Chapter Two: Ordinary Least Squares

Transcription

1 4 Studenmund Usng Econometrcs, Sxth Edton Chapter Two: Ordnary Least Squares -3. (a) 71. (b) 84. (c) 13, yes. (d) 155, yes -4. (a) The squares are least n the sense that they are beng mnmzed. (b) If R =, then RSS = TSS, and ESS =. If R s calculated as ESS/TSS, then t cannot be negatve. If R s calculated as 1 RSS/TSS, however, then t can be negatve f RSS > TSS, whch can happen f Y ˆ s a worse predctor of Y than Y (possble only wth a non-ols estmator or f the constant term s omtted). (c) Postve. (d) We prefer Model T because t has estmated sgns that meet expectatons and also because t ncludes an mportant varable (assumng that nterest rates are nomnal) that Model A omts. A hgher R does not automatcally mean that an equaton s preferred. -5. (a) Yes. The new coeffcent represents the mpact of HEIGHT on WEIGHT, holdng MAIL constant, whle the orgnal coeffcent dd not hold MAIL constant. We d expect the estmated coeffcent to change (even f only slghtly) because of ths new constrant. (b) One weakness of R s that addng a varable wll usually decrease (and wll never ncrease) the summed squared resduals no matter how nonsenscal the varable s. As a result, addng a nonsenscal varable wll usually ncrease (and wll never decrease) R. (c) R s adjusted for degrees of freedom and R sn t, so t s completely possble that the two measures could move n opposte drectons when a varable s added to an equaton. (d) The coeffcent s ndeed equal to zero n theory, but n any gven sample the observed values for MAIL may provde some mnor explanatory power beyond that provded by HEIGHT. As a result, t s typcal to get a nonzero estmated coeffcent even for the most nonsenscal of varables. -6. (a) Postve; both gong to class and dong problem sets should mprove a student s grade. (b) Yes. (c) >..6, so gong to class pays off more. (d) <.1.6, so dong problem sets pays off more. Snce the unts of varables can dffer dramatcally, coeffcent sze does not measure mportance. (If all varables are measured dentcally n a properly specfed equaton, then the sze of the coeffcent s ndeed one measure of mportance.) (e) An R of.33 means that a thrd of the varaton of student grades around ther mean can be explaned by attendance at lectures and the completon of problem sets. Ths mght seem low to many begnnng econometrcans, but n fact t s ether about rght or perhaps even a bt hgher than we mght have expected. (f) The most lkely varable to add to ths equaton s the th student s GPA or some other measure of student ablty. We d expect both R and R to rse.

2 Answers to Text Exercses 5-7. (a) Even though the ft n Equaton A s better, most researchers would prefer Equaton B because the sgns of the estmated coeffcents are as would be expected. In addton, X 4 s a theoretcally sound varable for a campus track, whle X 3 seems poorly specfed because an especally hot or cold day would dscourage ftness runners. (b) The coeffcent of an ndependent varable tells us the mpact of a one-unt ncrease n that varable on the dependent varable holdng constant the other explanatory varables n the equaton. If we change the other varables n the equaton, we re holdng dfferent varables constant, and so the ˆβ has a dfferent meanng. -8. (a) Yes. (b) At frst glance, perhaps, but see below. (c) Three dssertatons, snce (978 3) = $934 > ( ) = $48 > ($46 1) = $46 (d) The coeffcent of D seems to be too hgh; perhaps t s absorbng the mpact of an ndependent varable that has been omtted from the regresson. For example, students may choose a dssertaton advser on the bass of reputaton, a varable not n the equaton. -9. As we ll learn n Chapters 6 and 7, there s a lot more to specfyng an equaton than maxmzng R. -1. (a) V : postve. H : negatve (although some would argue that n a world of perfect nformaton, drvers would take fewer rsks f they knew the state had few hosptals). C : ambguous because a hgh rate of drvng ctatons could ndcate rsky drvng (rasng fataltes) or zealous polce ctaton polces (reducng rsky drvng and therefore fataltes). (b) No, because the coeffcent dfferences are small and the data wll dffer from year to year. We d be more concerned f the coeffcents dffered by orders of magntude or changed sgn. (c) Snce the equaton for the second year has smlar degrees of freedom and a much lower R, no calculaton s needed to know that the equaton for the frst year has a hgher R. Just to be sure, we calculated R and obtaned.65 for the frst year and.565 for the second year (a) It mght seem that the hgher the percentage body fat, the hgher the weght, holdng constant heght, but muscle weghs more than fat, so t s possble that a lean, hghly muscled man could wegh more than a less well-condtoned man of the same heght. (b) We prefer Equaton 1.4 because we don t thnk F belongs n the equaton on theoretcal grounds. The meanng of the coeffcent of X changes n that F now s held constant. (c) The fact that R drops when the percentage body fat s ntroduced to the equaton strengthens our preference for Equaton 1.4. (d) Ths s subtle, but snce.8 tmes 1. equals 3.36, we have reason to beleve that the mpact of bodyfat on weght (holdng constant heght) s very small ndeed. That s, movng from average bodyfat to no bodyfat would lower your weght by only 3.36 pounds.

