Outline. Zero Conditional mean. I. Motivation. 3. Multiple Regression Analysis: Estimation. Read Wooldridge (2013), Chapter 3.

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1 Outlne 3. Multple Regresson Analyss: Estmaton I. Motvaton II. Mechancs and Interpretaton of OLS Read Wooldrdge (013), Chapter 3. III. Expected Values of the OLS IV. Varances of the OLS V. The Gauss Markov Theorem I. Motvaton Zero Condtonal mean Example A drawback of SLR model: It may not be realstc to draw concluson that only x affects y when there are other factors that affect y. Example: y = wages and x = educ y = x wage = educ + exper E(u educ,exper) = 0 u are other factors (eg. nnate ablty, etc.) Some Good Ponts of Multple Regresson Analyss (MRA) 1) MRA can control many factors. ) More of the varaton n y can be explaned. 3) MRA can have farly general functonal form relatonshps. How s u related to educ and exper? Gven the values of educ and exper, the average value of nnate ablty s zero; E(u educ,exper) = 0 I. Motvaton 3 I. Motvaton 4

2 General Form: k explanatory varables. A Model wth k explanatory varables y = x + + k x k 0 s the ntercept 1,,, k are slope parameters What s u? Consder the model y = x 1 + x subscrpt : observaton number nd subscrpt: ndependent varable 1 or Zero Condtonal Mean Assumpton E(u, x k ) = 0 That mples that u and any of the ndependent varables, x k s uncorrelated. We can fnd the OLS regresson lne or SRF = + + x I. Motvaton 5 6 Interpretaton Interpretaton: Regresson Analyss Total dfferentate the equaton. Bvarate : wage = educ 1 the margnal effect of educaton on wage. = + x 1 + x Ceters parbus nterpretaton of 1^ = let x =0 When ncreases by one unt, y changes by unts, controllng for the varable x. Multple : wage = educ + exper 1 : the ceters parbus effect of educaton on average hourly wage. : the ceters parbus effect of experence on average hourly wage. What s ceters parbus? 7 8

3 General Model k regressors Interpretaton k regressors = + + x + + x k Consder the model wth k regressors y = x + + k x k Ceters parbus nterpretaton of Let x =0,... x k =0 We can fnd the OLS regresson lne or SRF = + + x + + x k = When ncreases by one unt, y changes by unts, holdng x, x 3,..., x k fxed Example: Regress wage on educ, exper, and tenure log(wage) = educ + exper + 3 tenure Sample: 1 56 Included observatons: 56 Consder the model: Varable Coeffcent Std. Error t-statstc Prob. C log(wage) = educ + exper + 3 tenure From estmaton: EDUC EXPER TENURE log() = educ exper + 0.0tenure (0.104) (.0073) (.0017) (.0031) n=56; R = Interpret =0.09. What f exper and tenure each ncreases by one year? R-squared Mean dependent var Adusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd Schwarz crteron Log lkelhood F-statstc Durbn-Watson stat Prob(F-statstc)

4 Method of OLS estmaton Method of OLS estmaton Consder the populaton model k regressors y = x 1 + x + + k x k Choose the estmates to mnmze the sum of squared resduals: subscrpt : observaton number or ndvdual =1,.,n nd subscrpt: regressor = 1,., k = (y x 1 x + + x k ) Sample counterpart for observaton y = + x 1 + x + + x k + We can fnd the OLS regresson lne or SRF = + x 1 + x + + x k Methods to fnd estmates (1) the method of moments, and () the method of OLS least squares There are k+1 equatons and k+1 unknowns Algebrac Propertes of OLS Statstcs Partallng Out Interpretaton = + + x + + x k If 1 > 0, then 1 underpredcts y 1 If < 0, then overpredcted y 1) The sample average of the resduals s zero (( ) = 0). ) The sample covarance between resduals and x s zero ( x = 0) 3) The sample averages of the dependent varable and regressors ( and ) are on the OLS regresson lne. : the effect on y of, holdng x,, x k constant the effect on y of after takng out the effect of x,, x k Defne : Can you fnd,...,? after takng out the effect of x,, x k rˆ 1 y ˆ 1 rˆ

5 Goodness-of-Ft R Squared Revsted We can thnk of each observaton as beng made up of an explaned part, and an unexplaned part, y yˆ uˆ Defnton R = SSE/SST R = 1 SSR/SST We then defne the followng: y s the total sum of squares (SST) yˆ y s the explaned sum of squares (SSE) y ˆ s the resdual sum of squares (SSR) u Then SST SSE SSR Interpret: It s the proporton of the varaton n y explaned by the explanatory varables (the OLS regresson lne). Words of Cauton Algebrac fact: R never decreases when any varable s added to a regresson. Thus, t s a poor tool for decdng whether a varable should be added to the model Goodness of Ft (contnued) Goodness-of-Ft s the dependent varable We can also thnk of R as beng equal to the squared correlaton coeffcent between the actual y and the values yˆ R y ˆ ˆ y y y y ˆ ˆ y y y We can thnk of each observaton as beng made up of an explaned part, and an unexplaned part, x x r We then defne the followng: r x x x x s the total sum of squares (SST ) s the explaned sum of squares (SSE ) s the resdual sum of squares (SSR ) 1 1 Then SST SSE SSR

