Estimation of Price Response Functions

Transcription

1 Estmaton of Prc Rspons Functons Pag Outln Rgrsson Lnar Dmand Constant Elastcty Logt» Gvn D» Jont D Maxmum Lklhood Estmaton Logt Multnomal Logt Basd on Commonalty of lnar dmand modl Bsbs and Zv On th (surprsng) suffcncy of lnar modls for dynamc prcng wth dmand larnng. Managmnt Scnc, Vol.6:

2 Lnar Dmand and Lnar Rgrsson Pag 2 Constant wllngnss to pay d(p)=d-mp Estmat markt sz D and slop m Us standard lnar rgrsson Excl Offc Button Excl Optons Add-Ins Analyss ToolPak Go Wat for nstallaton Excl Toolbar Data Clck on Data Analyss Pck Rgrsson Gvn data (d,p ) Mn D, m ( D mp d ) Us standard Solvr functon Excl Offc Button Excl Optons Add-Ins Solvr Go Wat for nstallaton Excl Toolbar Data Clck on Solvr 2 <s lnar.xls>

3 Constant Elastcty Pag 3 Constant lastcty: d(p)=cp -ε. Estmat paramtrs C and lastcty ε. Mak log transformaton: log d(p) = log C ε log p. Mak chang of varabl: y=log d(p), x=log p, ntrcpt=log C, slop = ε so that y=ntrcpt-slop*x. Us standard Lnar Rgrsson to fnd ntrcpt and slop. Gvn data (y,x ) us Solvr to fnd Mn Intrcpt, Slop (ntrcpt slop* x y ) 2 <s constant_lastcty.xls>

4 Logt Prc Rspons Gvn Markt Sz D Pag 4 Logt Prc Rspons functon. d Mn Intrcpt, Slop D ( + ( p) = a+ bp) For gvn markt sz D, stmat paramtrs a and b. Mak ln transformaton: ln ((D-d(p))/d(p)) = a+bp. Mak chang of varabl: y=ln (D/d(p)-), x=p, ntrcpt=a, slop = b so that y=ntrcpt+slop*x. Us standard Lnar Rgrsson to fnd ntrcpt and slop. Gvn data (y,x ) us Solvr to fnd (ntrcpt + slop* x y ) 2 D = D d( p) ( a+ bp) + D ( a+ bp) = + D d( p) d( p) = D d( p) ( a+ bp) <s logt_gvnd.xls>

5 Logt Prc Rspons Unknown Markt Sz D Pag 5 Logt Prc Rspons functon. d p D ( + ( ) = a+ bp) Estmat paramtrs D, a and b. No log or ln transformaton possbl to cast th stmaton as a lnar rgrsson. Gvn data (d,p ) us fnd Mn D D, a, b + abp d 2 Solvr dos not solv ths problm.

6 Pag 6 Logt Prc Rspons Unknown Markt Sz D Squntal Approach Suppos that D s gvn. Estmat a, b. Updat D such that t solvs = + + = + bp a bp a bp a d D d D D So + + = bp a bp a d b a p d D 2 ), ;, ( Squntal Algorthm 0. Start wth an ntal guss for D. Us currnt D to stmat a and b by usng th procdur of Logt wth gvn D. 2. Updat D so that t sts th drvatv of th sum of squars of rrors to zro 3. Stop or go to.

7 Logt Prc Rspons Squntal Algorthm Pag 7 Squntal algorthm s mplmntd n logt_jontd.xls. Unfortunatly, D-D(d,p ;a,b) s small: In othr words, updats do not chang th markt sz much. Hypothss: a and b ar suffcnt to rprsnt markt sz. Graph actual dmand vs. stmatd dmand for Squntal Actual Est- Est-2 Est-3 LSE D= a= b=

8 Logt Prc Rspons Squntal Algorthm Graphs Pag Actual Est- Est-2 Est-3 LSE Actual Est- Est-2 Est-3 LSE Most possbl prcs Prcs usd n stmaton Basd on ths computatons Concluson: LSE stmats ar sgnfcantly wors than th Squntal ons. Rason: Excl Solvr fals to mnmz sum of squars, a hghly nonlnar objctv. Suggston: Slct 2-3 valus of D and us Squntal algorthm.