3 6 Studenmund Usng Econometrcs, Sxth Edton -1. (a) Σ(e )/ ˆ β = Σ(Y ˆ ˆ β β1x )( 1) Σ(e )/ ˆ β = Σ(Y ˆ β ˆ β X )( ˆ β X ) = Σ(Y ˆ β ˆ β X ) (b) 1 = ˆ β Σ(Y ˆ β ˆ β X )(X ) or, rearrangng: 1 1 Σ Y nβˆ + ˆ β ΣX = 1 ΣYX ˆ β Σ X + ˆ β X = 1 These are the normal equatons. (c) To get ˆ β 1, solve the frst normal equaton for ˆ β, obtanng ˆ β = ( Σ Y X )/N β1σ and substtute ths value n for ˆβ where t appears n the second normal equaton, obtanng Σ Y X = ( ΣY ˆ ˆ β1σx )( Σ X )/N + β1x, whch becomes ˆ β = (NΣY X ΣY X )/[NΣX ( Σ X ) ]. Wth some algebrac manpulaton 1 (n part usng the fact that Σ X = NX), ths smplfes to Equaton.4. (d) To get Equaton.5, solve the frst normal equaton for ˆ β, usng X =Σ X /N (a) Yes. We d expect bgger colleges to get more applcants, and we d expect colleges that used the common applcaton to attract more applcants. It mght seem at frst that the rank of a college ought to have a postve coeffcent, but the varable s defned as 1 = best, so we d expect a negatve coeffcent for RANK. (b) The meanng of the coeffcent of SIZE s that for every ncrease of one n the sze of the student body, we d expect a college to generate.15 more applcatons, holdng RANK and COMMONAP constant. The meanng of the coeffcent of RANK s that every one-rank mprovement n a college s U.S. News rankng should generate 3.1 more applcatons, holdng SIZE and COMMONAP constant. These results do not allow us to conclude that a college s rankng s 15 tmes more mportant than the sze of that college because the unts of the varables SIZE and RANK are qute dfferent n magntude. On a more phlosophcal level, t s rsky to draw any general conclusons at all from one regresson estmated on a sample of 49 colleges. (c) The meanng of the coeffcent of COMMONAP s that a college that swtches to usng the common applcaton can expect to generate 1 more applcatons, holdng constant RANK and SIZE. However, ths result does not prove that a gven college would ncrease applcatons by 1 by swtchng to the common applcaton. Why not? Frst, we don t trust ths result because there may well be an omtted relevant varable (or two) and because all but three of the colleges n the sample use the common applcaton. Second, n general, econometrc results are evdence that can be used to support an argument, but n and of themselves they don t come close to provng anythng. (e) If you drop COMMONAP from the equaton, R falls from.681. Ths s evdence (but not proof) thaommonap belongs n the equaton.

4 8 Studenmund Usng Econometrcs, Sxth Edton 3-9. (a) Negatve; postve; none. (b) Holdng all other ncluded explanatory varables constant, a car wth an automatc transmsson gets.76 mles less per gallon than a model wth a manual transmsson, and a car wth a desel engne gets 3.8 mles more per gallon than one wthout a desel engne. (c) Lovell added the EPA varable because he wanted to test the accuracy of EPA estmates. If these estmates were perfectly accurate, then the EPA varable would explan all the varaton n mles per gallon (a) All postve except for the coeffcent of F, whch n today s male-domnated move ndustry probably has a negatve expected sgn. The sgn of ˆβ certanly s unexpected. (b) Fred, because $5, < ($4,, $3,7,). (c) Yes, snce 15.4 = $3,8, > $1,,. (d) Yes, snce $1,77, > $1,,. (e) Yes, the unexpected sgn of the coeffcent of ˆ βb (a) The best way to handle three dscrete condtons s to specfy two dummy varables. For example, one dummy varable could = 1 f the Pod s new (and otherwse) and the other dummy varable could = 1 f the Pod s used but unblemshed (and otherwse). The omtted condton, that the Pod s used and scratched, would be represented by both dummy varables equalng zero. (b) Postve; negatve; postve. (c) In theory, the narrower the tme spread of the observatons, the better the sample, but 3 weeks probably s a short enough tme perod to ensure that the observatons are from the same populaton. If the 3 weeks ncluded a major shock to the Pod market, however, then the frend would be rght, and the sample should be splt nto before the shock and after the shock subsamples. (d) Yes, they match wth the answer to part b. (e) (f) R s mssng! R s.431. Chapter Four: The Classcal Model 4-. (a) An addtonal pound of fertlzer per acre wll cause corn yeld (bushels/acre) to ncrease by.1 bushel/acre, holdng ranfall constant. An addtonal nch of ran wll ncrease corn yeld (bushels/acre) by 5.33 bushels/acre holdng fertlzer/acre constant. (b) No. (Ths s a typcal student mstake.) Frst, snce t s hard to magne zero nches of ran fallng n an entre year, ths ntercept has no real-world meanng. In addton, recall that the OLS estmate of the ntercept ncludes the nonzero mean of the error term n order to valdate Classcal Assumpton II (as explaned n the text), so even f ranfall were zero, t wouldn t make sense to attempt to draw nferences from the estmate of the β term unless t was known that the mean of the (unobservable) error term was zero. (c) No; ths could be an unbased estmate..1 s the estmated coeffcent for ths sample, but the mean of the coeffcents obtaned for the populaton could stll equal the true β. F (d) Not necessarly; 5.33 could stll be close to or even equal to the true value. An estmated coeffcent produced by an estmator that s not BLUE could stll be accurate. If the estmator s based, ts bas could be small and ts varance smaller stll. B

5 Answers to Text Exercses Par c clearly volates Assumpton VI, and par a probably volates t for most samples (a) Most experenced econometrcans would prefer an unbased nonmnmum varance estmate. (b) Yes; an unbased estmate wth an extremely large varance has a hgh probablty of beng far from the true value. In such a case, a slghtly based estmate wth a very small varance would be better. (c) The most frequently used possblty s to mnmze the mean square error (MSE), whch s the sum of the expected varance plus the square of any expected bas (a) Classcal Assumpton II. (b) Classcal Assumpton VI. (c) R: A one-unt ncrease n yesterday s R wll result n a.1% ncrease n today s Dow Jones average, holdng constant the other ndependent varables n the equaton. M: The Dow Jones wll fall by.17% on Mondays, holdng constant the other ndependent varables n the equaton. (d) Techncally, C s not a dummy varable because t can take on three dfferent values. Saunders assumed (at least mplctly) that all levels of cloud cover between % and % have the same mpact on the Dow and also that all levels of cloud cover between 1% and 99% have the same mpact on the Dow. In addton, by usng the same varable to represent both sunny and cloudy days, he constraned the coeffcent of sun and cloud to be equal. (e) In our opnon, ths partcular equaton does lttle to support Saunders concluson. The poor ft and the constraned specfcaton combne to outwegh the sgnfcant coeffcents of R t 1 and M (a) The estmated coeffcent of C shows that (for ths sample) a one percent ncrease n the nonwhte labor force n the th cty adds. percentage ponts to the overall labor force partcpaton rate n that cty, holdng constant all the other ndependent varables n the equaton. The estmated coeffcent of the dummy varable, D, shows that f a cty s n the South, the labor force partcpaton rate wll be.8 percentage ponts lower than n other ctes, holdng constant the other explanatory varables n the equaton. (b) Perfect collnearty s vrtually mpossble n a cross-secton lke ths one because no varable s a perfect lnear functon of another; some are closely related, but none s a perfect lnear functon. (c) Ths does not mply that one of the estmates s based. The estmates were taken from two dfferent samples and are qute lkely to dffer. In addton, the true value may have changed between decades. (d) Dsagree. Begnners often confuse the constant term wth the mean of the dependent varable. Whle the estmated constant term shows the value of the dependent varable n the unlkely case that all of the explanatory varables equal zero, t also ncludes the mean of the observatons of the error term as mentoned n Queston 4- (b).

