Explaining and Forecasting Online Auction Prices and … 2008... · Explaining and Forecasting...

Explaining and Forecasting Online AuctionPrices and Their Dynamics UsingFunctional Data Analysis

Shanshan WANGDepartment of Mathematics University of Maryland College Park MD 20742 (shanshanmathumdedu)

Wolfgang JANKDepartment of Decision and Information Technologies Robert H Smith School of Business University ofMaryland College Park MD 20742 (wjankrhsmithumdedu)

Galit SHMUELIDepartment of Decision and Information Technologies Robert H Smith School of Business University ofMaryland College Park MD 20742 (gshmuelirhsmithumdedu)

Online auctions have become increasingly popular in recent years and as a consequence there is a growingbody of empirical research on this topic Most of that research treats data from online auctions as cross-sectional and consequently ignores the changing dynamics that occur during an auction In this article wetake a different look at online auctions and propose to study an auctionrsquos price evolution and associatedprice dynamics Specifically we develop a dynamic forecasting system to predict the price of an ongoingauction By dynamic we mean that the model can predict the price of an auction ldquoin progressrdquo and canupdate its prediction based on newly arriving information Forecasting price in online auctions is challeng-ing because traditional forecasting methods cannot adequately account for two features of online auctiondata (1) the unequal spacing of bids and (2) the changing dynamics of price and bidding throughout theauction Our dynamic forecasting model accounts for these special features by using modern functionaldata analysis techniques Specifically we estimate an auctionrsquos price velocity and acceleration and usethese dynamics together with other auction-related information to develop a dynamic functional fore-casting model We also use the functional context to systematically describe the empirical regularities ofauction dynamics We apply our method to a novel set of Harry Potter and Microsoft Xbox data and showthat our forecasting model outperforms traditional methods

KEY WORDS Autoregressive model Dynamics Electronic commerce Exponential smoothing Func-tional data Nonparametric Online auctions Smoothing spline

1 INTRODUCTION

Electronic commerce and in particular online auctions hascreated much public interest in recent years One of the maindrivers of this interest is eBay (wwweBaycom) On any givenday there are several million items dispersed across thousandsof categories for sale on eBay eBayrsquos popularity among thepublic is evidently quantified in the following numbers In 2004alone $342 billion was reported in gross merchandise volumeup from $238 billion in 2003 the cumulative confirmed regis-tered users totaled a record 1355 million which was a 43 in-crease over the 949 million users reported in 2003 and eBayhosts approximately 254000 stores worldwide with approxi-mately 161000 stores hosted in the United States alone Ac-cording to the Forrester Technographics survey close to 30 ofall US households had bid in an eBay online auction in 2004

The popularity of online auctions is also one reason for anincreasing amount of scholarly research Most of this researchhas been published in the economics marketing and informa-tion systems literature (eg Lucking-Reiley Bryan Prasad andReeves 2000 Roth and Ockenfels 2002 Ba and Pavlou 2002Bajari and Hortacsu 2003 2004 Bapna Goes Gupta andJin 2004) We find it surprising though that the statisticscommunity has hardly been involved to date because onlineauctions arrive with a huge amount of new and different data-related challenges and problems In particular a main fea-ture of online auction data is the change in dynamics of the

bidding process Most approaches to date tend to ignore thisdynamic information and treat data as cross-sectional by ag-gregating over the temporal dimension However such ap-proaches lead to a great loss in information Moreover stud-ies investigating bidding regularities also tend to be limitedto reporting summary statistics (such as the percent of bidsplaced in the last minute of the auction) A first step beyondthose limitations is via the use of advanced data visualiza-tion For instance Shmueli and Jank (2005) introduce graph-ical methods such as profile plots and statistical zooming andShmueli Jank Aris Plaisant and Shneiderman (2006) developa tool for interactive visualization of bidding data a tool thatcouples temporal with cross-sectional data Another step is tomodel bidding regularities directly Shmueli Russo and Jank(in press) for instance take a probabilistic approach for mod-eling the bid arrival process and proposed the BARISTA (BidARrivals In STAges) model a class of three-stage nonhomo-geneous Poisson processes that captures different bidding dy-namics including the self-similarity property that has been em-pirically observed in Roth and Ockenfels (2002) And finallytaking a different approach Jank and Shmueli (2006) propose

copy 2008 American Statistical AssociationJournal of Business amp Economic Statistics

April 2008 Vol 26 No 2DOI 101198073500106000000477

144

Wang Jank and Shmueli Forecasting Online Auction Prices 145

state-of-the-art functional data methodology for directly model-ing temporal bidding information and its dynamic change Forexample Jank and Shmueli (in press) use functional data analy-sis to find that the price dynamics change quite sharply over thecourse of an auction even for auctions for the same item Infact functional data analysis proves to be a very suitable toolfor the analysis of online auction data and more discussion onthe versatility of this toolset in the broader context of electroniccommerce research can be found in Jank and Shmueli (2006)We build upon the general versatility of functional data analysisin this work