6 Example: Regress wage on educ, exper, and tenure III. Expected Values of the OLS Estmators Consder the model: log(wage) = educ + exper + 3 tenure Goal: to show that OLS estmators are unbased for the populaton parameters under these four assumptons. (MLR) From estmaton: log() = educ exper + 0.0tenure (0.104) (.0073) (.0017) (.0031) n=56; R = Interpret R Assumptons MLR.1 lnear n parameters MLR. random samplng MLR.3 no perfect collnearty. MLR.4 zero condtonal mean 1 MLR.1 lnear n parameters MLR. Random Samplng y = x 1 + x + + k x k MLR. Random Samplng A random sample of n observatons {y, x 1,..., x k : =1,,...,n} Random samplng defnton If Y 1, Y,, Y n are ndependent random varables wth a common pdf f(y;, ), then {Y 1, Y,, Y n } s a random sample from the populaton represented by f(y;, ) =1 y 1 1 k = y x x We also say that Y are..d. (ndependent, dentcally dstrbuted) random varables from a dstrbuton. =n y n x n1 x n... X n1 3 4

7 MLR.3 No Perfect Collnearty There are no exact lnear relatonshp among the ndependent varables and ntercept. MLR.3 allows ndependent varables to be correlated. Example: relate hourly wage to experence wage = exper + expermt exper: years of experence expermt = 1exper (perfect collnearty) Example: the effect of campagn expendtures on campagn outcomes votea : percent of the vote for canddate A expenda: expendtures by canddate A expendb: expendtures by canddate B totalexpend = expenda + expendb Example: the effect of educaton expendtures and ncome on test score. avgscore: expen: ncome: test score gov t spendng per student famly ncome avgscore = expen + ncome Mcronumerosty MLR.3 also fals f the sample sze n s too small. 5 6 MLR.4: Zero Condtonal Mean E(u,x,,x k ) = 0 Example: Correct Specfcaton wage = educ 1 + exper y = x E(u,x ) = 0 and x are exogenous explanatory varables. Theorem 3.1: Under assumptons MLR.1 MLR.4, The OLS estmators are unbased estmators of the populaton parameters. E( ) = = 1,,k Incorrect Model y = v (omttng exper) v = x E(v, x ) 0 s the endogenous explanatory varable. for any value of the populaton parameter 7 8

8 Unbasedness A sngle estmate cannot be sad to be unbased. OLS estmates from all possble repeated samples are obtaned. Ths procedure leads to unbased OLS estmators. Irrelevant Varables and Unbasedness Suppose the correct model s y = x but we msspecfy the model as y = x + 3 x 3 There s no effect on the unbasedness of E( ) = 0 E( ) = 1 E( ) = E( ) = 0 Overspecfyng a model does not affect the unbasedness, but has an effect on varances Omtted Varable Bas Omtted Varable Bas Correct Specfcaton: log(wage) = educ + ablty y = x y: wage : educaton x : nnate ablty x x x x x u x x x x = + + x Due to lack of data on nnate ablty, we omt x. y = v v = x snce E( u ) 0, takng expectatons we have E x1 x1x 1 1 E x1 x1 = + E( ) = 1 + What s the sze of the bas due to omtted relevant varable? where s the slope coeffcent of the smple regresson of x on 31 3

9 s based Postve or negatve bas: E( ) 1 = + The omtted varable bas s > 0 :postve bas (when?) < 0 :negatve bas (when?) E( ) 1 = + > 0 f corr(,x ) > 0 < 0 f corr(,x ) < 0 E( ) > 1 : has an upward bas. E( ) < 1 : has a downward bas. Two cases where s unbased: Summary of Drecton of Bas 1) f = 0, or ) f = 0 Corr(, x ) > 0 Corr(, x ) < 0 > 0 Postve bas Negatve bas < 0 Negatve bas Postve bas Postve or negatve bas: E( ) 1 = + IV. The Varance of the OLS Estmators Example: Regress log(wage) on educ log( ) = educ n=56, R =.0186 What can you say about the OLS slope estmate? MLR.5 homoskedastcty assumpton VAR(u,,x k ) = The varance n the error term u s the same for all combnatons of outcomes of the explanatory varables