6 1 Studenmund Usng Econometrcs, Sxth Edton 4-8. We know that ˆ ˆ ˆ Σ e = Σ(Y Y ) = Σ(Y β β1x ). To fnd the mnmum, dfferentate Σ e wth respect to ˆβ and ˆβ 1 and set each dervatve equal to zero (these are the normal equatons ): δ ( Σ e ) / δβˆ = [ Σ(Y ˆ β ˆ β X )] = 1 Y N( ˆ ) ˆ ( X ) or Σ = β + β1 Σ δ ( Σ e ) / δβˆ = [ Σ(Y ˆ β ˆ β X )X ] = 1 1 or Σ YX = ˆ β (X ) + ˆ β ( Σ X ) 1 Solve the two equatons smultaneously and rearrange: where x = (X X) and y = (Y Y). To prove lnearty: ˆ β = [N( ΣY X ) ΣY X ]/N( ΣX ) ( Σ X ) ] 1 ˆ β = [ ΣX ΣY ΣX ΣX Y ]/[N( ΣX ) ( Σ X ) ] = Y ˆ β X 1 ˆ β =Σx y / Σ x =Σx (Y Y)/ ΣX 1 =Σx Y / Σx Σx (Y)/ ΣX =Σx (Y )/ Σx YΣx / ΣX =Σx (Y )/ Σx snceσ x = =Σ k Y where k = x / Σx 1 ˆβ s a lnear functon of Y, snce ths s how a lnear functon s defned. It s also a lnear functon of the β s and, whch s the basc nterpretaton of lnearty. ˆ β1 = ˆ βσ k1 + β1kx + Σk 1. ˆ β ˆ = Y β1(x) where Y = ˆ β ˆ + β1(x), whch s also a lnear equaton. To prove unbasedness: ˆ β =Σ k Y =Σ k ( β + β X + ) =Σ k β +Σ k β X +Σk 1 Snce k = x / Σ X = (X X)/ ΣX X), then Σ k =, Σ X = 1/ ΣX, Σ k x = Σ k X = 1 So, ˆ β k, = β1 + Σ 1 and gven the assumptons of, unbased. 1 ˆ E( β 1) = β1 + Σ k1e( 1) = β1, provng 1 ˆβ s

7 Answers to Text Exercses 11 To prove mnmum varance (of all lnear unbased estmators): ˆ β 1 =Σ ky. Snce k = x / Σ x = (X X)/ Σ(X X)/ Σ(X X), ˆβ 1 s a weghted average of the Ys, and the k are the weghts. To wrte an expresson for any lnear estmator, substtute w for k, whch are also weghts but not necessarly equal to k : β =Σ w Y,soE( β ) =Σ x E(Y ) =Σ w ( β + β X ) * * = β Σ w + β Σw X * * In order for β 1 to be unbased, Σ w1 = and Σ wx = 1. The varance of β : 1 VAR( β ) = VAR Σ w Y = Σ w VAR Y = σ Σw * 1 [VAR(Y ) = VAR( ) = σ ] = σ Σ(w x / ΣX x / ΣX ) = σ Σ(w x / Σ X ) + σ Σx( Σx ) + σ Σ(w x / ΣX )(x / ΣX ) = σ Σ(w x / Σ X ) + σ /( ΣX ) * The last term n ths equaton s a constant, so the varance of β 1 can be mnmzed only by manpulatng the frst term. The frst term s mnmzed only by lettng w = x / Σ X, then: VAR( β ) = σ / Σ X = VAR( β ) * * 1 1 When the least-squares weghts, k, equal w, the varance of the lnear estmator β 1 s equal to the varance of the least-squares estmator, ˆ β 1. When they are not equal, ˆ * VAR( β ) > VAR( ˆ β ) Q.E.D (a) Ths possbly could volate Assumpton III, but t s lkely that the frm s so small that no smultanety s nvolved. We ll cover smultaneous equatons n Chapter 14. (b) Holdng constant the other ndependent varables, the store wll sell more frozen yogurts per fortnght f t places an ad. If we gnore long-run effects, ths means that the owner should place the ad as long as the cost of the ad s less than the ncrease n profts brought about by sellng more frozen yogurts. (c) The result doesn t dsprove the owner s expectaton. School s not n sesson durng the prme yogurt-eatng summer months, so the varable mght be pckng up the summer tme ncreased demand for frozen yogurt from nonstudents. (d) Answers wll vary wldly, so perhaps t s best just to make sure that all suggested varables are tme-seres for -week perods. For students who have read Chapters 1 4 only, the best answer would be any varable that measures the exstence of, prces of, or advertsng of local competton. Students who have read Chapter 6 mght reasonably be expected to try to fnd a varable whose expected omtted-varable bas on the coeffcent of C s negatve. Examples nclude the number of rany days n the perod or the number of college students returnng home for vacaton n the perod.

8 1 Studenmund Usng Econometrcs, Sxth Edton 4-1. (a) Yes; Yes. In partcular, there s no measure of prces n the equaton. (b) Yes. (c) Yes; very unlkely. (d) No. (e) No. (f) No. (g) The nghtclub should hre a dancer, because the estmated coeffcent s hgher (a) The coeffcent of DIVSEP mples that a dvorced or separated ndvdual wll drnk.85 more drnks than otherwse, holdng constant the other ndependent varables n the equaton. The coeffcent of UNEMP mples that an unemployed ndvdual wll drnk 14. more drnks than otherwse, holdng constant the other ndependent varables n the equaton. The sgns of the estmated coeffcents make sense, but we wouldn t have expected the coeffcent of UNEMP to be fve tmes the sze of the coeffcent of DIVSEP. (b) The coeffcent of ADVICE mples that an ndvdual wll drnk more drnks, holdng constant the other ndependent varables n the equaton, f a physcan advses them to cut back on drnkng alcohol. Ths coeffcent certanly has an unexpected sgn! Our guess s that DRINKS and ADVICE are smultaneously determned, snce a physcan s more lkely to advse an ndvdual to cut back on hs or her drnkng f that ndvdual s drnkng qute a bt. As a result, ths equaton almost surely volates Classcal Assumpton III. (c) We d expect each sample to produce dfferent estmates of β ADVICE. Ths entre group s called a samplng dstrbuton of β-hats. (d) The estmated β ADVICE for ths subsample s 8.6, whch s a lttle lower than the coeffcent for the entre sample. The other coeffcents for ths subsample dffer even more from the coeffcents for the entre sample, and the estmated coeffcent of EDUC actually has an unexpected sgn. These results are clear evdence of the advantages of large samples. Chapter Fve: Hypothess Testng 5-3. (a) H : β1, H : β1 > A (b) β1 A β1 β A β H :, H : < ; H :, H : > ; H : β3, H A : β3 > (The hypothess for β 3 assumes that t s never too hot to go joggng.) (c) β1 A β1 β A β H :, H : > ; H :, H : > ; H : β3,h A : β3 < ; (The hypothess for β 3 assumes you re not breakng the speed lmt.) (d) βg A βg H : = ; H : (G for grunt.) 5-5. For β : Reject N H : β f 4.4 > tc and 4.4 s negatve. For β : Reject H : β f 4.88 > tc and 4.88 s postve. For β I : Reject H : β f.37 > tc and.37 s postve. (a) = 1.943; reject the null hypothess for all three coeffcents. (b) = 1.311; reject H for all three coeffcents. (c) = 6.965; cannot reject the null hypothess for any of the three coeffcents.

