CHAPTER 10 DUMMY VARIABLE REGRESSION MODELS

Transcription

1 CHAPTER 10 DUMMY VARIABLE REGRESSION MODELS QUESTIONS (a) and (b) These are varables ha canno be quanfed on a cardnal scale. They usually denoe he possesson or nonpossesson of an arbue, such as naonaly, relgon, sex, color, ec. (c) Regresson models n whch explanaory varables are qualave are known as ANOVA models. (d) Regresson models n whch one or more explanaory varables are quanave, alhough ohers may be qualave, are known as ANCOVA models. (e) In a regresson model wh an nercep, f a qualave varable has m caegores, one mus nroduce only (m 1) dummy varables. If we nroduce m dummes n such a model, we fall no he dummy varable rap, ha s, we canno esmae he parameers of such models because of perfec (mul)collneary. (f) They ell wheher he average value of he dependen varable vares from group o group. (g) If he rae of change of he mean value of he dependen varable vares beween caegores, he dfferenal slope dummes wll pon ha ou (a) Quanave (b) qualave (c) quanave (d) qualave (e) quanave (f) qualave, f expressed n broad caegores, bu quanave f expressed as years of schoolng (g) qualave (h) qualave () qualave (j) qualave (a) If here s an nercep erm n he model, 11 dummes. (b) If here s an nercep erm n he model, 5 dummes. 79

2 10.4. (a) Here we wll fall no he dummy varable rap, because he four columns of he dummy varables wll add up o he frs column (represenng he nercep). (b) Ths equaon can be wren as: GNP = B 1 + ( B + B4 ) M + ( B B4 ) = B + A M 1 + A M where A = B + ) and A = B ). ( B4 ( B4 Alhough we can esmae B 1, A, and A, we canno esmae B, B, and B4 unquely. The problem here s ha he hrd explanaory varable n he orgnal model, ( M M 1), s a lnear combnaon of M and M 1, hereby leadng o perfec collneary (a) False. Leng D ake he values of (0, ) wll halve boh he esmaed B and s sandard error, leavng he rao unchanged. (b) False. Snce he dummy varables do no volae any of he assumpons of OLS, he esmaors obaned by OLS are unbased n small as well as large samples (a) Each regresson coeffcen s expeced o be posve. (b) B ells us by how much he average salary of a Harvard MBA dffers from he base caegory, whch s non-harvard and non-wharon MBAs. (c) I probably suggess ha he Harvard MBA has a premum over he Wharon MBA (a) The model gven n he prevous queson assumes ha he average sarng salares of Harvard and Wharon MBAs are dfferen from ha of he oher MBAs, bu he rae of change of salary wh respec o years of servce s he same for all graduaes. On he oher hand, he model gven n hs queson assumes ha he average sarng salary as well as he progresson of salary (.e., he rae of change) over years of servce s dfferen among Harvard, Wharon, and oher MBAs. (b) B4 and B 5 are dfferenal slopes. 1 + u M 1 + u 80

3 (c) Yes, oherwse, we wll be commng he omsson of relevan varable bas. (d) Ths can be esed by he F es. PROBLEMS (a) The coeffcen s he own-prce elascy, s he ncome elascy, and s he cross-prce elascy. (b) I s nelasc because, n absolue value, he coeffcen s less han one. (c) Snce he cross-prce elascy s posve, coffee and ea are subsue producs. (d) and (e) The rend coeffcen of suggess ha over he sample perod coffee consumpon had been declnng a he quarerly rae of 0.89 percen. Among oher hngs, he sde effecs of caffene may have somehng o do wh he declne. (f) (g) The esmaed value of he ncome elascy coeffcen s 1., whch s no sascally sgnfcan. Therefore, does no make much sense o es he hypohess ha s no dfferen from one. (h) The dummes here perhaps represen seasonal effecs, f any. () Each dummy coeffcen ells by how much he average value of lnq s dfferen from ha of he base quarer, whch s he fourh quarer. The acual values of he nerceps n he varous quarers are, respecvely, 1.188, 1.119, 1.69, and Takng he anlogs of hese values, we oban:.65,.0707,.5580, and.597 as he average pounds of coffee consumed per capa n he frs, second, hrd, and he fourh quarer, holdng he values of he logs of all explanaory varables zero. Noe: On he general nerpreaon of he dummy varables n a sem-log model, see Rober Halvorsen and Raymond Palmqus, "The Inerpreaon of Dummy Varables n Semlogarhmc Equaons," The Amercan Economc Revew, vol. 70 (June 1980), no., pp

