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.
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 = 95 -------------+------------------------------ F( 4, 9) = 95.3 Model 5.34699 4.3356748 Prob > F =. Resdual.6998 9.49 R-squared =.89 -------------+------------------------------ Adj R-squared =.85 Total 6.668 94.734766 Root MSE =.837 lsalary Coef. Std. Err. t P> t [95% Cof. Iterval] lcost -.7438.3643 -.9.846 -.788483.64767 lsat.78983.4339 4.3..94868.6397 rak -.3689.43-8.39. -.44635 -.7543 age.676.3653.73.466 -.458.9934 _cos 8.38384.73479.. 6.666 9.4757 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..493. 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.8994. d) Below are the results whe LSAT s removed from the model. Note that the correlato coeffcet betwee lsat ad rak s -.73. 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 = 95 -------------+------------------------------ F( 3, 9) = 58.78 Model 4.35484336 3.456445 Prob > F =. Resdual.47465 9.4694776 R-squared =.6596 -------------+------------------------------ Adj R-squared =.6484 Total 6.668 94.734766 Root MSE =.575 lsalary Coef. Std. Err. t P> t [95% Cof. Iterval]
lcost.847587.45737.85.67 -.687.75599 lsat.38855.45385 8.56..98399.47873 age.59.446 3.44..648.4 _cos 3.469744.633767 5.49..365 4.7588 5. 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 = 4 -------------+------------------------------ F( 4, 9) = 8.3 Model 945.7 4 3556.678 Prob > F =. Resdual 38643.86 9 83.39 R-squared =.339 -------------+------------------------------ Adj R-squared =.58 Total 4894.57 3 35648.646 Root MSE = 68.6 prce Coef. Std. Err. t P> t [95% Cof. Iterval] bedrooms 6.58 8.6.44.53-9.86649 6.9685 bathrooms 9.769 7.953 3.93. 54.3685 65.696 otherrooms 3.349 3.73668.33. 4.8949 59.657 age.3756.49649.66.5 -.6555788.3699 _cos -4.3946 7.7339 -.9.846-57.848 9.58 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 = 4 -------------+------------------------------ F( 5, 8) = 4.9 Model 644.53 5 3848.36 Prob > F =. Resdual 4453.5 8 444.9356 R-squared =.398 -------------+------------------------------ Adj R-squared =.374 Total 4894.57 3 35648.646 Root MSE = 49.8 prce Coef. Std. Err. t P> t [95% Cof. Iterval] bedrooms -.9485 8.39449 -.9.36-58.3759 4.546 bathrooms -.963856 3.737 -.3.976-64.7377 6.8 otherrooms -5.383 4.355 -.38.76-33.8.595 age -.375338.449888 -.3.76 -.99.754 sq_feet.7686.373365 5.43..876.76776 _cos 8.73887 66.58876..8-5.56.793 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
Source SS df MS Number of obs = 4 -------------+------------------------------ F(, ) = 35.53 Model 5768.9 78634.448 Prob > F =. Resdual 4565.68 6.3575 R-squared =.393 -------------+------------------------------ Adj R-squared =.3793 Total 4894.57 3 35648.646 Root MSE = 48.75 prce Coef. Std. Err. t P> t [95% Cof. Iterval] age -.359865.478868 -.56.573 -.6457.5984 sq_feet.796559.4987 8.36..37547.57 _cos 4.3538 46.3445.87.386-5.4696 3.4 5. 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
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