9 Answers to Text Exercses (a) t = ( 16)/5. = 1.6; =.5; therefore we cannot reject H. (Notce the volaton of the prncple that the null contans that whch we do not expect.) (b) t 3 =.37; =.756; therefore we cannot reject the null hypothess. (c) t = 5.6; =.447; therefore we reject H f t s formulated as n the exercse, but ths poses a problem because the orgnal hypotheszed sgn of the coeffcent was negatve. Thus the alternatve hypothess ought to have been stated: H A : β <, and H cannot be rejected Ths s a concern for part (a) but not for parts (b) and (c). In part (a), 16 probably s the coeffcent we expect; after all, f our expectaton was somethng else, why dd we specfy 16 n the null? In parts (b) and (c), however, t seems unlkely that we d expect zero (a) For both H : β and H A : β >. For M, we can reject the null hypothess because t M = 5. and 5. > 1.761, the 5% one-sded crtcal t-value for 14 degrees of freedom, and because 5. s postve. For Y, we cannot reject the null hypothess because t Y = 1.5 and 1.5 < (b) Here, H : β A = and H A : βa. We cannot reject the null hypothess because t A =.8 and.8 <.861, the 1% two-sded crtcal t-value for 19 degrees of freedom. (c) We thnk that B should have a negatve effect on mssed payments whle C seems lkely to have a postve effect (thought some students wll argue that havng more chldren ndcates that the father lkes chldren and wll therefore mss fewer payments). Thus, for B, H : βb and H A : β B <. We cannot reject ths null hypothess because t B = 1. and 1. < 1.363, 1.363, the crtcal 1% one-sded t-value for 11 degrees of freedom. For C, H : βc. Even though = 3. and 3. > 1.363, we cannot reject the null hypothess for C because s negatve, not postve (a) For all three, H : β, H A: β >, and the crtcal 5% one-sded t-value for 4 degrees of freedom s For LOT, we can reject H because + 7. > and +7. s postve. For BED, we cannot reject H because + 1. < even though +1. s postve. For BEACH, we can reject H because + 1. > and +1. s postve. (b) H : β, H A: β <, and the crtcal 1% one-sded t-value for 4 degrees of freedom s 1.318, so we reject H because. > and. s negatve. (c) H : β =, H A: β, and the crtcal 5% two-sded t-value for 4 degrees of freedom s.64, so we cannot reject H because 1. <.64. Note that we don t check the sgn because the test s two-sded and both sgns are n the alternatve hypothess. (d) The man problems are that the coeffcents of BED and FIRE are nsgnfcantly dfferent from zero. (e) Gven that we weren t sure what sgn to expect for the coeffcent of FIRE, the nsgnfcant coeffcent for BED s the most worrsome. (f) Unless the students have read Chapter 6, ths wll be a dffcult queston for them to answer. It s possble that the dataset s unrepresentatve, or that there s an omtted varable causng bas n the estmated coeffcent of BED. Havng sad that, the most lkely answer s that BED s an rrelevant varable f LOT also s n the equaton. Beach houses on large lots tend to have more bedrooms than beach houses on small lots, so BED mght be rrelevant f LOT s ncluded.

10 14 Studenmund Usng Econometrcs, Sxth Edton 5-1. (a) For both, H : β and H : β >. For WIN, we cannot reject H A, even though the sgn agrees wth the sgn mpled by H A, because + 1. < 1.697, the 5 percent one-sded crtcal t-value for 3 degrees of freedom. For FREE, we can reject H at the 5 percent level of sgnfcance because. > and because. has the sgn mpled by H A. (b) H : βweek and H A : β WEEK <. We can reject H at the 1 percent level because 4. >.457, the 1 percent, one-sded crtcal t-value for 3 degrees of freedom and because 4. has the sgn mpled by H A. (c) H : β DAY = and H A : βday. We cannot reject H because 1. <.4, the 5 percent two-sded crtcal t-value for 3 degrees of freedom. (d) The coeffcents of DAY and WIN are nsgnfcantly dfferent from zero. In addton, t s hard to rule out the possblty that a varable that belongs n the equaton mght have been omtted. (e) A potental omtted varable s more worrsome than an nsgnfcant coeffcent. (f) We d suggest addng a varable that measures the weather (lke nches of ranfall that day) to the equaton. Even gven San Dego s wonderful weather, there s a good chance that rany or cold weather could cut down on attendance at an outdoor event (a) For the t-tests: Coeffcent: βp βm Hypotheszed sgn: + + t-value: = reject reject do not reject (5% one-sded reject wth 6 d.f., as close to 73 as Table B 1 goes) βs (b) No. We stll agree wth the authors orgnal expectatons despte the contrary result. (c) Keynes pont s well taken; emprcal results wll ndeed allow an econometrcan to dscover a theoretcal mstake now and then. Unfortunately, far too many begnnng researchers use ths loophole to change expectatons to get rght sgns wthout enough thnkng or analyss. (d) Holdng all other ncluded explanatory varables constant, an ncrease n wnnng percentage of 15 ponts wll ncrease revenues by $7,965, ($53.1 tmes 15 tmes 1) and thus t would be proftable for ths team to hre a $4,, free agent who can rase ts wnnng percentage to 5 from (a) NEW: H : β, H A : β >. Reject H snce 4. > and has the sgn of H A. SCRATCH: H : β, H A : β <. Reject H snce 4. > and 4. has the sgn of H A. (b) BIDRS: H : β, H A : β >. Cannot reject H snce 1.3 <.358 even though +1.3 has the sgn of H A. (c) Some experenced econometrcans mght drop BIDRS from the equaton because of ts low t-score, but we d be nclned to keep the varable. The theory s strong, and the estmated coeffcent s n the expected drecton. As we ll see n Chapter 6, consstently droppng varables wth low t-scores wll result n coeffcent bas. (d) Most suggestons wll be attrbutes of the Pod, but attrbutes of the aucton of that Pod (lke the length of tme of the aucton or whether there was a buy t now opton avalable) also make sense. βt