9 Maxmum Lklhood Estmaton (MLE) Maxmzng to Match Modl and Data Probablty of sllng s WW(pp; aa, bb) wth th prc p and th WTP functon W, paramtrs a, b. Pag 9 Consdr customr wth WTP functon WW and conduct th prc xprmnt: Offr prc pp to ths customr and rcord a sal as yy = and no-sal as yy = 0 Collct (scannr) data { yy, pp } Whn yy = n th data, w dally want th consstnt outcom from th modl WW(pp; aa, bb) = Howvr, WW(pp; aa, bb) = hardly happns xcpt whn pp s qual to th lowst WTP. Gv up dalsm and pragmatcally match [Sals n th modl] wth [Sals n th data] [Modl s WW(pp; aa, bb) to b hgh] whn [data yy = ] max yy( WW pp; aa, bb ) [No-sals n th modl] wth [No-sals n th data] [Modl s WW(pp; aa, bb) to b hgh] whn [data yy = 0] max ( yy)ww pp; aa, bb Snc yy {0,}, combn two objctvs to obtan max yy( WW pp; aa, bb )+ yy WW pp; aa, bb Consdr sals to th customr as a bnary random varabl YY {0,} wth probablts P YY = = ( WW(pp; aa, bb)) and P YY = 0 = WW(pp; aa, bb) P YY = yy = yy WW pp; aa, bb + ( yy)ww pp; aa, bb for yy {0,}. Th objctv w arrvd at through consstncy argumnt s max P YY = yy Maxmz th lklhood of th ralzd vnt [YY = yy]

10 MLE wth NN 2 Customrs 2 ndpndnt customrs wth dntcal WTP functons: th sam W, a, b. Rpat prc xprmnt twc and collct data Data = { yy, pp, (yy 2, pp 2 )} max P YY = yy, YY 2 = yy 2 = max P YY = yy P YY 2 = yy 2 Maxmz th lklhood functon L whos LL(aa, bb; yy, pp, yy 2, pp 2 ) = [yy ( WW pp ; aa, bb )+ yy WW pp ; aa, bb ] [yy 2 ( WW pp 2 ; aa, bb )+ yy 2 WW pp 2 ; aa, bb ] varabls (aa, bb) ar th paramtrs of WTP and paramtrs (yy nn, pp nn ) ar th varabls of WTP & ts consqunc WW ; aa, bb. Indpndnt WTPs Idntcal WTPs Chang of prspctv Varabls Paramtrs Pag 0 For NN ndpndnt customrs wth contnuous WTP, LL aa, bb; Data For NN ndpndnt customrs wth dscrt WTP, LL aa, bb; Data NN = [yy nn P Sal at prc pp nn + ( yy nn )P No sal at pp nn ] nn= NN = nn= yy nn ( WW(pp nn ; aa, bb)) + yy nn WW pp nn ; aa, bb. NN = nn= yy nn WW pp nn ; aa, bb + ( yy nn WW(pp nn ; aa, bb)], whr pp nn = th largst WTP valu strctly lss than pp nn. Lklhoods ar multplcatons of probablty LL aa, bb; Data Dscrt WTP Exampl Porton wllng 40% to pay th prc 20% 20% 20% Prc % 80% 60% 40% and LL(aa, bb; Data) NN. Cumulatv WTP WW(pp) Prob. of no-sal wth prcs 0, 2 & 27 WW 0 = WW 0 = 0% WW 2 = WW 0 = 40% WW 27 = WW 20 = 60% Prob. of sal wth prcs 6 and 23 WW 6 = WW 0 = 60% WW 27 = WW 20 = 40% pp