In this article we develop a dynamic forecasting model to pre-dict price in online auctions By dynamic we mean a model thatoperates during the live auction and forecasts price at a futuretime point of the ongoing auction and as a by-product also atthe auction end This is in contrast to static forecasting modelsthat predict only the final price and that take into considerationonly information available at the start of the auction Such infor-mation may involve the length of the auction its opening priceproduct characteristics or the sellerrsquos reputation and may bemodeled using standard least squares regression analysis How-ever a static approach cannot account for information that be-comes available after the start of the auction (eg the amountof competition or current price level) and it cannot incorporatesuch information ldquoon the flyrdquo As we explain throughout thisarticle we find functional data analysis a very suitable tool fordeveloping dynamic price predictions

Forecasting price in online auctions can have benefits to dif-ferent auction parties For instance price forecasts can be usedto dynamically score auctions for the same (or similar) itemby their predicted price On any given day there are severalhundreds or even thousands of open auctions available espe-cially for very popular items such as Apple iPods or MicrosoftXboxes Dynamic price scoring can lead to a ranking of auc-tions with the lowest expected price Such a ranking could helpbidders focus their time and energy on only a few select auc-tions that is those that promise the lowest price Auction fore-casting can also be beneficial to the seller or the auction houseFor instance the auction house can use price forecasts to of-fer insurance to the seller This is related to the idea by Ghaniand Simmons (2004) who suggested offering seller insurancethat guarantees a minimum selling price In order to do sohowever it is important to correctly forecast the price at leaston average Whereas Ghani and Simmonsrsquo method is static innature our dynamic forecasting approach could potentially al-low more flexible features such as an ldquoInsure-It-Nowrdquo optionwhich would allow sellers to purchase insurance either at thebeginning of the auction or during the live auction (with a time-varying premium) Price forecasts can also be used by eBay-driven businesses that provide services to buyers or sellers Re-cently the authors were contacted by a company that providesbrokerage services for eBay sellers about using the dynamicforecasting system to create a secondary market for eBay-basedderivatives

While there has been some work related to price forecastingin online auctions our approach is novel particularly becauseof its dynamic nature (see also Hortacsu and Cabral 2005 forthe dynamics of seller reputation) As pointed out earlier Ghaniand Simmons (2004) using data-mining methods also predict

the end-price of online auctions however that method is staticand cannot account for newly arriving information in the liveauction Structural models to recover the bid distribution (Bajariand Hortacsu 2003) although able to more explicitly accountfor mechanism design are also focused on the final price Thedynamic nature of our forecasting approach is founded withinthe framework of functional data analysis (FDA) In FDA thecenter of interest is a set of curves shapes or objects or moregenerally a set of functional observations This is in contrastto classical statistics where the interest centers around a set ofdata vectors Although the concept of FDA has been around fora longer time the field has gained fast momentum lately dueto the work of Ramsay and Silverman (2000) There is quite abit of interest in the statistics literature to generalize classicalmodels and methods to the functional context Examples areregression analysis for a functional response (Farawary 1997)functional logistic regression (Escabias Aguilera and Valder-rama 2004) and generalized linear models for functional data(James 2002)

Our forecasting approach presents several methodologicaladditions to this stream of literature First to the best of ourknowledge forecasting functional data is a topic that has notbeen sufficiently addressed in the FDA literature to date Infact the use of functional data analysis presents several prac-tical and conceptual advantages for online auction data Tradi-tional methods for forecasting time series such as exponentialsmoothing or moving averages cannot be applied in the auc-tion context at least not directly because of the special datastructure Traditional forecasting methods assume that data ar-rive in evenly spaced time intervals such as every quarter orevery month In such a setting one trains the model on dataup to the current time period t and then uses this model topredict at time t + 1 Implied in this process is the importantassumption that the distance between two adjacent time peri-ods is equal which is the case for quarterly or monthly dataNow consider the case of online auctions Bids arrive in veryunevenly spaced time intervals determined by the bidders andtheir bidding strategies and the number of bids within a shortperiod of time can sometimes be very sparse and other times beextremely dense In this setting the interval between bids cansometimes be more than a day and at other times only secondsSecond online auctions even for the same product can expe-rience price paths with very heterogeneous price dynamics Byprice dynamics we mean the speed at which price travels dur-ing the auction and the rate at which this speed changes Tradi-tional models do not account for instantaneous change and itseffect on the price forecast This calls for new methods that canmeasure and incorporate this important dynamic information

Unevenly spaced data have always been very challenging fortraditional forecasting methods Some of the more recent andinfluential work in this area includes Engle and Russel (1998)who propose the use of the so-called autoregressive conditionalduration model for unevenly spaced transaction data such ashigh-frequency asset price data Similarly Shmueli et al (inpress) develop a class of three-stage nonhomogeneous Pois-son processes the BARISTA to model the changing bid arrivalprocess In both of these cases the underlying assumption isthat the discrete observations are manifestations of a continu-ous process Following this idea we take a functional data ap-proach Specifically we assume an underlying price curve that