11 Answers to Text Exercses (a) DIVSEP: H : β, H A : β >.. Cannot reject H snce 1.11 < even though has the sgn of H A. UNEMP: H : β, H A : β >. Reject H snce.75 > and +.75 has the sgn of H A. (b) EDUC: H : β =, H A : β. Cannot reject H snce.65 <.617. (c) ADVICE: H : β, H A : β <. Cannot reject H snce doesn t have the sgn of H A, even though 5.37 < (d) No. We d stll expect ADVICE to have a negatve mpact on DRINKS n ths structural equaton. The problem s that the two varables almost surely are smultaneously determned, snce a physcan would be more lkely to advse a patent to drnk less f that patent was drnkng qute a bt. Ths smultanety volates Classcal Assumpton III. We ll learn how to estmate smultaneous equatons n Chapter (a) All fve tests are one-sded, so = 1.76 throughout. GDPN: H : β, H A : β >. Reject H because > 1.76 and 6.69 s postve as n H A. CVN: H : β, H A : β <. Reject H because.66 > 1.76 and.66 s negatve as n H A. PP: H : β,h A : β >. Do not reject H because < DPC: H : β, H A : β <. Reject H because.5 > 1.76 and.5 s negatve as n H A. IPC: H : β, H A : β <. Do not reject H because 1.59 < * (b) Our confdence nterval equaton s ˆ β ± t ˆ C SE( β ), and the 1% two-sded t = 1.76 C (the same as a one-sded 5% ), so the confdence nterval equals ˆ β ± 1.76 SE( ˆ β ), or: GDPN: 1.7 < ˆ β < 1.79 CVN: PP: DPC: IPC:.98 < ˆ β < < ˆ β < < ˆ β < < ˆ β <.83 (c) Yes. The mportant sgns were as expected and statstcally sgnfcant, and the overall ft was good. (d) The szes of the coeffcents would change, but not ther sgns or sgnfcance.

12 16 Studenmund Usng Econometrcs, Sxth Edton Chapter Sx: Specfcaton: Choosng the Independent Varables 6-3. (a) Coeffcent: βc βe β M Hypotheszed sgn: t-value: = reject reject do not (1% one-sded reject wth 7 d.f.) The problem wth the coeffcent of M s that t s sgnfcant n the unexpected drecton, one ndcator of a possble omtted varable. (b) The coeffcent of M s unexpectedly negatve, so we re lookng for a varable the omsson of whch would cause negatve bas n the estmate of β. M We thus need a varable that s negatvely correlated wth meat consumpton wth a postve expected coeffcent or a varable that s postvely correlated wth meat consumpton wth a negatve expected coeffcent. For the sx varables lsted, the expected bas s: Possble Omtted Varable Expected Sgn of β Correlaton wth M B + + * + F W + * + + R + H + O + * Indcates a weak expected sgn or correlaton. Drecton of Bas (c) The best suggested varables are annual aggregate varables, the omsson of whch would cause negatve bas. The expected bas equaton s dffcult to work wth the frst tme around, so some students surely wll suggest tme-seres varables the omsson of whch would cause postve expected bas. Wth luck, no students wll suggest a dsaggregate varable (a) Coeffcent: βe βi Hypotheszed sgn: + Calculated t-score: = 1.68, so: sg. nsg. nsg. sg. sg. but unexp. sgn (b) Both ncome and tax rate are potental rrelevant varables not only because of the szes of the t-scores but also because of theory. The sgnfcant unexpected sgn for β R s a clear ndcaton that there s a potental omtted varable. (c) It s prudent to attempt to solve an omtted varable problem before worryng about rrelevant varables because of the bas that omtted varables cause. (d) The equaton appears to show that televson advertsng s effectve and rado advertsng sn t, but you shouldn t jump to ths concluson. Improvng the specfcaton could change ths result. In partcular, although t s possble that rado advertsng has lttle mpact on smokng, t s very hard to beleve that a rado antsmokng campagn could cause a sgnfcant ncrease n cgarette consumpton! βt βv βr

13 Answers to Text Exercses 17 (e) Theory: Gven the farly prce-nelastc demand for cgarettes, t s possble that T s rrelevant. t-score: The estmated coeffcent sn t sgnfcantly dfferent from zero n the expected drecton. R : R remans constant, whch s exactly what wll happen whenever a varable wth a t-score wth an absolute value of 1 s removed from (or added to) an equaton. Do other coeffcents change?: None of the other estmated coeffcents change sgnfcantly when T s dropped, ndcatng that droppng T caused no bas. Concluson: Based on these four crtera, t s reasonable to conclude that T s an rrelevant varable. (f) You should not have been surprsed. If a varable s coeffcent has a t-score of exactly 1., then takng that varable out of an equaton wll not change R (b) Y ˆ = PC +.36 PB +.4 YD.7 PRP (.3) (.18) (.5) (.36) t = N = 9 R =.99 Theory: Pork s one of many substtutes for chcken, so the theoretcal case for PRP s ncluson n the equaton s good but not overwhelmng. t-score: The estmated coeffcent s nsgnfcantly dfferent from zero n the unexpected drecton, provdng evdence that PRP s rrelevant. R : R falls when PRP s added to the equaton, provdng evdence that t s rrelevant. Do other coeffcents change?: None of the other estmated coeffcents change sgnfcantly when PRP s added, ndcatng that addng PRP has corrected no bas and provdng evdence that t s rrelevant. Concluson: Based on the four crtera, t s reasonable to conclude that PRP s an rrelevant varable (a) Coeffcent β1 β β3 β4 Hypotheszed sgn: Calculated t-score: =.485 (1% level), so: sg. nsg. sg. unexpected sgn (b) The sgnfcant unexpected sgn of β 4 s evdence of a possble omtted varable that s exertng postve bas. The omtted varable must ether be correlated postvely wth X 4 and have a postve expected coeffcent or else be correlated negatvely wth X 4 and have a negatve expected coeffcent. (c) A second run mght add an ndependent varable that s theoretcally sound and that could have caused postve bas n β. 4 For example, the number of attractons lke theaters or malls n the area would have a postve expected coeffcent and be postvely correlated wth the number of nearby competng stores Some students wll come to the concluson that sequental specfcaton searches are perfectly reasonable n busness applcatons, and they need to be remnded that the regular use of such searches wll produce consstently based coeffcent estmates.

