CONOMICS 5* -- Introducton to NOT CON 5* -- Introducton to NOT : Multple Lnear Regresson Models Interpretng Slope Coeffcents n Multple Lnear Regresson Models: An xample Consder the followng smple lnear regresson model for the brth weght of newborn babes gven by the followng populaton regresson equaton (PR): bwght β + β cgs + u () where the observable varables are defned as follows: bwght brth weght of newborn baby born to mother, n grams; cgs average number of cgarettes smoked per day durng pregnancy by mother. Interpretaton of slope coeffcent β on explanatory varable cgs n smple regresson model (): d ( bwght cgs ) d cgs d ( β + βcgs ) d cgs β unadjusted margnal effect of cgs on mean brth weght of newborn babes the change n mean brth weght of newborn babes, n grams, assocated wth an ncrease n mother s cgarette consumpton durng pregnancy of one cgarette per day CON 5* -- Introducton to Note : The Multple Regresson Model Page of 5 pages
CONOMICS 5* -- Introducton to NOT Consder the followng multple lnear regresson model for the brth weght of newborn babes gven by the followng populaton regresson equaton (PR): bwght β + β cgs + β famnc + β male + β whte + u () where the new explanatory varables are defned as follows: fa mnc annual famly ncome of mother, n thousands of 988 dollars per year; male f newborn baby of mother s male, otherwse; whte f mother s whte, otherwse. Interpretaton of slope coeffcent β on explanatory varable cgs n multple regresson model (): Let x be the 5 row vector of regressor values for observaton : [ cgs famnc male whte ] ( bwght x ) cgs ( β + β cgs + βfamnc + βmale cgs + β whte ) x. adjusted (partal) margnal effect of cgs on condtonal mean brth weght of newborn babes the change n condtonal mean brth weght of newborn babes, n grams, assocated wth an ncrease n mother s daly cgarette consumpton durng pregnancy of one cgarette, holdng constant the famly ncome and race of the mother and the sex of the newborn chld the change n condtonal mean brth weght of newborn babes, n grams, assocated wth an ncrease n mother s daly cgarette consumpton durng pregnancy of one cgarette, for newborns of the same sex whose mothers have the same famly ncome and are of the same race β CON 5* -- Introducton to Note : The Multple Regresson Model Page of 5 pages
CONOMICS 5* -- Introducton to NOT Compare the slope coeffcent β n Model and Model Model s the smple lnear regresson model gven by populaton regresson equaton (.) and the correspondng populaton regresson functon (.): bwght β + β cgs + u (.) ( bwght cgs ) β + βcgs (.) Model s the multple lnear regresson model gven by populaton regresson equaton (.) and the correspondng populaton regresson functon (.): bwght β + β cgs + β famnc + β male + β whte + u (.) ( bwght cgs, famnc, male, whte ) β + βcgs + βfamnc + βmale + βwhte (.) Queston: How does the slope coeffcent β n the smple lnear regresson model gven by equatons (.) and (.) dffer from the slope coeffcent β n the multple lnear regresson model gven by equatons (.) and.)? CON 5* -- Introducton to Note : The Multple Regresson Model Page of 5 pages
CONOMICS 5* -- Introducton to NOT Analytcal Answer The slope coeffcent β n the smple lnear regresson model gven by equatons (.) and (.) s the unadjusted or total margnal effect of cgarette consumpton on mean brth weght, because PR (.) and PRF (.) do not account for, or control for, the effects on brth weght of any other explanatory varables apart from cgs. Analytcally, ths means that the slope coeffcent β n the smple lnear regresson model (.)/ (.) corresponds to the total dervatve of mean brth weght wth respect to cgs : d ( bwght cgs ) d cgs d ( β + β cgs ) β n Model d cgs the unadjusted or total margnal effect of cgs on the mean brth weght of newborns the change n mean brth weght, n grams, assocated wth a -cgarette-per-day ncrease n daly cgarette consumpton of the mother durng pregnancy CON 5* -- Introducton to Note : The Multple Regresson Model Page of 5 pages