146 Journal of Business amp Economic Statistics April 2008

describes the price increase or price evolution of an onlineauction However limitations in human perception and mea-surement capabilities allow us to observe this curve only at dis-crete time points Following functional principles we recoverthis curve from the observed data using appropriate smoothingtechniques

Another appeal of the functional data framework is the obser-vation that the price dynamics change quite significantly overthe course of an auction (Jank and Shmueli in press) By treat-ing auction price as a functional object and recovering the un-derlying price curve we obtain reliable estimates of the pricedynamics via derivatives of the smooth functional object andwe can consequently incorporate these dynamics into the fore-casting model This results in a novel and potentially very pow-erful forecasting system Although one may also approximatedynamics differently for example by using the first forwarddifference or the central difference such an approach is likelyto be much less accurate especially for applications with veryunevenly spaced data (as in the case of an online auction) andeven more so for approximating higher order derivatives

The article is organized as follows In Section 2 we brieflyintroduce the auction mechanism on eBay its data availabilityand the data used in this work In Section 3 we provide a sys-tematic description of the empirical regularities in online bid-ding dynamics Section 4 develops the forecasting model andwe apply the method to our data in Section 5 Section 6 con-cludes with final remarks

2 EBAYrsquoS ONLINE AUCTIONS

21 Auction Mechanism and Data Availability

The dominant auction format on eBay is a variant of thesecond-price sealed-bid auction (Krishna 2002) with ldquoproxybiddingrdquo This means that individuals submit a ldquoproxy bidrdquowhich is the maximum value they are willing to pay for theitem The auction mechanism automates the bidding processto ensure that the person with the highest proxy bid is inthe lead of the auction The winner is the highest bidderand pays the second highest bid (plus an increment) Un-like other auctions eBay has strict ending times ranging be-tween 1 and 10 days from the opening of the auction as de-termined by the seller eBay posts information on closed auc-tions for a duration of at least 15 days on its web site (seehttplistingsebaycompool1listingslistcompletedhtml) These publicly available postings make eBay an invalu-able source of rich bidding data

A typical bid history for a closed auction (see Fig 1) in-cludes information about the magnitude and time when eachbid was placed Additional information that is made availableincludes information about the seller and the bidders (eg username feedback ratings) information about the item sold (egname description) and information about the auction format(eg auction duration magnitude of the opening bid)

Every day on eBay there are several million items for salewhich means that a large amount of data are publicly avail-able Although such data can be collected manually simply bybrowsing through individual web pages in practice this is very

time consuming Therefore data are often collected automati-cally using web agents or web crawlers Web crawlers are soft-ware programs that visit a number of pages automatically andextract required information In that way high-quality informa-tion on a large number of auctions can be gathered in a shortperiod of time

22 Data Used in This Study

The data used in this study are 190 7-day auctions of Mi-crosoft Xbox gaming systems and Harry Potter and the Half-Blood Prince books obtained via a web crawler during themonths of August and September of 2005 At the time of writ-ing Xbox systems were very popular items on eBay and soldfor about $17998 (based on Amazoncom) Harry Potter bookswere also very popular items and sold for about $2799 onAmazoncom We can thus consider Xbox systems high-valueditems and can compare the performance of our method relativeto the lower valued Harry Potter books

For each auction in our dataset we collected the bid historywhich reveals the temporal order and magnitude of bids andwhich forms the basis of the functional forecasting model Fig-ure 2 shows a scatterplot of the bid history for a typical auctionWe can see that bids arrive at very unevenly spaced time inter-vals Although the number of incoming bids is sparse duringsome periods of the 7-day auction (especially in the middle) itcan be very dense at other times such as at the very beginningand especially at the end of the auction Figure 3 shows the scat-ter of bids aggregated over all 190 auctions Notice that mostof the bids arrive in the last minutes of the auction which as wehave pointed out earlier is a typical feature of eBayrsquos auctions

We also collected information on the auction format theproduct characteristics and bidder and seller attributes This in-formation is summarized in Tables 1 and 2 Unsurprisingly thehigh-valued items (Xbox) have on average a higher openingbid and a higher final auction price However it is noteworthythat the high-valued items see on average more competition(ie a larger number of bids) but feature auction participantswith a lower average bidder and seller rating Only few auc-tions in our dataset had made use of the secret reserve priceoption so the (numerical) difference between high- and low-valued items with respect to this feature may not be of practicalimportance More interestingly though most of the high-valueditems are used (over 90) which compares to only 46 usedHarry Potter books Table 2 also shows the distribution for thetwo variables early bidding and jump bidding Note that thesetwo variables are not directly observed but are derived from thebid history We comment on how we derived these two variablesnext

Early Bidding The timing of a bid plays an important rolein biddersrsquo strategic decision making For example Roth andOckenfels (2002) find evidence that many bidders place theirbids very late in the auction resulting in what is often calledldquobid snipingrdquo Shmueli et al (in press) found that an auction of-ten consists of three relatively distinct parts an early part withsome bidding activity a middle part with very little biddingand a final part with intense bidding In particular they foundthat the early bidding part of the auction typically extends untilabout day 15 of a 7-day auction Bapna et al (2004) charac-terized biddersrsquo strategies by among other things the timing