14 18 Studenmund Usng Econometrcs, Sxth Edton 6-8. Expected bas n ˆ β = ( β ) f( ) omtted romtted, ncluded (a) Expected bas = ( ) (+) = ( ) = negatve bas. (b) (+) (+) = (+) = postve bas; ths bas wll be potentally large snce age and experence are hghly correlated. (c) (+) (+) = (+) = postve bas. (d) ( ) () = = no bas; t may seem as though t rans more on the weekends, but there s no theoretcal relatonshp between the two (a) In a supply equaton, the coeffcent of prce wll have a postve expected sgn because the hgher the prce, holdng all else constant, the more supplers would be wllng to produce. (b) The prce of nputs (such as labor, seeds, transporaton, machnery, and fertlzer), the prce of a producton substtute (a crop or product that could be produced nstead of the crop or product beng modeled), and exogenous factors (lke local growng condtons, local strkes that don t have an mpact on the prce, etc.) are just a few examples of mportant varables n a supply-sde equaton. (c) Lag those ndependent varables that nfluence the producton decson on a delayed bass. In partcular, lag them by the length of tme t takes for that partcular event to have an mpact on producton. For example, f growers must make producton decsons a year before the crop s harvested, then prce should be lagged one year, etc. If a product can be stored at a farly low cost, then such a lag mght not be approprate because producers could choose to wat untl prces rose before gong to market (a) Consumers and producers can react dfferently to changes n the same varable. A rse n prce causes consumers to demand a lower quantty and producers to supply a greater quantty. (b) Include varables affectng demand ( demand-sde varables ) only n demand equatons and varables affectng supply ( supply-sde varables ) only n supply equatons. (c) Revew the lterature, decde whether the equaton you wsh to estmate s a supply or a demand equaton, and when specfyng the model, thnk carefully about whether an ndependent varable s approprate for a demand or supply equaton (a) Coeffcent β PARENT β HSRANK Hypotheszed sgn: + Calculated t-score: = (5% level), so: reject H reject H (b) There are no obvous sgns of an omtted or rrelevant varable, but t seems probable that more than two varables determne fnancal ad grants n most colleges, so an omtted varable s very lkely from a theoretcal pont of vew. (d) The estmated coeffcent of MALE mples that a male fnancal ad applcant wll receve $157 less n grant ad than a female applcant, holdng constant PARENT and HSRANK.

15 Answers to Text Exercses 19 (e) Theory: When asked, most colleges wll state that they award fnancal ad wthout regard to gender, but lberal arts colleges attract more females than males, so t s possble that a partcular college mght try to tlt ts fnancal ad toward males. Gven gven ths possblty and even gven the charge of bas, however, the theory behnd MALE s farly weak. t-score: The absolute value of the t-score s greater than the new crtcal t-value of 1.68, but the sgn of the t-score s opposte that mpled by H A, so we cannot reject the null hypothess. R : R ncreases when MALE s added, provdng evdence that the varable belongs n the equaton. bas: Nether estmated slope coeffcent changes by anythng close to a standard error when MALE s added to the equaton, provdng evdence that omttng MALE from the equaton does not cause any bas. Three of the four specfcaton crtera favor Equaton 6., so we prefer Equaton 6. to Equaton 6.3. However, the sgnfcant unexpected sgn n Equaton 6.3 cannot be gnored. It ndcates that there very lkely s an omtted varable n Equaton 6.. Snce we were concerned about the possblty of an omtted varable on theoretcal grounds already, ths emprcal evdence s very convncng. In essence, nether equaton s the best equaton! Most begnnng econometrcans wll not be very happy wth ths answer, but t s an mportant learnng opportunty (a) No bas (+ ) unless weather patterns ndcate a correlaton between ranfall and temperature. If t tends to ran more when t s cold, then there would be a small negatve bas (+ ). (b) Postve bas (+ +). (c) Postve bas (+ +). (d) Negatve bas (+ ) gven a lkely negatve correlaton between hours studed for the test and hours slept (a) X 1 = ether dummy varable X = ether dummy varable X 3 = Parents educatonal background X 4 = Iowa Test score (b) We have two varables for whch we expect postve coeffcents (Iowa score and Parents educaton) and two postve estmated coeffcents ( ˆβ 3 and ˆβ 4 ), so we d certanly expect X 3 and X 4 to be those two varables. In choosng between them, t s far to expect a larger and more sgnfcant coeffcent for Iowa than for Parents. Next, we have two varables for whch we expect a zero coeffcent (the dummes) and two estmated coeffcents ( 1 ˆβ and ˆβ ) that are not sgnfcantly dfferent from zero, so we d certanly expect X 1 and X to be the dummes. There s no evdence to allow us to dstngush whch dummy s X 1 and whch s X. (Students who justfy ths answer by expectng negatve sgns for coeffcents of the two dummes are gnorng the presence of the Iowa test score varable n the equaton that holds constant the test-takng sklls of the student.)