CONOMICS 5* -- Introducton to NOT In contrast, the slope coeffcent β n the multple lnear regresson model gven by equatons (.) and (.) s the adjusted or partal margnal effect of cgarette consumpton on mean brth weght, because PR (.) and PRF (.) account for, or control for, the effect on brth weght of other explanatory varables apart from cgs, namely famly ncome (famnc ), sex of the newborn (male ), and race of the mother (whte ). Analytcally, ths means that the slope coeffcent β n the multple lnear regresson model (.)/ (.) corresponds to the partal dervatve of mean brth weght wth respect to cgs : ( bwght cgs, famnc, male, whte, mpg ) cgs (β + β cgs + βfamnc + βmale cgs the adjusted or partal margnal effect of cgs on the mean brth weght of newborns + βwhte) β n Model the change n condtonal mean brth weght, n grams, assocated wth an ncrease of cgarette-perday n cgarette consumpton durng pregnancy, holdng constant famly ncome (famnc), the sex of the newborn (male), and the race of the mother (whte). the change n condtonal mean brth weght, n grams, assocated wth a -cgarette-per-day ncrease n cgarette consumpton durng pregnancy for mothers of the same famly ncome and race and newborns of the same sex. CON 5* -- Introducton to Note : The Multple Regresson Model Page 5 of 5 pages
CONOMICS 5* -- Introducton to NOT Interpretng the slope coeffcent estmate of cgs n Model Wrte the OLS sample regresson functon (SRF) for Model as bwg~ ht ~ ~ β + β cgs (.) ~ where β j denotes the OLS estmate of β j n Model, and bwg ~ ht s the OLS estmate of mean brth weght gven by the populaton regresson functon (PRF) bwght cgs β + β cgs for Model, the smple lnear regresson model gven by PR (.) and PRF (.). ( ) The OLS SRF (.) for Model mples that the estmated change n mean brth weght assocated wth a change n cgarette consumpton cgs of Δ cgs s ~ Δ bwg~ ht βδcgs (.) In Model, the estmated effect on mean brth weght of a -cgarette-per-day ncrease n a mother s cgarette consumpton durng pregnancy can be obtaned by settng Δcgs n equaton (.): Δ bwg~ ~ ht β when Δ cgs (.5) ~ The slope coeffcent estmate β n Model s therefore an estmate of the change n mean brth weght assocated wth a -cgarette-per-day ncrease n cgs, holdng constant no other explanatory varables that may be related to brth weght. CON 5* -- Introducton to Note : The Multple Regresson Model Page 6 of 5 pages
CONOMICS 5* -- Introducton to NOT Interpretng the slope coeffcent estmate of cgs n Model Wrte the OLS sample regresson functon (SRF) for Model, obtaned by OLS estmaton of the multple lnear regresson equaton (.), as bwĝht βˆ cgs famnc male whte (.) where ˆβ j denotes the OLS estmate of β j n Model, and bwĝht s the OLS estmate of mean brth weght gven by the populaton regresson functon (PRF) for Model, the multple lnear regresson model gven by PR (.) and PRF (.). ( bwght cgs, famnc, male, whte ) β + βcgs + βfamnc + βmale + βwhte The OLS SRF (.) mples that the estmated change n mean brth weght assocated wth a change n daly cgarette consumpton cgs of Δ cgs and smultaneous changes n famly ncome of Δ famnc, n sex of the newborn of Δ male, and n race of the mother of Δ whte s Δbwĝht βˆ Δcgs famnc ˆ male ˆ Δ + βδ + β Δ whte (.) We can hold constant famly ncome, the sex of the newborn, and the race of the mother by settng Δ famnc, Δ male and Δ whte n equaton (.); the resultng change n estmated mean brth weght s then Δ bwĝht βˆ Δcgs when Δ famnc, Δ male and Δ whte (.5) CON 5* -- Introducton to Note : The Multple Regresson Model Page 7 of 5 pages
