BENFORD'S LAW AND PSYCHOLOGICAL BARRIERS IN. CERTAIN ebay AUCTIONS

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1 Eoomeris Workig Pper EWP0606 ISSN eprme of Eoomis BENFOR'S LAW AN PSYCHOLOGICAL BARRIERS IN CERTAIN eby AUCTIONS Oe F Lu & vid E. Giles* Asr eprme of Eoomis, Uiversiy of Viori Viori, B.C., Cd V8W Y Sepemer, 006 Usig geerlizios of Beford s Lw we es for he see of psyhologil rriers vrious prie levels i eby uios for professiol fooll ikes. Our empiril resuls idie h his hypohesis o e rejeed. Keywords: JEL Clssifiios: Psyhologil rriers, uio pries, Beford s lw C, C6, 44 * We re greful o Sophie Heslig for her ssise wih d olleio, d o o Ferguso for his helpful ommes.. Iroduio Auhor Co: vid Giles, ep. of Eoomis, Uiversiy of Viori, P.O. Box 700, STN CSC, Viori, B.C., Cd V8W Y; e-mil: dgiles@uvi.; FAX: (50) 7-64

2 . Iroduio Severl sudies hve ivesiged he possiiliy of psyhologil rriers or resise levels i fiil mrkes. os of hese sudies (e.g., oldso d Kim, 993; Ley d Vri, 994; Koedijk d Sork, 994; d Cyree e l., 999) ssume h he digis of he ssoied pries re uiformly disriued. However, e Ceuser e l. (997) use Beford s Lw o show h his ssumpio is ipproprie, d heir lysis quesios previous fidigs of suh rriers. I is url o sk wheher here re similr resise levels i oher mrkes, d uio prie d provides oe soure of iformio o ddress his quesio. Why wo eri idders go higher h some prie level? Wh re he hresholds for he ids? We lyze eby uio d for professiol fooll ikes d fid o sisil evidee of psyhologil rriers vriey of levels. The mehism ehid his reserh is o exr he sigifi digis of he uio pries o exmie wheher heir disriuios follow geerlized versios of Beford s Lw, whih re irodued i seio. Seio 3 desries our d d he use of -vlues defied y e Ceuser e l. (997) o pure iformio ou he relive proximiy o psyhologil rrier. Our empiril es resuls pper i seio 4, d some oludig remrks re give i seio 5.. Beford s Lw Beford (938) re-disovered Newom s (88) fidig h my urlly ourrig umeril d exhii speil feures wih regrd o he firs sigifi digi. He showed h he disriuio of firs sigifi digis is o-uiform. Beford s Lw idies h i umers from my rel-life d soures, he firs sigifi digi ours wih proiliy of lmos 3%, o.% (/9). The igger he digi, he less likely i is o our s he firs sigifi digi. Speifilly, le deoe he firs sigifi digi i umer N. If N = 95, he = 9; if N = , he = 4. Beford s Lw ses h Pr 0 k.[ = k] = log [ + (/ )] ; k =,,., 9. Wh is less well kow is h he joi disriuio for seod d higher sigifi digis is k i Pr.[ = d,..., k = dk ] = log0[ + ( di 0 ) ], k i=

3 for d {,,...,9} d d {0,,,...,9}, j >. So, he proiliy of he firs hree sigifi i j digis eig 5, d 8 is log 0 [ + ] = This disriuio is ivri o sle 58 (Pikhm, 96). Sok pries d x d (Geyer d Willimso, 004; Nigrii, 99; Nigrii d iermier 997) oey Beford s Lw, s do wiig ids for eri eby uios (Giles, 006). Beford s Lw provides foudio for esig for resise levels i mrkes. e Ceuser e l. (997) proposed es for psyhologil rriers usig ylil permuios of he oserved dily sok reurs d oluded h here ws o evidee of psyhologil rriers i vrious U.S., U.K. d Jpese sok mrke idies. Aggrwl d Luey (006) lysed gold pries i similr wy d foud evidee h psyhologil rriers he 00 s digis (prie levels suh s $00, $300, e.). 3. d -sisis Giles (006) hs show h he losig pries of suessful eby uios for pro-fooll gme ikes sisfy Beford s Lw, whih suggess he see of mrke ollusio mog idders or ierveio y sellers. Usig his d, our smple omprises ll,59 suessful uios for ikes for professiol U.S. fooll gmes i he eby eve ikes egory ewee 5 Novemer d eemer 004. As Beford s Lw is sisfied, we use hese d o osru sisis for esig for he exisee of vrious psyhologil rriers. We deoe he suessful uio pries s P, =,,, wih =,59. We osider poeil rriers he levels, 300, 400,, e. (oldso, 990; oldso d Kim, 993; Ley d Vri, 994; e Ceuser e l., 997), or: k 00, k =,,.., e. () I order o represe psyhologil rriers ll levels, we should osider rriers he levels, 0, 0,, 00, 00,,000, 0000,, e., for k 0, k =,,, 9; =, -, 0,, ; () or he more omprehesive se of levels, 0,,,, 00, 0,, 000, 00,, e., for k 0, k = 0,,, 99; =, -, 0,,. (3) We he eed o defie -vlues, whih re me o rry he iformio o he relive loseess o rrier. Correspodig o he ove levels defied i equio (), 3

