CHAPTER 2. = y ˆ β x (.1022) So we can write

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1 CHAPTER SOLUTIONS TO PROBLEMS. () Let y = GPA, x = ACT, ad = 8. The x = 5.875, y = 3.5, (x x )(y y ) = 5.85, ad (x x ) = From equato (.9), we obta the slope as ˆβ = = 5.85/ , rouded to four places after the decmal. From (.7), ˆ β = y ˆ β x 3.5 (.) So we ca wrte GPA = ACT = 8. The tercept does ot have a useful terpretato because ACT s ot close to zero for the populato of terest. If ACT s 5 pots hgher, GPA creases by.(5) =.5. () The ftted values ad resduals rouded to four decmal places are gve alog wth the observato umber ad GPA the followg table: = GPA GPA û You ca verfy that the resduals, as reported the table, sum to., whch s pretty close to zero gve the heret roudg error. () Whe ACT =, GPA= ().6. 3 Ths edto s teded for use outsde of the U.S. oly, wth cotet that may be dfferet from the U.S. Edto. Ths may ot be resold, coped, or dstrbuted wthout the pror coset of the publsher.

2 (v) The sum of squared resduals, = uˆ, s about.4347 (rouded to four decmal places), ad the total sum of squares, (y y ), s about.88. So the R-squared from the regresso s = R = SSR/SST (.4347/.88).577. Therefore, about 57.7% of the varato GPA s explaed by ACT ths small sample of studets..3 () Icome, age, ad famly backgroud (such as umber of sblgs) are just a few possbltes. It seems that each of these could be correlated wth years of educato. (Icome ad educato are probably postvely correlated; age ad educato may be egatvely correlated because wome more recet cohorts have, o average, more educato; ad umber of sblgs ad educato are probably egatvely correlated.) () Not f the factors we lsted part () are correlated wth educ. Because we would lke to hold these factors fxed, they are part of the error term. But f u s correlated wth educ the E(u educ), ad so SLR.4 fals..4 () We would wat to radomly assg the umber of hours the preparato course so that hours s depedet of other factors that affect performace o the SAT. The, we would collect formato o SAT score for each studet the expermet, yeldg a data set {( sat, hours ) : =,..., }, where s the umber of studets we ca afford to have the study. From equato (.7), we should try to get as much varato hours as s feasble. () Here are three factors: ate ablty, famly come, ad geeral health o the day of the exam. If we thk studets wth hgher atve tellgece thk they do ot eed to prepare for the SAT, the ablty ad hours wll be egatvely correlated. Famly come would probably be postvely correlated wth hours, because hgher come famles ca more easly afford preparato courses. Rulg out chroc health problems, health o the day of the exam should be roughly ucorrelated wth hours spet a preparato course. () If preparato courses are effectve, β should be postve: other factors equal, a crease hours should crease sat. (v) The tercept, β, has a useful terpretato ths example: because E(u) =, β s the average SAT score for studets the populato wth hours =. 4 Ths edto s teded for use outsde of the U.S. oly, wth cotet that may be dfferet from the U.S. Edto. Ths may ot be resold, coped, or dstrbuted wthout the pror coset of the publsher.

3 .5 () Whe we codto o c computg a expectato, c becomes a costat. So E(u c) = E( c e c) = c E(e c) = c because E(e c) = E(e) =. () Aga, whe we codto o c computg a varace, Var(u c) = Var( c e c) = ( c ) Var(e c) = σ e c because Var(e c) = c becomes a costat. So () Famles wth low comes do ot have much dscreto about spedg; typcally, a low-come famly must sped o food, clothg, housg, ad other ecesstes. Hgher come people have more dscreto, ad some mght choose more cosumpto whle others more savg. Ths dscreto suggests wder varablty savg amog hgher come famles..8 () We follow the ht, otg that cy = cy (the sample average of cy s c tmes the sample average of y ) ad cx = cx. Whe we regress c y o c x (cludg a tercept) we use equato (.9) to obta the slope: ( cx cx)( cy cy) cc( x x)( y y) β = = = = ( cx cx ) c( x x) = = ( x x)( y y) c = = c = ( x x) c = c ˆ β. σ e. From (.7), we obta the tercept as β = (c y ) β (c x ) = (c y ) [(c /c ) ˆ β ](c x ) = ˆ ˆ ˆ c ( y β x ) = c β ) because the tercept from regressg y o x s ( y β x ). () We use the same approach from part () alog wth the fact that ( c + y) = c + y ad ( c + x) = c + x. Therefore, ( c+ y ) ( c+ y) = (c + y ) (c + y ) = y y ad (c + x ) ( c + x) = x x. So c ad c etrely drop out of the slope formula for the regresso of (c + y ) o (c + x ), ad β = ˆβ. The tercept s β = ( c + y) β ( c + x) = (c + y ) ˆ β (c + x ) = ( y ˆ βx) + c c ˆβ = ˆ β + c c ˆβ, whch s what we wated to show. () We ca smply apply part () because log( cy ) = log( c) + log( y). I other words, replace c wth log(c ), y wth log(y ), ad set c =. (v) Aga, we ca apply part () wth c = ad replacg c wth log(c ) ad x wth log(x ). ˆ β ad ˆ β are the orgal tercept ad slope, the β ˆ = β ad β = ˆ β log( c ) ˆ β. If 5 Ths edto s teded for use outsde of the U.S. oly, wth cotet that may be dfferet from the U.S. Edto. Ths may ot be resold, coped, or dstrbuted wthout the pror coset of the publsher.

