Reading 3: Model Misspecification

Page No. 121Date//Kading Model MisspeaficationTypes of Model MisspecificationInconsistent & biased regression purameterImportant Variable omitted Omission of imp independent variableBased on economy theory, one or move variablethatshouldhaveincluddedare omittedEffectBiased and in Coninsistent regression parameterMay leadtoserialcorrelation or heteroscdesticityin the residedVariable formVariable faim inappropriateVariables not transformed property Inapprovide transformationThe relationshipbet the dependent and independentvariables maybe non-linearEffectMayleadtohete rosced asticity in the residualVariablesnotScaledproperty inappropriatevariablescalingVariablesmayneedtobetran sformed beforeesttheregressionsMay lead to hede cossed asticityintheresidualorEffectmu licollinearitySamples has periods of dissimilar eco environments (thatData pooled impropely Data in Time seriesshould not be pooled)EffectMay bead to hete ro scedas Ficity or serial correlationin the residualfrisizoman Page No. 122Date/ExamplePredict monthy chinese folio stock (R) returns july '96 tojune '02 withfolio beta (B)Y,Natural log of price to book (In PB) X3Natural log of margket Cap (1nM) X2Free float (FF)R = b. t b,B t b, InM + bb In PB t byFF + EIMisspecification : O mitting a variableLeave out I.M (if correlated with B FF)2Misspecification : Inappropriate variable fromUse market Cup instead of InM3Misspecification : Inappropriate scaling of the variableUsing the no. of free float shares (Vs. proportion)4% free floatMisspecification : Incorrectly pooling dataWhat if '96 to 199 is diff than '00 to '02 ?Assumptions of Multiple RegressionViolation of theree assumption is investigatedFor each Know :- What is it ?- How do we detect it ?- What is its ellect on the regression output?- How do we correct for it? memorizing Cross SectionalorTime seriesPage No. 123HomeConstantDate//HeteroskedasticityVariance/fluctuatingvolatilityAriseswhenerrortermvarianceisnon - constantTwotypesofTypeIUuncoditionalheteroskedasticity.heteroskedasticityRegression Coefficients- Not relatedtovariableswill be unbiased butindependent- Point : CausesStd Error biasednomajorproblems@Type2:ConditionalheteroskedesticityFor financial data, most- Relatedtoindependentvariableslikely the standard- ThisISproblemerrors understated- Impact : t-stat(&F-start)areunreliableand t-statsConsitent / in biasedinflationtbi ==EstType IbiNot affectedeffur56Std errorUnreliable-Coefficientsstillconsistent&unbiasedYLowresidualsb,X;variance0Type IType IITw slow forToo high forHigh residual+ve serial-ve serialvarianieCorrelationCorrelationxor SerialAlso affectsErtestcorrelationlowerisgood Page No.125Date/If BP2Detecting Hete roske dasticity2n°RejjectM.4xConductudePconstitional heteroskScatter diagrams : Plot residual againsteachindependentvariableand against timeOne tailed2111°Breusch-PagantestRegressresiduals"X:squaredonVariables(BP)Point : TestsignificantofresultingR2(dotheindependentvariablesexplain a significant part of thevariation insquaredresiduals ?)Ho:Nothe no. of observationno. of indepent variableheteroskedasticityChi - -Square test ; BP =nResid (with k dfK dofDesimal2Presid:-R2 from a Second regression (of the of Squared residuals fromtheThefirstregresion)ontheindependentvariablesThis is aone tailedtest 7 7 because hetero skedosticity is onlyaproblem ifthe RRImpact :- Type I& the BP . Test stat are too largeerror5x12CorrectUse Robust SE wasExample=60Youhave been studing five yr of monthy returnsforaparticularmutualfundhopingtodrawconclusionaboutthefundsavgperformanceYoucalculatethemean return , the standard deviation, & thefund'sbetaby&EOregressionsthefund'sreturnsonS&P500Indexreturn (the independent variable)xKEP Hansen Method , / adj for problemsassociated ,leteroskedasticity with serial Correlation&Page No. 125Date / /The standard deviation of returns and the funds betadon't seems to fit the firm's Stated risk profileYou have a scatter plot of regression error terms (actualreturn - predicted return) for five yr of returnResidualIndependentvariableDetermine whether the residual plot indicates a data problemAns :- The residual plot indicates the presence of conditionalheteroske dasticityNotice how the variation in the regression residuals increasesas the independent variable increasesThisindicates that the variance of the fund's returnsabout the mean is related to the Jul of the independentVariableTo confirm your suspicious , you regress the squard residualsfrom the original regression on the independent variable5&p 500 Index return . The R2 from that regression is 8%Use the Breu sch - Pagan test to determine whetherheteroske dasticity is present at the 5% significant lvl.x2 = 60 X0.08= 4.80 Page No. 126Date/SolutionWithfiveyrThetestofmonthlyobservationsnisequaltotostatisnxR2=60X008= 4.84.8 > 3.841 f reject the nullCritical chi-squale one-tail, + a equal do 5,hypothesis and conclude thatIfCritical onlyhave3.84youaproblem withIConditionalhete