16 Studenmund Usng Econometrcs, Sxth Edton (c) Coeffcent: βd βd βpe βit Hypotheszed sgn: + + t-value: =.93 do not do not (5% two-sded reject reject wth 19 d.f.) = 1.79 reject reject (5% one-sded wth 19 d.f.) (d) As you can see, we used a one-sded test for those coeffcents for whch we had a specfc pror expectaton but a two-sded test around zero for those coeffcents for whch we dd not (a) Theory: If PERCENT s the best proxy avalable for the qualty and relablty of the seller, then t has a strong theoretcal bass untl a better varable can be found. t-score: The coeffcent s n the expected drecton, but t s nsgnfcant at the 5% level. R : R s not gven, but t turns out that the addton of any varable wth a t-score greater than one n absolute value wll ncrease R. Bas: None of the coeffcents change sgnfcantly. Thus the four crtera are nconclusve. Because PERCENT appears to be the best avalable measure of seller qualty, and because the sgn of the coeffcent s n the expected drecton, we d tend to keep PERCENT. (b) In theory, PERCENT seems lke the best we can do, but t mght be an unrelable measure f there are very few transactons. (c) When you drop PERCENT from the equaton, R falls from.434 to (a) () The coeffcent of CV s.19 wth a SE ( ˆ β ) of.3 and a t-score of.86. The R s.773, and the rest of the equaton s extremely smlar to Equaton 5.14 except that the coeffcent of CVN falls to.48 wth a t-score of () The coeffcent of N s.54 wth a SE ( ˆ β ) of.63 and a t-score of.86. The R s.766, and the rest of the equaton s dentcal (for all ntents and purposes) to Equaton 5.1. (b) Theory: P s a prce rato, and whle t s possble that a prce rato would be a functon of the sze of a market or a country, t s not at all obvous that ether varable would add anythng snce CVN s already n the equaton. t-score: Both t-scores are nsgnfcant. R : R falls when ether varable s added. Bas: None of the coeffcents change at all when N s added, so t clearly s rrelevant. The omsson of CV does change the coeffcent of CVN somewhat, makng t lkely thav s redundant snce CVN s n the equaton. (c) Snce CVN = f[cv/n], t would make lttle theoretcal sense to nclude all three varables n an equaton, even though techncally you don t volate Classcal Assumpton VI by dong so. (d) It s good econometrc practce to report all estmated equatons n a research report, especally those that were undertaken for specfcaton choce or senstvty analyss.

17 Answers to Text Exercses (a) Coeffcent: βpr β β PRCOMP ADS βyd Hypotheszed sgn: Calculated t-score: = 1.711, so: nsg./unexpected sg. sg. sg. sgn (b) PR s hardly rrelevant, but there could be an omtted varable. (c) The obvous addton s advertsng for the competng nstant oatmeal. (d) Amsh Oats competes wth regular oatmeal, other cereals, and other breakfast foods, not just wth the competng nstant oatmeal. Chapter Seven: Specfcaton: Choosng a Functonal Form 7-3. (a) Lnear n the coeffcents but not the varables. (b) Lnear n the coeffcents but not the varables. (c) Lnear n the coeffcents but not the varables. (d) Nonlnear n both. (e) Nonlnear n both (a) Coeffcent β 1 β Hypotheszed sgn: + + Calculated t-score: 4.. = 1.78 at the 5% level, so: sg. sg. (b) It s the sum of the constant effect of omtted ndependent varables and the nonzero mean of the sample error term observatons; t does not mean that salares (logged) could be negatve. (c) For ths semlog functon, the slopes are β1 SAL and β SAL, whch both ncrease as the Xs rse. Ths mples that a one-unt change n ED wll cause a β1 percent change n SAL, whch makes sense for salares. (d) R s cannot be compared because the dependent varables dffer (a) To avod confuson wth β, let s use α S as the coeffcents. Coeffcent α BETA α EARN α DIV Hypotheszed sgn: + + Calculated t-score: = (5% level), so: sg. nsg. sg. (b) It s unusual to have a lagged varable n a cross-sectonal model, but n ths equaton all the varables are for 1996 except for BETA, whch s for and therefore s ndeed lagged. Far assumed that the rsk characterstcs of companes don t change rapdly over tme and stated that fve observatons per company s not enough to get trustworthy estmates (p. 17).

18 Studenmund Usng Econometrcs, Sxth Edton (c) We don t beleve that any of Far s varables are potentally rrelevant, because the theory behnd each varable s exceptonally strong. Some students wll thnk that EARN mght be rrelevant because ts coeffcent has a low t-score, but we dsagree wth ths concern because earnngs growth s one of the most mportant determnants of stock prces. A student who drops EARN should conclude, based on the four specfcaton crtera, that the varable belongs n the equaton, because three of the four crtera support keepng EARN n the equaton, and the t-score s close to beng sgnfcant n the expected drecton. (d) The functonal form s a semlog left, whch s ndeed approprate both on a theoretcal bass and also because two of the ndependent varables are expressed as percentages. (e) Ths optonal queston s ntentonally dffcult. EARN and DIV both nclude negatve values, so many students wll gve up. Snce the negatve values are extremely small, one possble way to estmate the equaton s to set all the negatve values equal to +.1, obtanng: LNPE = LN BETA +.71 LN EARN +.98 LN DIV (.11) (.35) (.8) t = N = 65 R =.3 However, these results, whle completely reasonable, shed very lttle lght on whether to use a double-log functonal form, because we urge researchers to focus on theory, and not ft, to choose ther functonal forms. We thnk that Far s choce of a semlog left s supported by the lterature and by the fact that two of the ndependent varables are expressed n percentage growth terms. Hs optonal queston s ntentonally dffcult. EARN and DIV both nclude negatve values, so many students wll gve up. Snce the negatve values are extremely small, t s reasonable to set all the negatve values equal to +.1 (or to add to each varable the smallest amount necessary to make the most negatve observaton of that varable flp postve). However, even f you do ths, the results shed very lttle lght on whether to use a double-log functonal form, because we urge researchers to focus on theory, and not ft, to choose ther functonal forms. We thnk that Far s choce of a semlog left s supported by the lterature and by the fact that two of the ndependent varables are expressed n percentage growth terms (a) The Mdwest (the fourth regon of the country). (b) Includng the omtted condton as a varable wll cause the dummes to sum to a constant (1.). Ths constant wll be perfectly collnear wth the constant term. (c) Postve. (d) Most correct = III; least correct = I. (e) Any number of worker attrbutes make sense; for example the gender, age, or experence of the th worker (a) Snce the equatons are double-log, the elastctes are the coeffcents themselves: Industry Labor Captal Cotton.9.1 Sugar (b) The sum ndcates whether or not returns to scale are constant, ncreasng, or decreasng. In ths example, Cotton s experencng ncreasng returns to scale whle Sugar s experencng decreasng returns to scale.