4 (j) The dummy coeffcens D 1 and D are ndvdually sascally sgnfcan. (k) Tha seems o be he case n quarers one and wo. Among oher hngs, coffee prces and weaher may have somehng o do wh he observed seasonal paern n hese wo quarers. (l) The benchmark s he fourh quarer. If we choose anoher quarer for he base, he numercal values of he dummy coeffcens wll change. (m) The mplc assumpon ha s made s ha he paral slope coeffcens do no change among quarers. (n) We can ncorporae dfferenal slope dummes as follows: ln Q = B 1 + B ln P + B ln I + B ln P 4 + B 5 T + B D B D B D 8 + B D ln ) + B D ln ) + B D ln ) 9( 1 P 10( P 11( P + B D ln ) + B D ln ) + B D ln ) 1( 1 I 1( I 14( I + B D ln ) + B D ln ) + B D ln ) + u 15( 1 P 16( P 17( P Noe: The subscrp has been omed o avod cluerng he equaon. The frs wo rows of he equaon are he same as n he ex. The dfferenal slope dummes are n he las hree rows. (o) One could esmae he model gven n (n). If here are oher subsues for coffee, hey can be brough n he model (a) I s a way of fndng ou f here are economes or dseconomes of scale. In general, f a a gven pon, he frs dervave (.e., he slope) s negave bu he second dervave s posve, means he slope s negave and ncreasng, ha s, he negave slope ends o be less seep as he value of he varable ncreases. (b) The same reasonng as n (a), excep ha mles has a posve sgn and mles squared has a negave sgn. In general, f a a gven pon, he frs dervave s posve bu he second dervave s negave, means ha he value of he funcon s ncreasng a a decreasng rae. In he presen case, 8

5 hs s an ndcaon of economes of scale, for he longer he dsance n mles s, he lesser s he ncremenal fare. (c) Populaon may be a proxy for raffc volume. The negave sgn here ndcaes perhaps some ype of economes of scale. (d) Alhough negave, he coeffcen s sgnfcan only for he dscoun caegory. Ths sgn s raher puzzlng. (e) The negave sgn makes economc sense n he sense ha he hgher he number of sopovers, he greaer s he me spen ravelng. Hence, he fare s lower o nduce passengers o ravel wh several sopovers. (f) I suggess ha he average level of fare for Connenal Arlnes s lower han s compeors. (g) The crcal Z value s 1.96 (5%, wo-aled) or 1.65 (5%, one aled). If he compued Z value exceeds hese crcal values, he coeffcen n queson s sascally sgnfcan. (h) Alhough hs dummy coeffcen s expeced o be posve for all caegores, s no clear why s sgnfcan only for he dscoun caegory. () Yes, hese observaons can be pooled. In ha case, nroduce an addonal dummy for he coach or dscoun fares. (j) Overall, he resuls are a mxed bag. Alhough he R s are que hgh for hs sample sze, and alhough several coeffcens are sascally sgnfcan, some of he coeffcens have dubous sgns (a) Snce he coeffcen of he Dumsex dummy s sascally sgnfcan, Model s preferable o Model 1. (b) The error of omng a relevan varable. (c) Ceers parbus, he average wegh of males s greaer han ha of females. (d) There s an addonal varable, Dumh, n Model, whch s sascally nsgnfcan. As shown n Chaper 11, f an unnecessary varable s added o a model, he OLS esmaors, whle unbased and conssen, are generally neffcen. Ths can be seen from Model. In Model he Dumsex 8