11 Exampls of MLE Ex: Suppos that th WTP for a shrt s unformly dstrbutd btwn unknown paramtrs aa 40 and bb 60. As th rtalr you dalt wth two customrs, on bought at th prc $40 and th othr dd not at th prc $60. What ar th assocatd scannr data? St up th lklhood functon to stmat aa, bb. DD = { yy =, pp = 40, (yy 2 = 0, pp 2 = 60)} LL(aa, bb; yy, pp, yy 2, pp 2 ) =[yy ( WW pp ; aa, bb )+ yy WW pp ; aa, bb ] [yy 2 ( WW pp 2 ; aa, bb )+ yy 2 WW pp 2 ; aa, bb ] =[ bbpp + 0][0 + pp 2aa bb40 ]= 60aa bbaa bbaa bbaa 2 Maxmum valu of obtand at aa = 40 and bb = 60. Pag Ex: Suppos that th WTP for a shrt s unformly dstrbutd btwn unknown paramtrs aa 40 and bb 60. As th rtalr you dalt wth thr customrs, on bought at th prc $40, th othr bought at th prc $50 but th last dd not at th prc $60. What ar th assocatd scannr data? St up th lklhood functon to stmat aa, bb. DD = { yy =, pp = 40, yy 2 =, pp 2 = 50, (yy 3 = 0, pp 3 = 60)} LL(aa, bb; yy, pp, yy 2, pp 2, (yy 3, pp 3 ) ) =[ bbpp + 0][ bbpp 2 + 0][0 + pp 3aa bbaa bbaa bb40 (bb50) 60aa ]= bbaa bbaa 3 For ach fxd bb, w want aa as larg as possbl. Bcaus dnomnator s cubc, dcrasng n aa. W st aa = 40. Th objctv thn s (bb 40)(bb 50)/ bb Ths s largr whn bb s smallr. W st bb = 60. Th maxmum lklhood valu s 0.5 In comparson wth th last xampl, lklhood droppd wth th thrd data pont (yy 3, pp 3 ). b\a <s maxlklhood.xls>

12 Multnomal Logt Choc Modl Th most common consumr choc modl s multnomal logt (MNL) for KK products Probablty of buyng product kk = ff kk pp = pp, pp 2,, pp KK = bb kk pp kk KK bb jj pp jj. Paramtrs ar bb,, bb KK. Paramtr bb kk s larg whn product kk s prc snstv. jj= Pag 2 Ex: Suppos KK = 3 products A, B, C, ach wth th probablty of choc ff kk, kk {AA, BB, CC}. NN = 9 customrs bought products as follows. 2 bought A; 3 bought B and 4 bought C. What s th probablty of ths vnt? P 2AA; 3BB; 4CC = 9! ff 2!3!4! AA 2 ff 3 BB ff 4 CC. Ths s also known as multnomal dstrbuton. For customr, lt yy AA =, yy BB = 0, yy CC = 0. Smlarly dfn yy 2 = [yy 2AA, yy 2BB, yy 2CC ] for customr 2 such that yy 2 = yy. For customr 3, lt yy 3AA = 0, yy 3BB =, yy 3CC = 0. Smlarly dfn yy 4 = yy 5 yy 3 for customrs 4 and 5. For customr 6, lt yy 6AA = 0, yy 6BB = 0, yy 6CC =. Dfn y 7 = yy 8 = yy 9 yy 6 for customrs 7, 8 and 9. W can rwrt P yyyyy = Constant ff AA yy AA +yy 2AA ff BB yy 3BB +yy 4BB +yy 5BB ff CC yy 6CC +yy 7CC +yy 8CC +yy 9CC = Constant ff AA yy AA +yy 2AA +yy 3AA +yy 4AA +yy 5AA +yy 6AA +yy 7AA +yy 8AA +yy 9AA ff BB yy 3BB +yy 4BB +yy 5BB ff CC yy 6CC +yy 7CC +yy 8CC +yy 9CC 9 yy = Constant Π kk AA,BB,CC Π nn= ff nnnn kk Multnomal Logt maxmum lklhood stmators for a gvn sampl of, 2,, NN ndvduals Each ndvdual nn mad a choc yy nnnn n rspons to prcs [pp nnn, pp nnn,, pp nnnn ]. Indcator varabl yy nnnn = f ndvdual nn chooss product k; othrws zro. Data for ndvdual nn: [yy nnn, yy nnn,, yy nnnn ; pp nnn, pp nnn,, pp nnnn ]. What ar th most lkly valus of paramtrs [bb, bb 2,, bb KK ],.., what paramtr valus maxmz th probablty of chocs? <s Gnrat and Pur Data shts of concrt.xls>