4 = [ P]mod00, () where [P ] is he ieger pr of he pries, mod00 sds for modulo 00, d lerly, he vlues re mde up of he pir of rilig digis preedig he deiml poi. For rriers he levels defied y equio () d (3), he -vlues re (log p )mod [00 0 ]mod00 = (log P )mod [000 0 ]mod00 = (3) () seles he pir of digis i P preedig he deiml poi; seles he seod d hird sigifi digis; d piks he hird d fourh sigifi digis. y reserhers wrogly ssume h he -vlues re uiformly disriued i he see of psyhologil rrier, les i lrge smples. e Ceuser e l. (997) derived of he limi disriuios of he -vlues, whih we hve pplied i pr of he followig lysis: r + 0 limpr.( ) lim log i + + r = r k = k = L = (4) r i 00 = i= i = i + r = r k 9 i 0 + k + limpr.( = k) = log (5) i= i 0 + k i 0 + j 0 + k + limpr.( = k) = log. (6) 3 i= j = 0 i 0 + j 0 + k The limi proiliies i equios (4), (5) d (6) give us he relive frequeies over he smple =,,, of he -vlues whe here re o psyhologil rriers i he mrke, d he smple size eds o ifiiy. The frequeies for he -vlues i our d re expeed o e o-uiform for d, d uiform for. Alhough he uiformiy of seems o riolize he sdrd ssumpio of uiformiy esig for he presee of psyhologil rriers, oly pure psyhologil rriers he levels, 00, 300,, 3400, 3500,., e., d he series hose levels is o mulipliively regeerive (e Ceuser e l., 997). Therefore, resuls h re oied from ivesigig oly he he wrog olusio, d i is ruil h he d vlues ould led o vlues re lso osidered. 4

5 4. Tes resuls Firs, we es wheher our d exhii properies osise wih he limi disriuios i equios (4) (6). The ull hypohesis is h here re o psyhologil rriers i pries for pro-fooll ikes i he eby uio mrke. The Kolmogorov-Smirov (K-S) es d oher o-prmeri ess suh s iegred deviios ess of he Crmér-vo ises ype re o pproprie here euse of he irulr ure of our d (Giles, 006). Kuiper s es is similr o he fmilir K-S es, whih uses he differee sisis omies + d + d. Kuiper's es io oe sisi, mkig he es s sesiive i he ils s he medi, d mkig i ivri uder ylil d rsformios. If he empiril disriuio fuio for he smple is F (x), d he ull populio disriuio fuio is F ( ) 0 x, Kuiper s es sisi is: where V, + = + + = sup. ( F ( x) F0 ( x)) < x< = sup. ( F0 ( x) F ( x)) < x<. Aoher impor feure of Kuiper s es is h he ull disriuio of he es sisi is ivri o he hypohesized disriuio, for ll. Sephes (970) ules riil vlues for he ull disriuio of he rsformed sisi V / = V ( * /. ) The 0%, 5% d % riil vlues re.60,.747 d.00 respeively, for ll. Our vlues for, d re 0.947,.663 d 4.0 respeively. So, our vlues for d re osise wih he hypohesis of o psyhologil rriers, he 5% level, u our vlue for implies rejeio of his hypohesis gis he mos geerl of he lerive hypoheses osidered. * V Our goodess-of-fi ess re desiged o exmie wheher he empiril d hve hrerisis osise wih he limi disriuios for he -vlues disussed ove. e Ceuser e l. (997) 5