19 Answers to Text Exercses 3 (c) Ths queston contans a hdden dffculty n that the sample sze s purposely not gven. D students wll gve up, whle C students wll use an nfnte sample sze. B students wll state the lowest sample sze at whch each of the coeffcents would be sgnfcantly dfferent from zero (lsted below), and A students wll look up the artcle n Econometrca and dscover that there were 15 cotton producers and 6 sugar producers, leadng to the s and hypothess results lsted below. Coeffcent: β1c βc β1s βs Hypotheszed sgn: t-value: Lowest d.f. at whch sgnf. (5%) 1 7 5% gven actual d.f (So all four coeffcents are sgnfcantly dfferent from zero n the expected drecton.) 7-8. Let PCI = per capta ncome n the th perod, GR = rate of growth n the th perod, and = a classcal error term. (a) GR = + 1 PCI + PCI + where we d expect 1 > and <. (b) A semlog functon alone cannot change from postve to negatve slope, so t s not approprate. (c) GR = β + β1 PCI + β D + β3 D PCI +, where D = f PCI $, and D = 1 f PCI > $,. ($, s an estmate of the turnng pont.) (a) H : β ; H A: β > ; for both. (b) L: t =.; K: t = 5.86, snce = 1.717, we can reject H for both. (c) The relatve prces of the two nputs need to be known before ths queston can be answered (a) The expected sgns are β, + or?; β, + β, + ; β, (b) AD /SA : The nverse form mples that the larger sales are, the smaller wll be the mpact of advertsng on profts. CAP, ES, DC : The semlog rght functonal form mples that as each of these varables ncreases (holdng all others n the equaton constant), PR ncreases at a decreasng rate. (c) β, β, and 3 β, 4 all have postve expected sgns, so (+) (+) = (+) = postve expected bas on β 1 f one of the other Xs were omtted (a) Polynomal (second-degree, wth a negatve expected coeffcent for age and a postve expected coeffcent for age squared). (b) Double-log. (We would not qubble wth those who chose a lnear form to avod the constant elastcty propertes of a double-log.) (c) Semlog (lnx). (d) Lnear. (All ntercept dummes have a lnear functonal relatonshp wth the dependent varable by defnton.) (e) Inverse. (Most students wll remember from the text that a U-shaped polynomal typcally s used to model a cost curve and wll want to apply t here. The problem s that the telephone ndustry appears to be an ndustry n whch costs contnually decrease as sze ncreases, makng an nverse our choce.)

20 4 Studenmund Usng Econometrcs, Sxth Edton 7-1. (a) The estmated coeffcents all are n the expected drecton, and those for A and S are sgnfcant. R seems farly low, even for a cross-sectonal data set of ths nature. (b) It mples that wages rse and then fall wth respect to age but does not mply perfect collnearty. (c) Wth a semlog left functonal form (lny), a slope coeffcent represents the percentage change n the dependent varable caused by a one-unt ncrease n the ndependent varable (holdng constant all the other ndependent varables). Snce pay rases are often dscussed n percentage terms, such a functonal form frequently s used to model wage rates and salares. (d) It s a good habt to gnore ˆβ (except to make sure that one exsts) even f t looks too large or too small. (e) The poor ft and the nsgnfcant estmated coeffcent of unon membershp are all reasons for beng extremely cautous about usng ths regresson to draw any conclusons about unon membershp (a) Coeffcent: βlq βa βv Hypotheszed sgn: + t-value: = 1.75 reject reject do not (5% one-sded reject wth d.f.) (b) Q constant; A and V non-constant. (c) No. The coeffcent of V s qute nsgnfcant, and the equaton (smplfed from an unpublshed artcle) s flawed to boot. Note, however, that the volence may be causng the absentee rate to rse, so that the sgnfcant coeffcent for A does ndcate some support for the charge. (d) In our opnon, ths s a classc case of spurous correlaton because actual total output appears on both sdes of the equaton, causng almost all of the ft by defnton. If we could make one change, we d drop LQ from the equaton, but we worry that lttle wll be left when we do (a) Coeffcent βb βs βd Hypotheszed sgn: + + Calculated t-score: = 1.68, so: nsg. sg. nsg. The nsgnfcance of ˆβ B could be caused by an omtted varable, but t s lkely that the nteracton varable has soaked up the entre effect of beer consumpton. Although we cannot reject the null hypothess for ˆ β D, we see no reason to consder D to be an rrelevant varable because of ts sound theory and reasonable statstcs. (b) The nteracton varable s a measure of whether the mpact of beer drnkng on traffc fataltes rses as the alttude of the cty rses. For each unt ncrease n the multple of B and A, F rses by.11, holdng constant all the other ndependent varables n the equaton. Thus the sze of the coeffcent has no real ntutve meanng n and of tself.

21 Answers to Text Exercses 5 (c) H : βba H A : β BA > Reject H because > tc = 1.68 and 4.5 s postve and thus matches the sgn mpled by H A. (d) Although there s no ronclad rule (as there s wth slope dummes) most econometrcans nclude both nteracton-term components as ndependent varables. The major reason for ths practce s to avod the possblty that an nteracton term s coeffcent mght be sgnfcant only because t s pckng up the effect of the omtted nteracton-term component (bas). (e) The excepton to ths general practce occurs when there s no reason to expect the nteractonterm component to have any theoretcal valdty on ts own. We prefer Equaton 7.5 to 7.6 because we don t beleve that alttude typcally would be ncluded as an ndependent varable n a hghway fatalty equaton. Of our other three specfcaton crtera, only the ncrease n R supports consderng A to be a relevant varable. However, even moderate theoretcal support for the ncluson of A on ts own would result n our preferrng Equaton (a) Let s look at β. D The easy answer s that an ncrease of 1, resdental customers wll cause an ncrease of $5. n advertsng and promotonal expense per 1 resdental klowatt hours, holdng constant G, D, G*D, and S*D. However, ths techncally correct answer gnores the exstence of the nteracton varables. We d rather say that for duopoles, an ncrease of 1, resdental customers wll cause an ncrease of $5. n advertsng and promotonal expense per 1 resdental klowatt hours, holdng constant G (because the other terms fall out of the equaton), and for monopoles an ncrease of 1, resdental customers wll cause a decrease of $15. n advertsng and promotonal expense per 1 resdental klowatt hours, holdng constant G. The answers for the other coeffcents are smlarly annoyng. (b) Coeffcent: βs βg βd βsd βgd Hypotheszed sgn: ? + t-value: = reject do not reject do not reject (5% one-sded reject reject wth nfnte d.f.) (c) As Prmeaux puts t (on page 6 of hs artcle), A duopoly frm of small sze spends more than a monopoly frm of the same sze. However, as scale ncreases, eventually, the duopoly frm spends less. (d) Agan, from page 6, There s no dfference between monopoly and duopoly frms at zero rates of growth n sales. However, as growth takes place, the duopoly frms engage n more sales promoton actvty (a) Coeffcent βtop β WEIGHT βhp Hypotheszed sgn: + Calculated t-score: =.479, so: sg. nsg. sg. (b) At frst glance, all three problems seem possble. (c) Snce TIME and HP are nversely related by theory, an nverse functonal form should be used.