CONOMICS 5* -- Introducton to NOT In Model, the estmated effect on mean brth weght of a -cgarette-per-day ncrease n mother s cgarette consumpton durng pregnancy can be obtaned by settng Δ cgs n equaton (.5), or equvalently by settng Δ cgs, Δ famnc, Δ male and Δ whte n equaton (.): Δ bwĝht β ˆ when Δ cgs and Δ famnc, Δ male, and Δ whte (.6) The slope coeffcent estmate ˆβ n Model s therefore an estmate of the change n mean brth weght assocated wth a -cgarette-per-day ncrease n cgs, holdng constant famly ncome (famnc), the sex of the newborn (male), and the race of the mother (whte). The followng Stata exercse s desgned to llustrate the correct nterpretaton of the slope coeffcent estmate n the multple lnear regresson model, Model. It also llustrates the meanng of holdng constant other varables n multple lnear regresson models. These exercses ntroduce you to an mportant post-estmaton Stata command, the lncom command. ˆβ CON 5* -- Introducton to Note : The Multple Regresson Model Page 8 of 5 pages
CONOMICS 5* -- Introducton to NOT OLS stmaton of Models and Usng Stata OLS estmaton of the smple lnear regresson model for the brth weght of newborn babes gven by populaton regresson equaton () yelds the followng OLS sample regresson equaton: ~ ~ bwght + (*) β + β ~ cgs u Stata output for Model of OLS estmaton command regress:. regress bwght cgs Source SS df MS Number of obs 88 -------------+------------------------------ F(, 86). Model 9695. 9695. Prob > F. Resdual 5 86 565.96 R-squared.7 -------------+------------------------------ Adj R-squared. Total 6887 87 969.67 Root MS 57.65 bwght Coef. Std. rr. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- cgs -.565.5658-5.68. -9.59796-9.598 _cons 95.5 6.586 9.7. 6.7 7.6 CON 5* -- Introducton to Note : The Multple Regresson Model Page 9 of 5 pages
CONOMICS 5* -- Introducton to NOT OLS estmaton of the multple lnear regresson model for the brth weght of newborn babes gven by populaton regresson equaton () yelds the followng OLS sample regresson equaton: bwght βˆ cgs famnc male whte + û (*) Stata output for Model of OLS estmaton command regress:. regress bwght cgs famnc male whte Source SS df MS Number of obs 88 -------------+------------------------------ F(, 8) 6.8 Model 55.7 568.9 Prob > F. Resdual 7598 8 8.779 R-squared.65 -------------+------------------------------ Adj R-squared.7 Total 6887 87 969.67 Root MS 56.9 bwght Coef. Std. rr. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- cgs -..57758-5.. -8.9868-8.8687 famnc.7555.8655.97.9.998.9579 male 89.6755.56.9. 9.6868 8.76 whte 5.959 8.7656.96. 77.659 9.58 _cons 77.5.899 77.68. 96.88 57.8 CON 5* -- Introducton to Note : The Multple Regresson Model Page of 5 pages
CONOMICS 5* -- Introducton to NOT Stata xercse: Model Queston: What s the effect on the mean brth weght of newborns of an ncrease n ther mothers cgarette consumpton durng pregnancy from to cgarettes per day (Δcgs ), whle holdng constant the mother s famly ncome at $, per year (famnc ), the sex of the newborn at male (male ), and the race of the mother at whte (whte )? Analytcal Answer: Compare the expressons mpled by Model for () the mean brth weght of newborns for whom cgs, famnc, male and whte and () the mean brth weght of newborns for whom cgs, famnc, male and whte. That s, compare and ( bwght cgs, famnc, male, whte ) β + β+ β + β + β (.) ( bwght cgs, famnc, male, whte ) β + β + β + β + β (.) Subtract the second functon from the frst functon: t s obvous that ths dfference s smply β : ( bwght cgs, famnc, male, whte ) ( bwght cgs, famnc, male, whte ) β + β + β + β + β β β β β ( ) β β β (.) CON 5* -- Introducton to Note : The Multple Regresson Model Page of 5 pages