13 MLE of Multnomal Logt Choc Modl Pag 3 From th multnomal probablty mass functon, th lklhood s yy LL bb, bb 2,, bb KK = Const. ff nnnn kk = Const. nn,kk nn,kk bb kkpp nnkk KK jj= bb jjpp nnjj yy nnnn Instad maxmz th logarthm of th lklhood wthout th constant log LL bb, bb 2,, bb KK = yy nnnn log nn,kk bb kkpp nnnn KK jj= bb jjpp nnnn Us a solvr to fnd paramtrs [bb, bb 2,, bb nn ] that maxmz log LL([bb, bb 2,, bb nn ]). <s Gnrat and Pur Data, MLE shts of concrt.xls>

14 R Estmaton Exampl: Anothr Multnomal Exampl: Fshng Mod Pag 4 Consdr th Fshng mod and Incom xampl on Pag 494 of Mcroconomtrcs: Mthods and Applcatons (2005) by Camron, Adran Coln.; Trvd, P. K. Book avalabl as an book from UTD Lbrary. Fshng mods ar:, Pr, Prvat Boat, Chartr (boat). Indvduals choos on of ths. Each ndvdual rports own ncom and choc. Data: For ndvdual n: [yy nn,bbbbbbbbb, yy nn,pppppppp, yy nn,pppppppppppppp, yy nn,ccccccccccccc ; nn ] Statstcal modl wth paramtrs aa and bb for an ndvdual: P( a + b*incom y = ) = a + b *Incom a + b *Incom a + b *Incom a + b + Pr Pr + Pr vat Pr vat + Chartr Chartr *Incom P( apr + bpr *Incom y Pr = ) = a + b *Incom a + b *Incom a + b *Incom a + b + Pr Pr + Pr vat Pr vat + Chartr Chartr *Incom

15 Downloadng R Pag 5. Download standard R vrson from 2. Ths should crat R drctory and, undrnath t, drctors bn; doc; tc; nclud; lbrary; moduls; shar; src; Tcl 3. Among thos drctors, lbrary s mportant to us as w shall add stmaton spcfc and othr packags to ths lbrary. Lbrary drctory should hav 27 subdrctors. Downloadng Usful Packags Download th followng packags from xlsradwrt.zp, maxlk.zp, mlogt.zp, Ecdat.zp; nls2.zp nto nw drctors that you crat undr R-2.9.2\lbrary wth nams xlsradwrt, maxlk, mlogt, Ecdat. Th rols of ths packags ar: xlsradwrt; ## Rqurd for m(x)portng xcl fls maxlk; ## Rqurd for Maxmum Lklhood Estmaton mlogt; ## Rqurd for Multnomal Logt modl Ecdat; ## Intrstng Economtrc data fls nls2; ## Nonlnar last squars Chck th drctors n R-2.9.2\lbrary You should hav drctors 27 standard R drctors, plus 5 that you hav manually addd abov. At ths pont th ntr R drctory taks 86,392,832 byts on my hard dsk.

16 Startng R Pag 6. Start R (clck on th con on your dsktop, on th quck start button on your Start mnu, or clck on R-2.9.2\bn\Rgu.x). Ths wll start R wth a hom drctory of R I suggst that you kp your data n a dffrnt drctory, say C:\Dmrman\R\. 3. You hav to tll R whch drctory you want to work n. Go to Fl mnu and thn clck on to Chang dr(ctory) Mak R rad your packags In R, ssu commands > lbrary(xlsradwrt); > lbrary(maxlk); > lbrary(mlogt); > lbrary(ecdat); > lbrary(nls2);

17 Multnomal Exampl: Fshng Mod Pag 7 To rad th Fshng data, ssu command > data("fshng",packag="mlogt"); Brfly Fshng data ar about 82 ndvduals' fshng mod chocs. Data com from a survy conductd by Thomson and Crook (99). Issu > fx(fshng); to s what s nsd th Fshng datafram structur. It has row for ach ndvdual and 82 rows n total. It has 2 columns = 3 columns + 4 columns + 4 columns + column. Th frst 3 columns hav th chosn mod of fshng. Its prc and th probablty of catchng a fsh. Th nxt four columns hav th prc for ach mod of fshng. Ths prcs chang from on ndvdual to anothr as thy can dpnd on th locaton and accss of th ndvdual. Th nxt four columns hav th catch probablty for ach mod of fshng. Th last column contans th monthly ncom of th ndvdual.