6 varable was sascally sgnfcan, bu s nsgnfcan n Model because of he apparenly superfluous Dumh varable. Also, keep n mnd he possbly of mulcollneary. (e) Choose Model. No only s he Dumsex varable sascally sgnfcan n hs model, bu he coeffcen of he hegh varable s abou he same n boh Models and. On he oher hand, neher dummy varable s sascally sgnfcan n Model. (f) We observe from he correlaon marx ha he coeffcen of correlaon beween Dumsex and Dumh s very hgh, almos uny. As we show n he chaper on mulcollneary, n cases of very hgh collneary, OLS esmaors, alhough unbased, have relavely large sandard errors. Also, he sgns and magnudes of he coeffcens can change wh slgh aleraons n he daa or n he specfcaon of he model (a) Sa ˆes l = 4, D + 1, D +, = (14.714) (1.991) (.05) (6.50) R = (b) The average sales n he frs quarer was abou $4,768 mllon. In he second quarer was hgher by abou $91 mllon, n he hrd quarer by abou $ 1,99 mllon, and n he fourh quarer by abou $,910 mllon, all hese dfferences beng sascally sgnfcan. The acual values of he nerceps n he varous quarers can be obaned by addng he dfferenal nercep dummes o he base quarer value. The ndvdual nercep values are, respecvely, (all n mllons of dollars): D 4 1 s Quarer nd Quarer rd Quarer 4 h Quarer 4, , , , (c) To deseasonalze he daa, subrac from each quarer's sales fgure he dummy coeffcen of ha quarer. For nsance, f you subrac from he sales fgure for he fourh quarer of each year he number, , he resulng fgure for ha quarer wll ndcae he seasonally adjused sales for 84

7 ha quarer. Thus, he seasonally adjused fgures for he fourh quarer of 198, 1984, 1985, and 1986 are as follows: 4 h Quarer 4 h Quarer 4 h Quarer 4 h Quarer , , , , (a) In hs model, we have assgned a dummy coeffcen for each quarer. Bu noce ha, o avod he dummy varable rap, we have omed he nercep erm from hs model. (b) Yes. (c) ˆes Sa l = 4, D 1 + 5, D + 6, D + 7, D 4 = (14.714) (17.59) (19.00) (.69) Ths model gves drecly he nercep values for all he four quarers, whereas, as shown n problem 10.11, he nercep values for he second, hrd, and fourh quarers were obaned by addng he dfferenal nercep dummy values o he nercep value of he base quarer. Of course, boh procedures gve dencal resuls, as hey should. Noe: The R value of hs model s no presened for he reasons explaned n he ex (Ch. 9) In hs case he model wll be: AAS = B + B D + B D + B PPS + B PPS ) + u 1 The EVews regresson resuls are as follows: 4 5( DPPS ) + B6 ( D Dependen Varable: PAY Sample: 1 51 Varable Coeffcen Sd. Error -Sasc Prob. C D D PPS D*PPS D*PPS R-squared F-sasc

8 Noe: The dependen varable, Pay, s he same as AAS: AAS s labeled Pay n Table 10-4, on whch he example s based. The F sasc s also shown here as par of he EVews oupu for he frs me. Compared wh Equaon (10.17), hese resuls sugges ha here s no regonal varaon n he coeffcen of PPS. Hence, he resuls of Equaon (10.17) seem accepable Ths can be accomplshed by addng as varables he produc of X and and he produc of X and D Usng EVews, and suppressng he nercep o avod he problem of perfec mulcollneary, we oban he followng resuls: D Dependen Varable: FRIG Sample: 1978:1 1985:4 Varable Coeffcen Sd. Error -Sasc Prob. DUM1MINE DUM DUM DUM R-squared Here he varous dummes represen he average sale of refrgeraors n each quarer (a) By neracon we mean when boh effecs (sex and race) are presen smulaneously. (b) B = dfferenal effec of beng a male B = dfferenal effec of beng whe (c) B 4 = dfferenal effec of beng a whe male E( Y ) = ( B1 + B + B + B4) + B5 X gven ha D D 1. Thus, a whe male's mean annual salary s hgher = = by B 4 as compared o he mean salary of a male alone or a whe alone. 86