CONOMICS 5* -- Introducton to NOT Step : Use a Stata lncom command to compute for Model the estmated mean brth weght of a male newborn (for whom male ) who s born to a whte mother (for whom whte ) who smoked cgarettes per day durng pregnancy (for whom cgs ) and whose famly ncome s $, per year (famnc ),.e., to compute an estmate of the condtonal mean functon ( bwght cgs, famnc, male, whte ) β + β + β + β + β. Our estmate of ths condtonal mean functon can be wrtten as Ê ( bwght cgs, famnc, male, whte ) βˆ ˆ ˆ ˆ ˆ + β + β + β + β nter the Stata lncom command:.. lncom _b[_cons] + _b[cgs]* + _b[famnc]* + _b[male]* + _b[whte]* ( ) cgs + famnc + male + whte + _cons bwght Coef. Std. rr. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- () 6.66.569 9.7. 76.59 95.8 CON 5* -- Introducton to Note : The Multple Regresson Model Page of 5 pages
CONOMICS 5* -- Introducton to NOT Step : Next, ncrease cgs by cgarette-per-day, from to, whle holdng constant famly ncome at a value of thousand dollars per year (famnc ), the sex of the newborn at male (male ), and the race of the mother at whte (whte ). Use another Stata lncom command to compute for Model the estmated mean brth weght of a male newborn (for whom male ) who s born to a whte mother (for whom whte ) who smoked cgarettes per day durng pregnancy (for whom cgs ) and whose famly ncome s $, per year (famnc ),.e., to compute an estmate of the condtonal mean functon ( bwght cgs, famnc, male, whte ) β + β+ β + β + β. Our estmate of ths condtonal mean functon can be wrtten as Ê ( bwght cgs, famnc, male, whte ) βˆ ˆ ˆ ˆ ˆ + β+ β + β + β nter the Stata lncom command:.. lncom _b[_cons] + _b[cgs]* + _b[famnc]* + _b[male]* + _b[whte]* ( ) cgs + famnc + male + whte + _cons bwght Coef. Std. rr. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- ().7.89.. 59.579 85.868 CON 5* -- Introducton to Note : The Multple Regresson Model Page of 5 pages
CONOMICS 5* -- Introducton to NOT Step : Fnally, use a Stata lncom command to compute the dfference between () the estmated mean brth weght of newborns for whom cgs, famnc, male and whte and () the estmated mean brth weght of newborns for whom cgs, famnc, male and whte,.e., to compute Ê ( bwght cgs, famnc, male, whte ) Ê( bwght cgs, famnc, male, whte ) βˆ βˆ βˆ ( βˆ ) βˆ βˆ βˆ βˆ β ˆ ˆβ ( ) nter on one lne the Stata lncom command:. lncom _b[_cons] + _b[cgs]* + _b[famnc]* + _b[male]* + _b[whte]* - (_b[_cons] + _b[cgs]* + _b[famnc]* + _b[male]* + _b[whte]*) ( ) cgs bwght Coef. Std. rr. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- () -..57758-5.. -8.9868-8.8687 CON 5* -- Introducton to Note : The Multple Regresson Model Page of 5 pages
CONOMICS 5* -- Introducton to NOT Result: Compare the output of the lncom command n Step wth the slope coeffcent estmate ˆβ for the regressor cgs produced by the regress command used to estmate Model by OLS. You wll see that they are dentcal.. lncom _b[_cons] + _b[cgs]* + _b[famnc]* + _b[male]* + _b[whte]* - (_b[_cons] + _b[cgs]* + _b[famnc]* + _b[male]* + _b[whte]*) ( ) cgs bwght Coef. Std. rr. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- () -..57758-5.. -8.9868-8.8687. lncom _b[cgs] ( ) cgs bwght Coef. Std. rr. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- () -..57758-5.. -8.9868-8.8687 Result: The slope coeffcent estmate ˆβ of cgs n Model s an estmate of the change n mean brth weght of newborns assocated wth an ncrease of cgarette per day n the cgarette consumpton of mothers durng pregnancy (Δcgs ), whle holdng constant the other determnants of newborns brth weght, namely famly ncome (famnc), sex of the newborn (male), and race of the mother (whte). CON 5* -- Introducton to Note : The Multple Regresson Model Page 5 of 5 pages