Figure 1 Partial bid history of an eBay Palm-515 auction On the leftmost side of the table we can see the biddersrsquo user names followed bytheir rating The stars indicate that an eBay member has achieved 10 or more feedback points The amount and time of the bids appear on theright

of their bid and found that bidders of the ldquoearly evaluatorsrdquotype place their first bid on average on day 14 of a 7-day auc-tion Following this empirical evidence we define an auction ascharacterized by early bidding if the first bid is placed withinthe first 15 days Table 2 shows the distribution of auctionswith early bidding We can see that among the high-valued auc-tions (Xbox) over 50 experience early bidding whereas this

number is much lower (28) for the low-valued items (HarryPotter) It may well be that bidders for high-valued items aremore inclined to bid early in order to establish a time prioritybecause in the case of two bidders with identical bids the bid-der with the earlier bid wins the auction

Jump Bidding We also include information about jumpbidding To that end one has to define what exactly determines


Figure 2 Bids placed in auction 75 of a Microsoft Xbox auctionThe horizontal axis denotes time (in days) the vertical axis denotesbid amount (in $)

a ldquojump bidrdquo that is what magnitude of difference between twoconsecutive bids constitutes an unusually high increase in thebidding process There exists only little prior investigation onthat topic For instance Easley and Tenorio (2004) study jumpbidding as a strategy in ascending auctions and defined jumpbids as bid increments that are larger than the minimum incre-ment required by the auctioneer (see also Isaac Salmon andZillante 2002 Daniel and Hirshleifer 1998) Bid incrementslarger than the minimum increment are relatively common oneBay (see Fig 4) We therefore focus here on increments thatresult in a very unusual ldquojumprdquo In order to define ldquounusualrdquo wetake the following approach For all auctions in our dataset wefirst examine all differences in bid magnitudes between pairs ofconsecutive bids The difference in consecutive bids leads to astep function of bid increments Figure 4 shows this step func-tion for all Xbox and Harry Potter auctions We can see thatmost auctions are characterized by only very small bid incre-ments (ie only very small steps) But we can also see that the

relevance of a jump depends on the scale (ie the value of theitem) and should be considered relative to this value

The distribution of the relative jumps relative to the averagefinal price is displayed in Figure 5 We see that the distributionfor both high- and low-valued items is very skewed In additionthe great majority of relative jumps regardless of item valueare smaller than 30 [see Fig 5(c)] We therefore define ajump bid as a bid that is at least 30 higher than the previousbid We define a corresponding indicator variable for auctionsthat have at least one jump bid (ie the variable jump bidding inTable 2 equals 1 if and only if the auction has at least one jumpbid) Table 2 shows that over 25 of the low-valued auctions(Harry Potter) see jump bidding whereas this number is onlyabout 9 for the high-valued auctions

3 FUNCTIONAL REGRESSION ANDAUCTION DYNAMICS

To understand the motivation for our forecasting model it isuseful to first take a closer look at eBay auction data We havepointed out earlier that the data are characterized by rapidlychanging price dynamics We illustrate this phenomenon in thissection by investigating the relationship between eBayrsquos auc-tion dynamics and other auction-related information This willalso lay the ground for the forecasting model that we describein the next section

We investigate the empirical regularities in eBayrsquos auctiondynamics using functional regression analysis Functional re-gression analysis is similar to classical regression in that itrelates a response variable to a set of predictors Howeverin contrast to classical regression where the response and thepredictors are vector valued functional regression operates onfunctional objects which can be a set of curves shapes or ob-jects In our application we refer to the continuous curve thatdescribes the price evolution between the start and the end ofthe auction as the functional object More details on functionalregression can be found in Ramsay and Silverman (2005)

Functional regression analysis involves two basic steps Inthe first step the functional object is recovered from the ob-served data We describe this step in the next section After

(a) (b)

Figure 3 Data for the 190 7-day auctions Panel (a) shows the amount of the bid versus the time of the bid aggregated across all auctionsPanel (b) shows a histogram of the distribution of the bid arrivals over 7 days Each bin corresponds to a 12-hour time interval


Table 1 Summary statistics for all continuous variables

Variable Item Count Mean Median Min Max Standard deviation

Opening bid Xbox 93 3622 2499 01 17500 3796Harry Potter 97 413 400 01 1099 326

Final price Xbox 93 13458 12500 2800 40500 6603Harry Potter 97 1156 1150 700 2050 240

Number of bids Xbox 93 2001 1900 200 7500 1276Harry Potter 97 847 800 200 2400 430

Seller rating Xbox 93 23204 4900 00 460400 61307Harry Potter 97 32599 12600 00 951900 99578

Bidder rating Xbox 93 3033 400 minus100 273600 13506Harry Potter 97 8321 1400 minus100 225800 22621

recovering the functional object we model the relationship be-tween a response object and a predictor object in a way that isconceptually very similar to classical regression We describethat step in Section 32