10 Example: wage = educ + exper + 3 tenure VAR(u educ, exper, tenure) = Gauss Markov assumptons: Assumptons MLR. 1 MLR. 5 are collectvely known as the Gauss Markov assumptons for cross secton regresson. If we have the same varance, homoskedastcty 1 = 16 If we have dfferent varances, heteroskedastcty 1 16 (volaton). Exercse: What are E(y x) and VAR(y x)? y = x + + k x k E(y x) = x + + k x k VAR(y x) = Ths mples that the varance of y does not depend on the combnaton of outcomes of the explanatory varables. Theorem 3.: Under Gauss Markov assumptons, = 1,,k ˆ var( ) SST (1 R ) SST = (x x bar) (SST s the total sample varaton n x ) R s the R squared from regresson of x on all other explanatory varables (ncludng an ntercept) Three components: ˆ var( ) SST (1 R ) Example: two regressors and R 1 1 st component : error varance hgh hgh VAR( ) Remedy : One way to reduce error s to add more explanatory varables to the equaton. y = x ˆ var( ) SST (1 R ) nd Component SST low SST hgh VAR( ) Remedy : One way to ncrease SST s to ncrease sample sze. 3rd Component R hgh R hgh VAR( ) Remedy : One way to mtgate the problem of multcollnearty s to ncrease sample sze. R 1 : s the R squared from the regresson of on x R 1 1 means that and x are hghly correlated and VAR( )..e., x explans much of the varaton n 39 40

11 R and Multcollnearty Remedy: Multcollnearty Multcollnearty mples hgh R. Multcollneary refers to hgh correlaton between two or more of the ndependent varables. Eg. R = 0.9 Ths means that 90% of the sample varaton n x s explaned by the remanng ndependent varables appearng n the equaton. We may try to drop ndependent varables to reduce multcollnearty. Cauton: But droppng a varable that belongs n the populaton model leads to bas. Multcollnearty Problem?? Not Really, but lkely: VAR( ^) also depends on and SST 41 4 Dtch or Keep a Varable Example: Loan approval rate Dtch or Keep depends on the queston we want to answer (see next example). Model: y = x + 3 x 3 y loan approval rate poverty rate x average ncome average housng value x 3 If housng value(x )and ncome (x 3 ) are hghly correlated, then Var( ) and Var( ) are large. Goal: want to know the effect of poverty rate on loan approval rate. If (poverty rate) s uncorrelated wth x and x 3 (.e., R 1 = 0), then the varance of s accurate

12 Varance n Msspecfed Model: Two Conclusons: Model I: = + + x Model II: = + Tradeoff between bas and varance ( 0) Usng unbasedness as a crteron, s preferred to Usng varance as a crteron, s preferred to (1) When = 0, and are unbased, and Var( ) <Var( ) (Model II s a abetter model) () When 0, Tradeoff: s bas, s unbased, but Var( ) < Var( ) by gnorng that the error varance ncreases n Model II. (Model I could be a better model) When 0, two good reasons why x should be ncluded? Estmatng error varance : = E(u ) = (1) Bas n does not shrnk as the sample sze grows. But VAR( ) 0 as n. So multcollnearty nduced by addng x becomes less mportant. () When x s excluded from the model, the error varance ncreases. It s the average of squared errors, whch could be an unbased estmator of. But we cannot observe the errors u. Theorem 3.3 Under the Gauss Markov Assumptons MLR.1 MLR.5 E( ) =, where uˆ 1 ˆ n k 1 n 47 48

13 The unbased estmator of n the general multple regresson case s s.d. and s.e. of 1 ˆ n n uˆ Degrees of freedom = n (k+1) = n k 1 k 1 Standard devaton (s.d) of depends on that cannot be observed Standard error (s.e) of depends on the estmated value, ^ s.e. ( ) hnges on the assumpton of constant varance (homoskedastcty) can ether decrease or ncrease when another regressor s added to a regresson (for a gven sample). The change depends on the role of SSR and degrees of freedom. Note that heteroskedastcty volates MLR.5 and s.e.( ) s nvald, but ths does not mply that s based IV. Effcency of OLS Theorem 3.4 (Gauss Markov Theorem) OLS estmators ( = 1,,k) are the best lnear unbased estmators (BLUE) of the populaton parameters ( = 1,,k) Summary When MLR.1 MLR.4 hold, OLS estmators are unbased. In the class of competng lnear unbased estmators, OLS s the best wthn ts class of lnear unbased estmators. IOWs, OLS estmator s BLUE. (1) OLS estmator s lnear. () OLS estmator s unbased. (3) OLS has mnmum varance. Wth Assumptons MLR.1 MLR.5, Gauss Markov Theorem suggests that no lnear unbased estmators wll be better than OLS estmators. IV. Effcency of OLS 51 IV. Effcency of OLS 5

14 Recap of Multple Regresson Analyss Motvaton Mechancs and Interpretaton of OLS Expected Values of the OLS Varances of the OLS The Gauss Markov Theorem 53