18 Multnomal Exampl: Fshng Mod Data Manpulaton Issu command > Fsh <- mlogt.data(fshng,varyng=c(4:), shap="wd", choc="mod") to prpar Fshng data for multnomal logt rgrsson. Immdatly th numbr of rows bcom 4728 (=82*4), so on row for ach ndvdual and ach altrnatv. S th Fsh datafram by ssung >fx(fsh); Th frst four rows ar now for th frst ndvdual whos dntty s n th column namd chd. Th nxt column s th altrnatv. By puttng ndvdual d and altrnatv togthr w obtan th frst column, namd row.nams. Th mportant columns for our purpos ar namd mod and ncom. Manpulaton on th modl Pag 8 P( y P( y Pr P( y log P( y = ) = ) Pr = a a Pr + b + b = ) = a ) = Pr Pr *Incom *Incom a = xp( a + ( b Pr Pr b a + ( b Pr )* Incom b )* Incom), So a and b can b assumd to b zro to stmat th othr paramtrs wth rspct to ths two.

19 Multnomal Exampl: Fshng Mod Logt Rgrsson Issu command > summary(mlogt(mod~ ncom,data=fsh)); to stmat a,b paramtrs. R outputs Call: mlogt(formula = mod ~ ncom, data = Fsh) Frquncs of altrnatvs: bach boat chartr pr Nwton-Raphson maxmsaton gradnt clos to zro. May b a soluton 5 tratons, 0h:0m:0s g'(-h)^-g = 9.47E-30 Coffcnts : Estmat Std. Error t-valu Pr(> t ) altboat *** altchartr *** Paramtr a altpr *** altboat:ncom * altchartr:ncom Paramtr b altpr:ncom ** Sgnf. cods: 0 *** 0.00 ** 0.0 * Pag 9 ##Not that numbrs n rd ar th coffcnts n column MNL of Tabl 5.2 of Camron and Trvd (2005). Log-Lklhood: ; McFaddn R^2: ; Lklhood rato tst : chsq = 4.45 (p.valu= )

20 Multnomal Exampl: Fshng Mod Intrprtaton of Logt Rgrsson Pag 20 Insrtng th coffcnts log log log P( y P( y P( y P( y P( y P( y Boat Chartr Pr = ) ) = = ) ) = = ) ) = = * Incom /000 = * Incom /000 = * Incom /000 As th ncom ncrass, on s much lss lkly to fsh on th pr (-0.43), lss lkly to fsh on a chartr boat (-0.032) and mor lkly to fsh on a prvat boat (0.092).

21 Summary Pag 2 Outln Rgrsson Lnar Dmand Constant Elastcty Logt» Gvn D» Jont D Maxmum Lklhood Estmaton Logt Multnomal Logt

22 Anothr Exampl n R: Transportaton Mod Issu command > data("mod",packag="mlogt"); to rad th transportaton Mod data. Transportaton mods ar Car, Carpool, Bus, Ral. Data ar from 453 ndvduals. Issu > fx(mod) to s what s nsd th Mod datafram. It has row for ach ndvdual and 453 rows n total. It has 2 columns = column + 4 columns + 4 columns. Th frst column s th chosn mod of transportaton. Th nxt four columns ar th prc of ach mod. Th nxt four columns ar th duraton of ach mod. Issu command > markt <- mlogt.data(mod, alt.lvls=c("car", "carpool", "bus", "ral"), shap="wd", choc="choc"); to prpar Mod data for multnomal logt rgrsson. Pag 22

23 Issu command Transportaton Mod Rsults Pag 23 > summary(mlogt(choc~ cost.car+cost.carpool+cost.bus+cost.ral, data=markt)); to stmat paramtrs. R outputs Coffcnts : Paramtr b Paramtr a Estmat altcarpool altbus altral altcarpool:cost.car altbus:cost.car altral:cost.car altcarpool:cost.carpool altbus:cost.carpool altral:cost.carpool altcarpool:cost.bus altbus:cost.bus altral:cost.bus altcarpool:cost.ral altbus:cost.ral altral:cost.ral Estmatons ar wth rspct to Car probablty. If cost.car ncrass, th probablty of choosng anothr mod s hghr. If cost.carpool ncrass, probablty of carpool drops, thos of bus and ral ncras. If cost.bus ncrass, probablty of bus drops, thos of carpool and ral ncras. If cost.ral ncrass, probablty of carpool, bus both ral all drop wth rspct to car. That s th probablty of car ncrass.