The Simple Linear Regression Model

ModuleOutline•Thismodulewillintroduceanumberofeconometricmodels•Forexample,–TheSimpleLinearRegressionModel–MultivariateRegression–InstrumentalVariables•Thiswillbecombinedwithpractical(applied)exampleswhichrelyondatatotesttheoryandhypotheses•Usefultextbooks:–1.BasicEconometricsbyDamodarN.Gujarati(4thedition).Thelibraryholdscopiesofthisbuttheyarelimited.Aneweditionhascomeoutbutthe4theditionisavailableonAmazonfrom£12.50.–2.IntroductoryEconometrics:AModernApproachbyJefferyWooldridge•BasicEconometricscoversmostofthematerialtouchedoninthelectureslides.IntroductoryEconometricsisslightlymoretechnicalbutitscoverageofpracticalmattersisverygood.Lecture1.TheSimpleLinearRegressionModel•Whatiseconometrics?Economics(andfinanceandmanagement)postulatestheories.Forexample,holdingallelseequal,areductioninthepriceofagoodispredictedtoraisedemand.Butthetheoryitselfdoesnotprovideanynumericalmeasureoftherelationshipbetweensupplyanddemand.Econometricsistheapplicationofstatistics(especiallyregressiontechniques)toeconomicorfinancialdata,inordertoquantifyrelationshipsexpressedineconomicorfinancetheory.Mathematicalstatisticsprovidesmanyofthetoolsusedinthetrade.Manyofthesehavebeenadaptedbyeconometricianssothattheymaybeappliedtoeconomicdata.Thischangeisessentialsincemosteconomicandfinancialdataarenotgeneratedastheresultofacontrolledexperiment•ExampleWhyaresomefirmssomuchmoreproductivethanothers?Alternatively,whydosomefirmsproducesomuchmoreoutputwiththesameinputscomparedtoothers?FirmProductivityintheUK•Whatarethereasonsbehindthesedifferences?–thecapital-labourratio(machineryperworker)–computerusage–unionisationoftheworkforce–abilityofmanagerswithinfirms.sumproductivity,detproductivity-------------------------------------------------------------PercentilesSmallest1%.9493689.00646015%1.772064.007169510%2.071567.0106065Obs36844525%2.77089.0106065SumofWgt.36844550%3.523866Mean3.751184LargestStd.Dev.1.63070375%4.61116525.155390%5.63986125.15882Variance2.65919395%6.26034225.17633Skewness3.45166599%7.36539925.22529Kurtosis38.871460246810PercentageofFirms0510152025FirmProductivity(OutputperWorker)FirmProductivityintheUK•Wewanttousedataoneachofthesevariablestoconstructamathematicalequationthatallowsustoquantifytherelationship.Wecanthenusethisequationtoanswerquestionssuchas…•Whatdoweexpectwouldhappentoproductivityifthecapitalstockofthefirmwasdoubled?or•Whatdoweexpectwouldhappentoproductivityifthefirmemployedbettermanagers?•Inmostoftheeconometricmodelstobediscussedinthiscourse,wewillassumethattherelationshipsarelinear;i.e.oftheform:yi=1+2x2i+3x3i+4x4i+uiwhere:yi=Productivityoffirmi(oversomespecifictimeinterval)x2i=Capital-labourratiooffirmix3i=ICTexpenditurebyfirmix4i=Abilityofmanagersinfirmi•1,2,3,4areunknownconstants,whosevaluesreflecttheinfluenceofeachexplanatoryvariableorindependentvariable(x2i,x3i,x4i)onthedependentvariable(yi).1,2,3,4areknownastheregressioncoefficientsorparameters.•uiisanerrortermordisturbanceterm,whichcapturestheeffectofanyotherdeterminantsofyithatarenotincludedinthemodel(omittedexplanatoryvariables/randominfluences/measurementerroretc.).Theerrortermisassumedtobearandom,orstochastic,variable.InterpretationofRegression•Broadly,regressionisconcernedwithhowacertainexplanatoryvariableaffectsthedependentvariable•Weshallfirstconsiderasimpletwo-variablemodel•Supposeeconomictheoryproposesthatmoreeducatedindividualsearnhigherwagesyi=1+2xi•Inthedata,thereareneverexactrelationshipsasintheory.Toallowforthisinexactrelationshiptheeconometricianmodifiesthedeterministicfunctiontoincludeui,thedisturbanceorerrorterm.yi=1+2xi+ui•Theerrortermrepresentsfactorsthataffectwagesbutthatarenottakenintoaccountexplicitly•Theregressionallowsustoinvestigatetheaverageimpactoftheexplanatoryvariable(education)onthedependentvariable(wages)•Despitevariabilityinwagesforagivenlevelofeducation,theaveragevalueofwagesgenerallyincreasesasthenumberofyearsofeducationincreases.•Theslopeoftheregressionlineisequalto2.Thepointatwhichtheregressionlinemeetsthey-axisis1.0510152025HourlyWage($)81012141618NumberofYearsofEducationwageFittedvaluesWagesandEducationThreeTypesofDataSet•Toestimaterelationshipssuchasthese,werequiredata.Therearethreetypesofdatasetthatmaybeused.•(a)Cross-sectional(asassumedintheexample).Dataonacollectionofeconomicunits(individuals,households,firmsetc.)observedatonemomentintime.yi=1+2xi+ui•(b)Timeseries.Asetofobservationsoneachvariableovertime.Timeperiodmaybeyearly,quarterly,monthly,weekly,dailyorevenhourly.Timeseriesdatasetsarecommonlyusedinmacroeconomicsandinfinance,e.g.yt=1+2x2t+3x3t+...+utwhere:yt=aggregateconsumerexpenditureinChinainyeartx2t=aggregatepersonaldisposableincomeinChinainyeartx3t=interestrateinChinainyeartandsoon.•(c)Panel.Datawithbothacross-sectionaldimensionandatimedimension.Dataonacollectionofeconomicunits(individuals,households,firmsetc.)isobservedatseveraldifferentpointsintime,e.g.yi,t=1+2x2i,t+3x3i,t+...+ui,twhere:yi,t=profitabilityofbankiinyeartx2i,t=assetsizeofbankiinyeartx3i,t=capitalratioofbankiinyeart