31 Recovery of the Functional Object

Functional data consist of a collection of continuous func-tional objects such as the price which increases in an onlineauction Despite their continuous nature limitations in humanperception and measurement capabilities allow us to observethese curves only at discrete time points Moreover the pres-ence of error results in discrete observations that are noisy re-alizations of the underlying continuous curve In the case ofonline auctions we observe only bids at discrete times whichcan be thought of as realizations from an underlying continuousprice curve Thus the first step in every functional data analysisis to recover from the observed data the underlying continu-ous functional object This is typically done using smoothingtechniques

The recovery stage is often initiated by some data preprocess-ing steps (eg Ramsay 2000) We denote the time that the ithbid was placed i = 1 nj in auction j j = 1 N by tij

Table 2 Summary statistics for all categorical variables

Variable Item Case Count Proportion

Reserve price Xbox Yes 4 430No 89 9570

Harry Potter Yes 1 103No 96 9897

Condition Xbox New 8 860Used 85 9140

Harry Potter New 52 5361Used 45 4639

Early bidding Xbox Yes 53 5699No 40 4301


Jump bidding Xbox Yes 9 968No 84 9032


NOTE ldquoCaserdquo is the category for the particular variable

Note that because of the irregular spacing of the bids the tijrsquosvary for each auction In our data N = 190 and 0 lt tij lt 7 Let

y( j)i denote the bid placed at time tij To better capture the bid-

ding activity especially at the end of the auction we transformbids into log scores To account for the irregular spacing welinearly interpolate the raw data and sample it at a common setof time points ti 0 le ti le 7 i = 1 n Then we can representeach auction by a vector of equal length

y( j) = (y( j)

1 y( j)n

) (1)

where y( j)i = y( j)(ti) denotes the value of the interpolated bid

sampled at time tiOne typically recovers the underlying functional object us-

ing smoothing techniques (see Ramsay and Silverman 2005)One very flexible and computationally efficient choice is the

(a)

(b)

Figure 4 Step function of bid increments for Microsoft Xbox sys-tems (a) and Harry Potter books (b)


(a) (b) (c)

Figure 5 Distribution of relative jumps for Xbox alone (a) Harry Potter alone (b) and both combined (c)

penalized smoothing spline (eg Simonoff 1996) Consider apolynomial spline of degree p

f (t) = β0 + β1t + β2t2 + middot middot middot +Lsum

l=1

βpl(t minus τl)p+ (2)

where τ1 τL is a set of knots and u+ denotes the posi-tive part of a function u The choices of L and p strongly in-fluence the departure of f from a straight line The degree ofdeparture can be measured by the roughness penalty PENm =int

Dmf (s)2 ds The penalized smoothing spline minimizes thepenalized residual sum of squares that is the jth smoothingspline f ( j) satisfies

PENSS( j)λm =

nsum

i=1

(y( j)

i minus f ( j)(ti))2 + λPEN( j)

m (3)

where the smoothing parameter λ controls the trade-off be-tween data fit and the smoothness of f ( j)

We base the selection of the knots on the bid arrival distri-bution Consider again Figure 3 which shows that over 60of the bids arrive during the last day of the auction Moreoverthe phenomenon of bid sniping (Roth and Ockenfels 2002) sug-gests that auctions should be sampled more frequently at theirlater stages Also Shmueli et al (in press) found that the bid in-tensity changes significantly during the last 6 hours Motivated

by this empirical evidence we place a total of 14 knots and dis-tribute the first 50 equally over the first 6 auction days Thenwe increase the intensity by placing the next 3 knots at every6 hours between day 6 and day 675 We again increase theintensity over the final auction moments by placing the remain-ing 4 knots every 3 hours up to the end of the auction Thisresults in a total set of smoothing spline knots given by ϒ =012345662565675681256875693757

We use smoothing splines of order m = 5 because this choiceallows for a reliable estimation of at least the first three deriva-tives of f (Ramsay and Silverman 2005) Note that our resultsare robust to changes in the knot allocation and with respect tothe choice of λ (see the App for a sensitivity study)

Figure 6 shows the recovered functional object for a typi-cal auction Figure 6(a) shows the curve pertaining to the priceevolution f (t) on the log-scale (solid line) together with theactual bids (crosses) and Figures 6(b)ndash6(d) show the first sec-ond and third derivatives of f (t) respectively The price evo-lution shows that price as expected from an auction increasesmonotonically toward the end However the rate of increasedoes not remain constant Although the price evolution resem-bles almost a straight line the finer differences in the change ofprice increases can be seen in the price velocity f prime(t) [the firstderivative of f (t)] or in the price acceleration f primeprime(t) (its secondderivative) For instance whereas the price velocity increases

(a) (b)

(c) (d)