9 We defne he new Sex dummy varable as equal o 1 for female and 1 for male and name SEX1FN1M, o dsngush from he orgnal dummy varable SEX already n Table 10-. In SEX1FN1M, 1FN1M sands for 1 for female and negave 1 for male. The EVews oupu s as follows: Dependen Varable: FOODEXP Sample: 1 1 Varable Coeffcen Sd. Error -Sasc Prob. C SEX1FN1M R-squared Wh hs dummy seup, he consan erm represens he average nercep of he regresson lne from whch he female and male nerceps dffer by 51.58, n he oppose drecon. Thus, he nercep for males s (, ) =,176.8, and he one for females s calculaed as (, ) =, , whch are he values obaned for model (10.1) and shown n Equaon (10.4), save any mnor roundng errors In hs problem, we defne he new Sex dummy varable as equal o for female and 1 for male and name SEXF1M, o dsngush from he orgnal dummy varable SEX already n Table 10-. In SEXF1M, F1M sands for for female and 1 for male. The regresson resuls are: Dependen Varable: FOODEXP Sample: 1 1 Varable Coeffcen Sd. Error -Sasc Prob. C SEXF1M R-squared These resuls are precsely he same as n Equaon (10.4), by nong ha when D = (female), he female nercep s, ( ) =, and he male nercep s, =,

10 10.19 (a) Based on he 19 observaons, he EVews regresson resuls are: Dependen Varable: NDIV Sample: 1999:1 00: Varable Coeffcen Sd. Error -Sasc Prob. C ATPROFITS R-squared As hese resuls show, here s a sascally sgnfcan posve relaonshp beween he wo varables, an unsurprsng fndng. (b), (c),and (d) We can nroduce hree dummes o dsngush four quarers and can also nerac hem wh he profs varable. Ths exercse yelded no sasfacory resuls, snce boh he dummes and neracon erms were compleely nsgnfcan, suggesng ha perhaps here s no seasonaly nvolved. Ths makes sense, for mos corporaons do no change her dvdends from quarer o quarer. I seems ha here s no reason o consder explcly seasonaly n he presen case The reference caegory (.e., he caegory wh 0 value for all dummes) s unmarred whe male. Therefore, he nercep for hs caegory s All oher varables reman he same. The nercep for whe unmarred female s ( ) = Snce he coeffcen of DF s no sascally sgnfcan a he 5% level, seems ha here s no dfference beween he wo caegores n her nercep values. Oher varables reman he same You wll have o expand he model by ncludng he produc of each dummy varable wh he oher explanaory varables (6 n all). Thus you wll have o add (6 ) = 18 addonal varables o he model. Bu do no forge he prncple of parsmony (a) Snce he p value of he dummy coeffcen s abou 14%, seems ha produc-dfferenaon does no lead o a hgher rae of reurn. 88

11 (b) From (a) s obvous ha here s no sascal dfference n he rae of reurn for frms ha produc-dfferenae and he frms ha do no. (c) Perhaps. If we had he orgnal daa, we could verfy hs. Produc dfferenaon s he resul of adversng and markeng sraeges. For deals, see any ndusral organzaon exbook. (d) To he equaon gven, add he produc of D wh each of he explanaory varables. Thus, here wll be hree addonal varables n he model (a) Snce boh he dfferenal nercep and slope coeffcens are sascally sgnfcan, he Phllps curve has changed beween he wo me perods. The regresson models for he wo perods derved from hs regresson are: : Ŷ = ( ) + ( ) X = X : Ŷ = X Wha s srkng abou he laer perod s ha he slope coeffcen s negave! Ths would mply a posvely sloped Phllps Curve. (b) The orgnal Phllps curve may be dead bu several aemps have been made o revve. See any modern exbook on macroeconomcs From Table we observe ha of he 40 observaons, 6 observaons have negave predced values and 6 have predced values n excess of 1. Hence, here are 1 ncorrec predcons. Therefore, The convenonal Coun R value s R = 8 / 40 = (a) A scaer plo wll show ha he hree expendure caegores are lnearly relaed o PCE. (b) Snce he daa are seasonally adjused, f you regress each expendure caegory on PCE and nclude he dummy varables, he dummy coeffcens 89

12 are lkely o be nsgnfcan. Ths, n fac, urns ou o be he case. Bu keep n mnd ha he mehod of seasonal adjusmen used by he U.S. governmen s dfferen from he dummy varable mehod. Noe: EVews provdes seasonal adjusmen opons. (c) By ncludng he dummy varables unnecessarly, you wll be commng he bas of ncludng superfluous varables. As a resul, he sandard error of he PCE coeffcen s lkely o be overesmaed, whch wll lower he values. 90