伍德里奇计量经济学英文版第三版中关于回归模型交叉项引入的说明

回归模型中交叉项的说明伍德里奇计量经济学英文版第三版中关于回归模型交叉项引入的说明：其实就是因为一个变量固定的时候，其对因变量的影响还要与另外一个自变量有关。比如出行费用对出行方式选择行为的影响，还和人的收入水平有关系。以下为原文（pp207-209）ModelswithInteractionTermsSometimes,itisnaturalforthepartialeffect,lasticity,orsemi-elasticityofthedependentvariablewithrespecttoanexplanatoryvariabletodependonthemagnitudeofyetanotherexplanatoryvariable.Forexample,inthemodelPrice=β0+β1sqrft+β2bdrms+β3sqrft*bdrms+β4bthrms+μthepartialeffectofbdrmsonprice(holdingallothervariablesfixed)isΔprice/Δbdrms=β2+β3sqrft(6.17)Ifβ30,then(6.17)impliesthatanadditionalbedroomyieldsahigherincreaseinhousingpriceforlargerhouses.Inotherwords,thereisaninteractioneffectbetweensquarefootageandnumberofbedrooms.Insummarizingtheeffectofbdrmsonprice,wemustevaluate(6.17)atinterestingvaluesofsqrft,suchasthemeanvalue,orthelowerandupperquartilesinthesample.Whetherornotβ3iszeroissomethingwecaneasilytest.TheparametersontheoriginalvariablescanbetrickytointerpretwhenweincludeanInteractionterm.Forexample,intheprevioushousingpriceequation,equation(6.17)showsthatβ2istheeffectofbdrmsonpriceforapricewithzerosquarefeet!Thiseffectisclearlynotofmuchinterest.Instead,wemustbecarefultoputinterestingvaluesofsqrft,suchasthemeanormedianvaluesinthesample,intotheestimatedversionofequation(6.17).Often,itisusefultoreparameterizeamodelsothatthecoefficientsontheoriginalvariableshaveaninterestingmeaning.Consideramodelwithtwoexplanatoryvariablesandaninteraction:y=β0+β1x1+β2x2+β3x1*x2+μAsjustmentioned,isthepartialeffectofonwhen.Often,thisisnotofinterest.Instead,wecanreparameterizethemodelasy=η0+δ1x1+δ2x2+β3(x1-μ1)*(x2-μ2)+ξWhereμ1isthepopulationmeanofx1andμ2isthepopulationmeanofx2.Wecaneasilyseethatnowthecoefficientonx2δ2,isthepartialeffectofonatthemeanvalueofx2.(Bymultiplyingouttheinteractioninthesecondequationandcomparingthecoefficients,wecaneasilyshowthatδ2=β2+β3μ1.Theparameterδ1hasasimilarinterpretation.)Therefore,ifwesubtractthemeansofthevariables-inpractice,thesewouldtypicallybethesamplemeans—beforecreatingtheinteractionterm,thecoefficientsontheoriginalvariableshaveausefulinterpretation.Plus,weimmediatelyobtainstandarderrorsforthepartialeffectsatthemeanvalues.Nothingpreventsusfromreplacingμ1orμ2withothervaluesoftheexplanatoryvariablesthatmaybeofinterest.Thefollowingexampleillustrateshowwecanuseinteractionterms.