Figure 6 Price dynamics for Xbox auction 10 (a) Price evolution (b) price velocity (c) price acceleration (d) price jerk


at the beginning of the auction it stalls after day 3 and remainslow until the end of day 5 only to rise again and to sharply in-crease toward the end of the auction Acceleration precedes ve-locity and we can see that a price deceleration over the first dayis followed by a decline in price velocity after day 1 In a simi-lar fashion the third derivative [ f primeprimeprime(t)] measures the change inthe second derivative The third derivative is also referred to asthe ldquojerkrdquo and we can see that the jerk increases steadily overthe entire auction duration indicating that price acceleration isconstantly experiencing new forces that influence the dynamicsof the auction Similar changes in auction dynamics have alsobeen noted by Jank and Shmueli (in press)

32 The Mechanics of Functional Regression Models

In this section we briefly review the general mechanics offunctional regression models for a functional response variableFor a more detailed description see Ramsay and Silverman(2005 chap 11)

Our starting point is an N times 1 vector of functional objectsy(t) = [y1(t) yN(t)] where N denotes the sample size thatis the total number of auctions in this case We use the sym-bol yj(t) in a rather generic way To model the price evolutionof an auction we set yj(t) equiv fj(t) However one of the advan-tages of functional data is that we also have estimates of thedynamics For instance to model an auctionrsquos price acceler-ation we set yj(t) equiv f primeprime

j (t) and so forth Classical regressionmodels the response as a function of one (or more) predic-tor variables and that is no different in functional regressionLet xi = [xi1 xiN] denote a vector of p predictor variablesi = 1 p xij can represent the value of the opening bid inthe jth auction or alternatively its seller rating Time-varyingpredictors can also be accommodated in this setting For in-stance xij(t) can represent the number of bids in the jth auctionat time t which we refer to as the current number of bids at tOperationally one can include such a time-varying predictorinto the regression model by discretizing it over a finite grid Letxijt denote xij(t) evaluated at t for a suitable grid t = t1 tGWe collect all predictors (time varying and time constant) intothe matrix X Typically this matrix has a first column of onesfor the intercept Also we could write X = X(t) to emphasizethe possibility of time-varying predictors but we avoid it forease of notation We then obtain the functional regression model

y(t) = XTβ(t) + ε(t) (4)

where the regression coefficient β(t) is time dependent reflect-ing the potentially varying effect of a predictor at varying stagesof the auction

Estimating the model (4) can be done in different ways (seeRamsay and Silverman 2005 for a description of different esti-mation approaches) We choose a pointwise approach that iswe apply regular least squares to (4) for a fixed t = tlowast and re-peat that process for all t on a grid t = t1 tG By smoothingthe resulting sequence of parameter estimates β(t1) β(tG)we obtain the time-varying estimate β(t)

Although functional regression is at least in principle verysimilar to classical least squares regression attention has to bepaid to the interpretation of the estimate β(t) We re-emphasizethat because the response is a continuous curve so is β(t) Thismakes reporting and interpreting the results different from clas-sical regression and slightly more challenging We show howthis is done in the next section

33 Empirical Application and Results

We fit the functional regression model (4) to our data andinvestigate two different models The first model investigatesthe effect of different predictor variables on the price evolutionthat is we set yj(t) equiv fj(t) The results are shown in Figure 7The second model investigates the effect of the same set of pre-dictors on the price velocity that is yj(t) equiv f prime

j (t) Those resultsare shown in Figure 8 For both models we use the nine pre-dictor variables described in Tables 1 and 2 Figures 7 and 8show the estimated parameter curves β(t) (solid lines) togetherwith 95 confidence bounds (dotted lines) indicating the sig-nificance of the individual effects The confidence bounds arecomputed pointwise by adding plusmn2 standard errors at each pointof the parameter curve (Ramsay and Silverman 2005)

Interpretation of the parameter curves has to be done withcare At any time point t β(t) evaluated at t indicates the signand strength of the relationship between the response (ie pricein Fig 7 velocity in Fig 8) and the corresponding predictorvariable The time-varying curve underlines the time-varyingnature of this relationship The confidence bounds help in as-sessing the statistical significance of that relationship

The insight from Figures 7 and 8 is summarized as followsMechanism Design We see that the choices that a seller

makes regarding the opening bid and inclusion of a secret re-serve price affect price according to what auction theory pre-dicts Higher opening bids and inclusion of a secret reserveprice are associated with higher price at any time during theauction (see Fig 7) What has not been shown in previous stud-ies though is the fact that this relationship for both predictorsholds throughout the auction rather than only at the end Evenmore interesting is the observation that high opening bids andusage of a reserve price influence the price dynamics negativelytoward the end of the auction by depressing the price velocity(see the negative coefficients in Fig 8) In both cases this ismost likely because price has already been inflated by the highopening bid andor the driving bidding force of the unobservedreserve price We describe each of these two effects in moredetail below

Opening bid The coefficient for opening bid in the regres-sion on price evolution curves is shown in the middle-left panel in Figure 7 Throughout the entire auction thecoefficient is positive indicating a positive relationshipbetween the opening bid and the price at any time dur-ing the auction However the coefficient does decreasetoward the end of the auction indicating that althoughthe positive relationship between opening bid and priceis strong at the start of the auction it weakens as theauction progresses One possible explanation is that atthe start of the auction in the absence of other bids auc-tion participants derive of information from the open-ing bid about their own valuation As the auction pro-gresses this source of information decreases in impor-tance and participants increasingly look to other sources(eg number of competitors number of bids and theirmagnitude communication with the seller etc) for de-cision making In addition the coefficient for openingbid in the regression on price velocity (Fig 8) is neg-ative throughout the auction and strongest at the start


