z<1.67)P(-1.35<Z<0.8)ExercisesNoteP(Z≤-1.8)=P(Z<-1.8)

sinceP(Z=-1.8)=0

Therequiredprobabilityis

representedbytheshaded

areainthediagram

Lookingup“-1.8”intheleft-handcolumnand“0”inthetoprowwefindtherequiredprobabilitytobe: P(Z≤-1.8)=0.0359ExercisesP(Z>1.32)=ExercisesP(0.08<Z<1.67)=P(Z<1.67)-P(Z<0.08)

P(Z<1.67)=0.9525(fromtables) P(Z<0.08)=0.5319(fromtables) P(0.08<Z<1.67)=0.9525-0.5319 =0.4206ExercisesP(-1.35<Z<0.8)=PropertiesoftheNormalAlineartransformationofanormalvariableresultsinanothernormalvariable.Thatis,ifX~N(m,s2)andY=a+bX,thenYisalsoanormalvariableSpecifically Y~N(a+bm,b2s2)newmeannewv