Sociology 470. Bivariate Regression. Extra Points. Regression. Liying Luo Job talk on Thursday 11/3 at Pond 302
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1 Socology 470 Bvarate Regresson Lyng Luo.0 Extra Ponts Job talk on Thursday /3 at Pond 302 Another job talk on Tuesday /8 at Pond 302 A thrd job talk on Thursday /0 at Pond 302 Regresson Regresson Analyss: the procedure for esfmafng and tesfng the relafonshp between confnuous varables. Bvarate Regresson (Smple Lnear Regresson): two confnuous varables MulFvarate Regresson: three or more confnuous varables
2 Bvarate Regresson FOed regresson model for sample ntercept Usng the regresson equafon we can...esfmate the average value of Y for a gven value of X (to a less extent)predct an ndvdual s value of Y for a gven value of X Bvarate Regresson FOed regresson model for sample ntercept Learnng objecfves Understand the basc dea of esfmafng regresson coeffcents Be able to wrte down foed regresson model and nterpret regresson coeffcents Be able to test regresson coeffcents Be able to compute and nterpret coeffcent of determnafon Estmatng Bvarate Regresson Coeffcent FOed regresson model for sample ntercept EsFmaFng bvarate regresson coeffcent: a ntercept:? Regressng Exam3 on Exam2 00 b :? Exam 3 Score Exam 2 Score
3 Estmatng Bvarate Regresson Coeffcent EsFmaFng bvarate regresson coeffcent: The lne we draw... the values of ntercept a and b we choose... maxmzes our ablty to predct the value of Y and thus mnmzes the predc.on errors. 2 N ( ) = MathemaFcally, we choose the lne for whch Y Ŷ = That s, the sum of the squared verfcal dstance s smallest. N = 2 e s smallest. Ths least squares error sum crteron produces ordnary least squares (OLS) esfmates of a and b Games Exam 2 Score Won Regressng Exam 2 Scores on Exam Scores Exam Score Estmatng Bvarate Regresson Coeffcent EsFmaFng bvarate regresson coeffcent: b = Covarance Varance X XY = s YX 2 sx N = = N ( Y Y)( X X) = 2 ( X X) a large b vs a small b SXY SXY S 2 X S 2 Y S 2 X S 2 Y Estmatng Bvarate Regresson Coeffcent For the baseball example, the predcfon equafon s Ŷ =
4 Interpretng Bvarate Regresson Coeffcent FOed regresson model for sample ntercept InterpreFng bvarate regresson coeffcent: a ntercept: the foed value of Y when X=0 b : the amount of change n Y for every one unt change n X Interpretng Bvarate Regresson Coeffcent Ŷ = How do we nterpret ths regresson equaton? It says for every one unt ncrease n X (runs) we should observe a 0. unt ncrease n Y (wns) It also lterally says f a were to zero runs n a season such X=0 we should observe the would wn game (more on ths later) Interpretng Bvarate Regresson Coeffcent Example : An economst s nterested n the relafonshp between annual salary (Y) and heght n nches (X). He regressed annual salary on heght and found the esfmated ntercept a=30,000 and b=350. Wrte down the foed regresson model and nterpret the ntercept and the.
5 Interpretng Bvarate Regresson Coeffcent Example 2: A crmnologst s nterested n the relafonshp between number of homcde (Y) and medan household ncome (X) n neghborhoods. She regressed the number of homcde on medan household ncome. She found the esfmated ntercept a=0. and b=-0.5. Wrte down the foed regresson model and nterpret the ntercept and the. Estmatng a Regresson EquatonEstmatng a Regresson Equaton Estmatng a Regresson Equaton For the baseball example, the predcton equaton s: For Ŷ the = baseball example, the predcton equaton s: Ŷ = + 0.X For the baseball example, the predcton equaton s: Ŷ = Games Won Calculatng Expected Runs Values Scored Ŷ = Interpretng the Regresson Equaton Interpretng the Regresson Equaton Interpretng the Regresson Equaton Ŷ Usng = the regresson equaton we can Ŷ = How Ŷ = do estmate we nterpret + 0.X the ths average regresson value of equaton? Y for a gven value of X How do we nterpret ths regresson equaton? It says predct for every an ndvdual s one unt ncrease value of n Y X for (runs) a gven we should value of X How do we nterpret ths regresson equaton? It says for every one unt ncrease n X (runs) we should observe a 0. unt ncrease n Y (wns) It says for every one unt ncrease n X (runs) we should observe a 0. unt ncrease n Y (wns) observe a 0. unt ncrease n Y (wns) It also lterally says f a were to zero runs n a It also lterally says f a were to zero runs n a season such X=0 we should observe the would It It also also lterally lterally says says f f a were were to to zero zero runs runs n n a season such X=0 we should observe the would wn game (more on ths later) season season such such X=0 X=0 we we should should observe observe the the would would wn game (more on ths later) wn wn game game (more (more on on ths ths later) later) Usng the regresson equaton we can Usng the regresson equaton we can Usng estmate the regresson the average equaton value of Y we for can a gven value of X estmate the average value of Y for a gven value of X predct estmate estmatean the the ndvdual s average average value value of of of Y for for for a gven gven value value of of of X predct an ndvdual s value of Y for a gven value of X predct an ndvdual s value of Y for a gven value of X predct an ndvdual s value of for gven value of Calculatng Expected Values Interpretng the Regresson Equaton Interpretng the Interpretng the Regresson Equaton Regresson Equaton The The equaton equaton Ŷ = + 0.X 0.X means means (n (n Englsh) Englsh) The equaton Ŷ = means (n Englsh) The equaton Ŷ = The Expected Expected equaton Number Number of of Wns Wns 0.X = means (n Englsh) means + 0.Runs 0.Runs (n Englsh) Expected Number of Wns = + 0.Runs Expected Number of Wns = + 0.Runs Expected Number of Wns 0.Runs How many wns would we predct a to wn f they How many wns would we predct a to wn f they d How many 80 wns runs? would we predct a to wn f they d 80 runs? How many wns would we predct to wn f they d 80 runs? d d Expected 80 runs? 80 runs? Number of Wns 0.(80) Expected Number of Wns = + 0.(80) = Expected Number of Wns = + 0.(80) = Expected Number of Wns 0.(80) What s the average number of wns among s What s the average number of wns among s What What 750 s the s the runs? average number of wns among s average number of wns among s 750 runs? Expected 750 runs? 750 runs? Number of Wns 0.(750) Expected Number of Wns = + 0.(750) = Expected Number of Wns = + 0.(750) = Expected Number of Wns 0.(750) Socology 38 ~ 3/3/205 Socology 38 ~ 3/3/205 7 Socology 38 3/3/205
6 Worksheet A fnancal analyst would lke to know the relafonshp between the mutual fund fees (X, n %) and ts annual yeld (Y, n %). She regressed annual yeld on fees and found n her sample a=4 and b=-0... Wrte down the foed regresson model. 2. Interpret the esfmated ntercept and. 3. What s the expected return for a mutual fund chargng % fee? 5% fee?
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