Figure 7 Estimated parameter curves based on functional regression on the price evolution The x axis denotes the time of the 7-day auctionThe dotted lines correspond to 95 pointwise confidence bounds

Figure 8 Estimated parameter curves based on functional regression on the price velocity The x axis denotes the time of the 7-day auctionThe dotted lines correspond to 95 pointwise confidence bounds


and end of the auction This indicates that higher open-ing bids depress the rate of price increase especially atthe start and end of the auction Thus although higheropening bids are generally associated with higher pricesat any time in the auction (Fig 7) the auction dynamicsare slowed down by high opening bids In some sensehigher opening bids leave a smaller gap between the cur-rent price and a bidderrsquos valuation and therefore lessincentive to bid

Reserve price Auctions with a secret reserve price tend tohave a higher price throughout the auction but a reserveprice similar to the high opening bid appears to nega-tively influence price dynamics

Seller Characteristics The anonymity of the Internetmakes it hard to establish trust A sellerrsquos rating is typicallythe only sign that bidders look to in order to evaluate a sellerrsquostrustworthiness (Dellarocas 2003)

Seller rating Empirical research has shown that higherseller ratings are associated with higher final prices (egLucking-Reiley et al 2000 Ba and Pavlou 2002) Fig-ures 7 and 8 though show that higher seller ratingsare associated with lower prices during the entire auc-tion except for the end of the auction Moreover higherseller ratings are associated with faster price increasesbut again only toward the end of the auction

Item Characteristics The items in our dataset are charac-terized by condition (used vs new) and by value (high for Xboxand low for Harry Potter)

Usednew condition Overall new items achieve higherprices which is not surprising However the relationshipbetween item condition and price velocity is negativeThe price of new items appears to increase faster thanused items earlier but slows down later in the auctionwhen used item prices increase at a faster pace Perhapsthe uncertainty associated with used items leads biddersto search for more information (such as contacting theseller or waiting for other bids to be placed) therebyleading to delays in the price spurts

Item value As one might expect high-value items seehigher prices than low-value items throughout the auc-tion and this gap increases as the action proceeds Moreinteresting the price dynamics are very similar in bothlow- and high-value items until about day 6 but thenprice increases much faster for low-value items This isindicative of later bidding on low-value auctions a phe-nomenon that we observed in the exploratory analysisBidders are more likely to bid early on high-value itemsperhaps to establish their time priority

Bidding Characteristics Our data contain four variablesthat capture effects of bidders and bidding namely the cur-rent number of bids as a measure of level of competition thecurrent average bidder rating as a measure of bidder experi-ence and early and jump bidding as a measure of differentbidding strategies All variables share the feature that their im-pact changes sometime during the auction thereby creating twophases in how they affect the price evolution and the price ve-locity In some cases different strategies (such as early vs late

bidding) lead to direct impacts on the price but to more subtleeffects on the price dynamics For instance the current num-ber of bids affects the price evolution directly during the firstpart of the auction whereas during the second part of the auc-tion it affects price only through the price dynamics The op-posite phenomenon occurs with early bidding Thus functionalregression reveals that (1) bidding appears to have two phasesand (2) price can be affected either directly or indirectly by thebidding process

Current number of bids This factor influences the price evo-lution during the first part of the auction with more bidsresulting in higher prices However this effect decreasestoward the end of the auction where it only influencesprice through increasing price dynamics

Early bidding The effect of this factor switches its directionbetween the first and second part of the auction At firstauctions with early bidding have higher dynamics butlower price evolution but later this effect is reversedThis means that early bidding manifests itself as earlyincreased price dynamics which later turn into higherprice curves

Jump bidding Auctions with jump bidding tend to have gen-erally higher price curves and especially high price dy-namics close to the end of the auction The jump biddingobviously causes the price curve to jump and the pricevelocity to peak at the time of the jump bid When av-eraging over the entire set of auctions the effect of ajump bid has its highest impact on price at the end ofauction This is not necessarily in contrast to Easley andTenorio (2004) who examined the timing of jump bid-ding (rather than the time of its highest impact) Theyfound that jump bidding is more prevalent early in theauction which they explained by the strategic value ofjump bidding for bidders

Current average bidder rating It appears that higher ratedbidders are more likely to bid when the price at the startof the auction is high compared to lower rated bidders(as reflected by the positive coefficient during the firstday) But then they are able to keep the price lowerthroughout the auction (the coefficient turns negative)Toward the end of the auction though participation ofhigh-rated bidders leads to faster price increases whichreduces the final price gap due to bidder rating