roskedasticity25.99137.815If bet" upper &44.488lower value thenS11.07no sersal Correlation612.5928outsite the rangeSerial Correlationthen problem ofError terms arecorreletedwithserial correlationeach otherDurbintionofresidualerror+VeWatsonVeImpactTypeIstat +ve & -veTypeIIDuruin waison- only1lay.22x(1-r)ConductureCorrectRobust SEDoes not specify tve/iveS cital correlationBreusch - Godfrey (BG) testSerial Correlet- multiple lags - F-testindependent variableF>Fglagged*TheBreusch - God/rey (BG) Testegression cofficientare invalid2 degree of freedoslagsBGstatisticCFdistributionCoefficient std errorP&n-p-k-1dofdefectedsoftwarepackagesNallcorrelatibymostnoserialU-statinvalidTheBGtestusesresidualsfromtheorginthedependent variable. Page No. 127Date//These residualsarefunagainstinitialregressionpluslaggedresidualsHe: Etcoeficientsoftheleggedresidualsarezero== bo + 9, X1t +artiz7dk XktP1 Et-L+PzE6-CorrectionofSerialcorrelationPreferred metords-Use robust Stanard error- aka Newey - Westcorrectionstanarderrorsforserialcorrelationaka White - correctedstandarderrorsforConditionalheteroskedastity*Multicollinearity about indipendent variable (x)Unreliable est ofregressionpurametersHighlyDefine Twoormore"X"variablesarecorrelatedwitheachotherEffects:-.InflatesSEsreducest-stats;increaseschanceofType11errorsPoint : t- statartificiallysmallsovariablesfalsolylookunimportantMakes b unreliableR2Detecting MulticollinearityFtest siyNo of slope diseffientsaresignTell talesigns fromregressiondata:observation1 :HighR2,significantstat&insignificantt-statsObservation2:Highcorrelationbet"X"variables(forK=Waseonly)rCorrectionImpactType IIData im properlyPooled"IIOmitoneormoreoftheXvariables2missionofimp variable Page No. 128Date / /Determine if there is multicollinearity in this regression :VariablesbiP- valueF-statClowp-value)Avg P/B X,3.520.15with Significant insignificiant individual variables CoupleAug PIE X22.780.21>0.05Mkt Cap X34.03(high p-value) indicates multiallierary0.11is present.F-test34.6<0.001 < 5%005The /variance inflation factor (VIF)ismulticollincarity :No multicollineasity when Rj is 0110 a measure for quantifyIf VIF; is1forXj then there is noVIF; = 1Correlation bet" Xj & the other regressors1- R.21-01-VIF; >5 warrants further investigation-VIF; >10 indicates serious multicollinearity leg correction0.9Rj is 0.90.11Solutions to multicollinearity :- Dropping one or more of the regression variables- - Using a diff proxy for one of the variablesincreasing the sample size.Describe influence analysis and methods of detectinginfluential data pointsHigh - levelage points ( are the extreme observations of theindependent (or x') variablesHigh Loverage points , :- would be identified using a measureCalled leverage (Lij) which can bell provided to by Stat puckages Page No. 129Date / /Leverage : Dist bet" variable and Sample mean, scaledtobe bet" 0 & 1ledThehigher the value of leverage the greater the distance-and hence the higher the potential influence of theobservation - on the est regression parameters, The Sum of the individual leverage for all observationsis k t 1.Ifavariable leverage is higher than three times theavg [3 CK+1) / n], it is considered potentially influential.Outlier are extreme observations of the dependent (or 5'4') variableWe Can identify outliers using the studentized residuals TheSteps below outlience the procedure ;- Est the regression model using the original Sample of size n- Delete one observation and re-est the regression using (n+) observ- Perform this sequentially for all observations, deleting oneobservation at a time- Compare the actual y value of the deleted observation i tothe predicted Y-values using the model palameters est withthat observation deleted*E.= = Yi - YiThe studentized residual is the residual in step 2 divided byits standard deviationti = eist Page No. 130Date//We can then Compare this studentized residual to critial valuesfrom a t- distribution with n-k-2 degrees of freedom todetermine if the observation is influential.Detecting Influential data pointslook's distance (Di) is a Composite metric (i.e it takes into account boththe leverage and outliers) for evaluating if a specificobservation is influentialDi = e2hiix MSE(1-hayei = residual for ith observationK = no- of independent variablesMSE = mean square error of the regression modelhis = leverage value for ith observation Page No. 32Assumption ViolationsVVIMPDate / /HeteroPSCNSCMulticollinosistyWhat doesVariance of efforEsror terms areError term aicOne or more indi (it meanof term is Not constantCorrelated withcrelated withvar lineady Coffeeeach othereach otherwith each other+ve- VeHow toBreach Pagan -testDustin watsonDustinF-stat sign Non of thedetectChi bq testtestwatson testindividual slope 6alc sign + laig (R2)Type I errorType IType IIType IIIImpactt-stat "too large'FTRSE "too low"T-stat "too smallUnable to lejetSE "tou taigeHow toRobust SEHarsen SEHanse SEDrop one of thefix itwhite correted SECircleled variudHansen SE