Reading 1: Quants Multiple

QuantsPage No. 107Dependent - ExplainedvariableReadingIMallipleIndependent ExplanatoryVariableBasic Ided - Linear relationship with more than one inde pendentDependentvariable/ExplainagriableIndependent variables ExplanatoryYi = bo + b1 X1i + bzXzi t+ bkXKi + E;Error terminterceptSlop coefficientVaciable Variation of the dependent variable by using twoormore independent variableH Can improve plediction , (use incorretly create spuriousrelationships that Can undermine predictionsProblem addressed by multiple linear regressionIIdentify relationship bet" variables2Forecast variables3Test existing theoriesAtime series is covariance stationary if it satisfies the followingthree conditionsNote:-Formulate a multiple linear regression model describe therelation bet" the dependent variable and several independentVariables & interpret est regression coefficientsw Page No 108Date//Interpretingp-values((df=n-k-1)AnothertotestHo(alwayswitht-test)wayagreesplul of significantIfP - valueathen reject HoExample:x=15%95%confidencelvl)4If(i.etheaCalculated p-value = 0.07 t we fail to reject NoP-valueCanbeviewedassmallestsignificantelvl (d) atwhich we Can reject Hot=CofficientExamplestd error of co offecetVariablebiSbi+ - -StateP-valuebo-11.6%1.657 %-7<0.0001PR0.250.0327.8<0.0001YCSX20.140.280-50.62bo&PRaresignificant(hight-stat,lowp-value)whileYESinsignificantClow+stat&highp-value)FormulateRegressionEquationSupposewethinkS&P500+10yrrealRarningsgrowth(EG10)is explained by ;Trailing divident payout ratio of index (PR)Yield curve slope ( 4(S)Ex:Formulateamultipleregressioneq"todescribethisrelationshipAdapted fromArnott&Asness(2003) Page No. 109Date//yEG10=b.+b,xPR+b2XYCS+E(610 =10-yr real earnings growth in s&p 500PR , trailing div payout ratio of indexYCS =yielddiffbet10-yrT-bond&3-month-billRegressEG10on independent variableYou will need n &- 46 annual observation Cn =46)K for several- Two independent "x" variables (K=2)formulacRegressionOutput : FreqstartingpointforExam QueVariablebiSbib.-11.671.657%PR0.250.032YCS0.140.28Interpreting Coefficients&X20Y=-11.61.(y)X1isEGIOwillequal-11-6%iftheindependentvariablearebothzero..All elseConstantIncrease in- 1%inPR=0.257increaseinEG10Xresultsincrease0.14inEG 10 increasemy- 1%increaseinYCS=increaseIf Not LinearAssumptions of Multiple RegressionrelationshipX(dependentTransformuse otherearLinearbet"Y&model uriableIndependent variable are uncorre lated with th4 thereisno exactlinearrelationskinbetntwoormoreindependentvaridaM Fited value - actual valuePage No. 110Every observation will have (esidualDate //No exact linear relationship amg y'sThese areExpected value of error term = 0 No biasNot this TrelatedVariace Residual of error term is constant (Not 0) for all observation violationsError not serially correlated / Error should be correlatedError term normally distributedResidual are homoscedastic*Normal Q-Q plotHomoskedasticityA normal Q-Q plot (usually called a O-Q plot) Compares avariable! distribution to a normal distribution-This help us to visualize the distributionThe Q-Q plot is useful for exploring whether the residualsare normally distributed.-This is a key assumption of linear regressionExample907. N + 1.65 std devIr this example t the regression95% N11.96model error term is close tobeing normally distributedof(r00Though a handful of outfier lieoutlierabove of below the theoretical largegelsIIAResidual plots allow analysts to get preliminary indication ofviolation of regression assumption