6 show h exremely lrge smple sizes re eeded for hese limi disriuios o e pplile, d he fiie-smple disriuios re d-depede. We follow e Ceuser e l. (998) d lso es for psyhologil rriers usig ylil permuios of he d. Psyhologil rriers geere ormlly low umer of -vlues i he 00 regio. Le Ψ e se of -vlues i he eighourhood of psyhologil rrier, suh s Ψ = {00}, Ψ = {99, 00, 0}, e. For some Ψ, he uio prie P is i eighourhood of psyhologil rrier if Ψ. Le I Ψ ( ) = i his se, zero oherwise, d defie ψ = = I Ψ ( ). For y d Ψ, suffiiely smll ψ sigls psyhologil rrier. The disriuio of ψ is d-speifi, u p-vlues for he hypohesis of o psyhologil rrier e oosrpped s he proporio of oosrpped ψ vlues less h he empiril ψ. Our experimel resuls, for 0,000 repeiios, pper i Tle. The ull hypohesis of o psyhologil rriers i he uio pries for fooll ikes o e rejeed. 5. Colusios Usig some geerlizios of Beford s Lw we fid h psyhologil rriers re se i eby uios for pro-fooll ikes. This is osise wih e Ceuser e l. s (997) fidig vrious sok idies, u orss wih Aggrwl d Luey s (006) resuls for gold pries. Our resuls hve impliios for users of eby s proxy iddig servie. For exmple, offerig mximum id of $00.0 i iipio h oppoes hve psyhologil rrier or jus uder $00 my o e vile sregy. I would e ieresig o exed his lysis o oher eby egories. 6

7 Tle : Boosrp Resuls Ψ ψ-sisi Boosrp p-vlue e (%) {00} {99,.,0} {98,.,0} {97,.,03} {95,.,05} {90,.,0} {00} {99,.,0} {98,.,0} {97,.,03} {95,..05} {90,.,0} {00} {99,.,0} {98,.,0} {97,.,03} {95,.,05} {90,.,0} Noe: {97,, 03} implies {97, 98, 99, 00, 0, 0, 03}, e. 7

8 Referees Aggrwl, R. d B.. Luey, 006. Psyhologil rriers i gold pries?, Review of Fiil Eoomis, i press. Beford, F., 938. The lw of omlous umers, Proeedigs of he Ameri Philosophil Soiey 78, Cyree, K. B.,. L. omi,. A. Louo d E. J. Yoio (999). Evidee of psyhologil rriers i he odiiol momes of mjor world sok idies, Review of Fiil Eoomis 8, e Ceuser,. K. J., G. hee d T. Shem, 998. O he hypohesis of psyhologil rriers i sok mrkes d Beford s lw, Jourl of Empiril Fie 5, oldso, R. G. d H. Y. Kim, 993. Prie rriers i he ow Joes idusril verge, Jourl of Fiil d Quive Alysis 8, Geyer, C. L. d P. P. Willimso, 004. eeig frud i d ses usig Beford s lw, Commuiios i Sisis B 33, Giles,. E. A., 006. Beford s lw d urlly ourrig pries i eri eby uios, Applied Eoomis Leers, i press. Koedijk, K. G. d P. A. Sork (994). Should we re? Psyhologil rriers i sok mrkes, Eoomis Leers 44, Kuiper, N. H., 959. Alerive proof of heorem of Birhum d Pyke, Als of hemil Sisis 30, 5-5. Ley, E. d H. Vri, 994. Are here psyhologil rriers i he ow-joes idex?, Applied Fiil Eoomis 4, 7-4. Newom, S. (88), Noe o he frequey of use of differe digis i url umers, Ameri Jourl of hemis 4, Nigrii,. J., 99. The deeio of iome x evsio hrough lysis of digil frequeies, Ph.. isserio, Uiversiy of Ciii. Nigrii,. J. d L. I. iermier, 997. The use of Beford's lw s id i lyil proedures, Audiig: A Jourl of Prie d Theory 6, Pikhm, R. S., 96. O he disriuio of firs sigifi digis, Als of hemil Sisis