ε. Therefore, the estimate

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Edmund Richardson
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1 Suggested Aswers, Problem Set 3 ECON 333 Da Hugerma. Ths s ot a very good dea. We kow from the secod FOC problem b) that ( ) SSE / = y x x = ( ) Whch ca be reduced to read y x x = ε x = ( ) The OLS model chooses ad such that x s by costructo ucorrelated wth ε. Therefore, the estmate for γ wll be by costructo ad t does ot form us at all about whether x ad ε are correlated.. The three st order codtos are: () = ( y x x ) x = () = ( y x x ) x = (3) = ( y x x ) = Equato (3) ca be reduced to read ( y x x ) =. Dvdg by ad solvg for we fd that = y x x ad because we have assumed that y = x = x = the =. Equato () ca be re-wrtte to read y x x x x x =. Sce = ad x x = ths reduces to y x x same procedure, you ca also demostrate that = ad therefore y x = = 8 / 4 =. x y x = = 6 / = 3. x Usg the 3. True, by addg more varables, o matter how rrelevat the varables are, the R ca ever fall. Ths s because f the R was R a wth oly varable, the worst that would ever happe by addg more varables s that the computer would set the estmated coeffcets for the ew varables to zero ad obta ad R of R a.

2 4. A sample program that geerates results for ths questo s uder the Chapter 3 headg the STATA porto of the class web page. The program s called aps3_q3.do. Source SS df MS Number of obs = F( 4, 9) = 95.3 Model Prob > F =. Resdual R-squared = Adj R-squared =.85 Total Root MSE =.837 lsalary Coef. Std. Err. t P> t [95% Cof. Iterval] lcost lsat rak age _cos a) A % crease s cost s estmated to reduce salares by.7 percet. b) A oe ut crease rak (movg from 5 th to 6 th for example) s estmated to reduce salares by.36 percet. c) Below s the matrx of correlato coeffcets. Just lke s predcted by the frst order codtos, the covarace betwee the estmated resduals ad the x s s by costructo equato to zero res lsat lcost res. lsat.. lcost d) The correlato coeffcet betwee actual ad predcted y s.8994 ad ths umber squared s.98 whch s exactly the R the model lsalary pred lsalary. pred d) Below are the results whe LSAT s removed from the model. Note that the correlato coeffcet betwee lsat ad rak s We kow that l(salares) are egatvely related to rak ad egatvely correlated wth the lsat so takg rak ot of the model would put more weght o the lsat varable the regresso ad crease ts value, whch s exactly what happes. Notce that the coeffcet o lsat doubles whe school rak s elmated from the model. * ru model deletg lsat from basc model. reg lsalary lcost lsat age Source SS df MS Number of obs = F( 3, 9) = Model Prob > F =. Resdual R-squared = Adj R-squared =.6484 Total Root MSE =.575 lsalary Coef. Std. Err. t P> t [95% Cof. Iterval]

3 lcost lsat age _cos A sample program that geerates results for ths questo s uder the Chapter 3 headg the STATA porto of the class web page. The program s called aps3_q4.do. Model : Source SS df MS Number of obs = F( 4, 9) = 8.3 Model Prob > F =. Resdual R-squared = Adj R-squared =.58 Total Root MSE = 68.6 prce Coef. Std. Err. t P> t [95% Cof. Iterval] bedrooms bathrooms otherrooms age _cos a) Remember, house prces are measured thousads of dollars. Each addtoal bedroom crease house prces by $6,. Every year crease age crease house prces by $3. b) Notce that whe sq_feet s added to the model, the coeffcets o bedrooms, bathrooms ad otherrooms decle so much that the sgs are all ow egatve. Ths makes sese because sq_feet s postvely correlated wth these three varables so addg t to the model should decrease the coeffcets o these three varables. Model Source SS df MS Number of obs = F( 5, 8) = 4.9 Model Prob > F =. Resdual R-squared = Adj R-squared =.374 Total Root MSE = 49.8 prce Coef. Std. Err. t P> t [95% Cof. Iterval] bedrooms bathrooms otherrooms age sq_feet _cos Model 3 c) Notce that the R for model 3 s.393 whle the R for model s.398, ot much of a chage. I ths sample, oce oe cotrols for sq_feet, addg formato about the umber of rooms does ot add much explaatory power to the model. 3

4 Source SS df MS Number of obs = F(, ) = Model Prob > F =. Resdual R-squared = Adj R-squared =.3793 Total Root MSE = prce Coef. Std. Err. t P> t [95% Cof. Iterval] age sq_feet _cos a) If x s a lear combato of x where x =a+bx, the a regresso of x o x would geerate a R of ad the deomator the varace calculato would be zero, makg the varace udefed. We caot estmate models where covarates are lear combatos of each other. b) If a regresso of x o x produces a R lke.999, the the deomator approaches zero ad the varace would explode. Whe two hghly correlated varables are added to a model, t s dffcult to dscer aythg precse about the exact mpact of x o y because t s hard to separate the exact effect of x from that of x. 7. a) Sce x s radomly assged the we expect t to be ucorrelated wth all of the possble covarates. As a result, addg these ew varables to the model s ot expected to chage the estmate o. b) a smple bvarate model, the varace o would be σ V( ) = ε. I the multvarate model where σ V( ) = ε ( R ) possble covarates, the, sce we expect that x wll be ucorrelated wth all of the R should be pretty close to zero ad the varace the multvarate case should look a lot lke the varace the smple bvarate regresso model, or σ V( ) = ε. However, recall that σ ε = SSE / ( k ) ad addg covarates to the model should reduce the SSE ad therefore, reduce the estmated varace o. I Radom Assgmet Clcal Trals, we typcally add covarates because they reduce the objectve fucto (SSE) whch drectly reduces estmated varaces. 8. I a bvarate regresso model, we kow that Var( ) = regresso model, we kow that Var( ) = ( R ) 4 σ ε σ ε where whereas a multvarate R s the R from a regresso of

5 x o x. Note that top of the results o page 8, we see the correlato coeffcet betwee x o x s.9994 whch meas that R should be very close to. Therefore, by addg x to the model, a varable hghly correlated wth x, the umerator Var( ) model () blows up because R approaches zero. 5

ln( weekly earn) age age

ln( weekly earn) age age Problem Set 4, ECON 3033 (Due at the start of class, Wedesday, February 4, 04) (Questos marked wth a * are old test questos) Bll Evas Sprg 08. Cosder a multvarate regresso model of the form y 0 x x. Wrte