4 .9 () The tercept mples that whe c =, cos s predcted to be egatve $4.84. Ths, of course, caot be true, ad reflects that fact that ths cosumpto fucto mght be a poor predctor of cosumpto at very low-come levels. O the other had, o a aual bass, $4.84 s ot so far from zero. () Just plug 3, to the equato: cos = (3,) = 5,465.6 dollars. () The MPC ad the APC are show the followg graph. Eve though the tercept s egatve, the smallest APC the sample s postve. The graph starts at a aual come level of $, ( 97 dollars). MPC APC.9 MPC.853 APC c 6 Ths edto s teded for use outsde of the U.S. oly, wth cotet that may be dfferet from the U.S. Edto. Ths may ot be resold, coped, or dstrbuted wthout the pror coset of the publsher.

5 SOLUTIONS TO COMPUTER EXERCISES C. () The average prate s about ad the average mrate s about.73. () The estmated equato s prate = mrate =,534, R =.75. () The tercept mples that, eve f mrate =, the predcted partcpato rate s 83.5 percet. The coeffcet o mrate mples that a oe-dollar crease the match rate a farly large crease s estmated to crease prate by 5.86 percetage pots. Ths assumes, of course, that ths chage prate s possble (f, say, prate s already at 98, ths terpretato makes o sese). (v) If we plug mrate = 3.5 to the equato we get prate ˆ = (3.5) = Ths s mpossble, as we ca have at most a percet partcpato rate. Ths llustrates that, especally whe depedet varables are bouded, a smple regresso model ca gve strage predctos for extreme values of the depedet varable. (I the sample of,534 frms, oly 34 have mrate 3.5.) (v) mrate explas about 7.5% of the varato prate. Ths s ot much, ad suggests that may other factors fluece 4(k) pla partcpato rates. C.3 () The estmated equato s sleep = 3, totwrk = 76, R =.3. The tercept mples that the estmated amout of sleep per week for someoe who does ot work s 3,586.4 mutes, or about hours. Ths comes to about 8.5 hours per ght. () If someoe works two more hours per week the Δtotwrk = (because totwrk s measured mutes), ad so Δ sleep =.5() = 8. mutes. Ths s oly a few mutes a ght. If someoe were to work oe more hour o each of fve workg days, Δ sleep =.5(3) = 45.3 mutes, or about fve mutes a ght. C.5 () The costat elastcty model s a log-log model: log(rd) = β + β log(sales) + u, where β s the elastcty of rd wth respect to sales. () The estmated equato s 7 Ths edto s teded for use outsde of the U.S. oly, wth cotet that may be dfferet from the U.S. Edto. Ths may ot be resold, coped, or dstrbuted wthout the pror coset of the publsher.

6 log( rd ) = log(sales) = 3, R =.9. The estmated elastcty of rd wth respect to sales s.76, whch s just above oe. A oe percet crease sales s estmated to crease rd by about.8%. C.7 () The average gft s about 7.44 Dutch gulders. Out of 4,68 respodets,,56 dd ot gve a gft, or about 6 percet. () The average malgs per year s about.5. The mmum value s.5 (whch presumably meas that someoe has bee o the malg lst for at least four years) ad the maxmum value s 3.5. () The estmated equato s gft = malsyear = 4,68, R =.38 (v) The slope coeffcet from part () meas that each malg per year s assocated wth perhaps eve causes a estmated.65 addtoal gulders, o average. Therefore, f each malg costs oe gulder, the expected proft from each malg s estmated to be.65 gulders. Ths s oly the average, however. Some malgs geerate o cotrbutos, or a cotrbuto less tha the malg cost; other malgs geerated much more tha the malg cost. (v) Because the smallest malsyear the sample s.5, the smallest predcted value of gfts s. +.65(.5).67. Eve f we look at the overall populato, where some people have receved o malgs, the smallest predcted value s about two. So, wth ths estmated equato, we ever predct zero chartable gfts. 8 Ths edto s teded for use outsde of the U.S. oly, wth cotet that may be dfferet from the U.S. Edto. Ths may ot be resold, coped, or dstrbuted wthout the pror coset of the publsher.

ECON 482 / WH Hong The Simple Regression Model 1. Definition of the Simple Regression Model

ECON 482 / WH Hong The Simple Regression Model 1. Definition of the Simple Regression Model ECON 48 / WH Hog The Smple Regresso Model. Defto of the Smple Regresso Model Smple Regresso Model Expla varable y terms of varable x y = β + β x+ u y : depedet varable, explaed varable, respose varable,