4 DYNAMIC AUCTION FORECASTING VIAFUNCTIONAL DATA ANALYSIS

We now describe our dynamic forecasting model We haveshown in the previous section how unequally spaced data canbe overcome by moving into the functional context and alsothat online auctions are characterized by changing price dynam-ics Our forecasting model consists of four basic componentsthat capture price dynamics price lags and information relatedto sellers bidders and auction design First we describe thegeneral forecasting model which is based on the availability ofprice dynamics Then we describe how to obtain forecasts forthe price dynamics themselves


41 The General Forecasting Model

Our model combines all information that is relevant to priceWe group this information into four major components (1) sta-tic predictor variables (2) time-varying predictor variables(3) price dynamics and (4) price lags

Static predictor variables are related to information that doesnot change over the course of the auction This includes theopening bid the presence of a secret reserve price the sellerrating and item characteristics Note that these variables areknown at the start of the auction and remain unchanged overthe duration of the auction Time-varying predictor variablesare different in nature In contrast to static predictors time-varying predictors do change during the auction Examples oftime-varying predictors are the number of bids at time t or thenumber of bidders and their average bidder rating at time tPrice dynamics can be measured by the price velocity the priceacceleration or both Finally price lags also carry importantinformation about the price development Price lags can reachback to price at times t minus 1 t minus 2 and so on This correspondsto lags of order 1 2 and so forth

We obtain the following dynamic forecasting model Lety(t|t minus 1) denote the price at time t given all informationobserved until t minus 1 For ease of notation we write y(t) equivy(t|t minus 1) Our forecasting model can then be formalized as

y(t) = α +Qsum

i=1

βixi(t) +Jsum

j=1

γjD( j)y(t) +

Lsum

l=1

ηly(t minus l) (5)

where x1(t) xQ(t) is the set of static and time-varying pre-dictors D( j)y(t) denotes the jth derivative of price at time t andy(t minus l) is the lth price lag The resulting h-step-ahead predic-tion given information up to time T is then

y(T + h|T) = α +Qsum

i=1

βixi(T + h|T) +Jsum

j=1

γjD( j)y(T + h|T)

+Lsum

l=1

ηly(T + h minus 1|T) (6)

The model (6) has two practical challenges (1) price dy-namics appear as coincident indicators and must therefore beforecasted before forecasting y(T + h|T) and (2) the static pre-dictor variables among the xirsquos do not change their value overthe course of the auction and must therefore be adapted torepresent time-varying information We explain these two chal-lenges in more detail in the following discussion and presentsome solutions

42 Forecasting Price Dynamics

The price dynamics D( j)y(t) enter (6) as coincident indica-tors This means that the forecasting model for price at time tuses the dynamics from the same time period However be-cause we assume that the observed information extends onlyuntil t minus 1 we must obtain forecasts of the price dynamics be-fore forecasting price This process is described next

We model D( j)y(t) as a polynomial in t with autoregressive(AR) residuals We also allow for covariates xi The rationalefor these covariates is that dynamics are strongly influenced by

certain auction-related variables such as the opening bid (seeFig 8) This results in the following model for the price dy-namics

D( j)y(t) =Ksum

k=0

aktk +Psum

i=1

bixi(t) + u(t) t = 1 T (7)

where u(t) follows an autoregressive model of order R

u(t) =Rsum

i=1

φiu(t minus i) + ε(t) ε(t) sim iid N(0 σ 2) (8)

To forecast D( j)y(t) based on (7) we first estimate theparameters a0a1 aK b1 bP and estimate the resid-uals Then using the estimated residuals u(t) we estimateφ1 φR This results in a two-step forecasting procedureGiven information until time T we first forecast the next resid-ual via

u(T + 1|T) =Rsum

i=1

φiu(T minus i + 1) (9)

and then use this forecast to predict the corresponding pricederivative

D( j)y(T + 1|T)

=Ksum

k=0

ak(T + 1)k +Psum

i=1

bixi(T + 1|T) + u(T + 1|T) (10)

In a similar fashion we can predict D( j)y(t) h steps ahead

D( j)y(T + h|T)

=Ksum

k=0

ak(T + h)k +Psum

i=1

bixi(T + h|T) + u(T + h|T) (11)

43 Integrating Static Auction Information

The second structural challenge that we face is related to theincorporation of static predictors into the forecasting modelTake for instance the opening bid The opening bid is staticin the sense that its value is the same throughout the auctionthat is x(t) equiv x forall t Ignoring all other variables we can rewritemodel (5) as

y(t) = α + βx (12)

Because the right-hand side of (12) does not depend on t theleast squares estimates of α and β are confounded

The problem outlined previously is relatively uncommon intraditional time series analysis because it is usually only mean-ingful to include a predictor variable in an econometric modelif the predictor variable itself carries time-varying informationHowever the situation is different in the context of forecast-ing online auctions and may merit the inclusion of certain staticinformation The opening bid for instance may in fact carryvaluable information for predicting price in the ongoing auc-tion Economic theory suggests that sometimes bidders deriveinformation from the opening bid about their own valuationbut the impact of this information decreases as the auction pro-gresses What this suggests is that the opening bid can influ-ence biddersrsquo valuations and therefore also influence price


Figure 9 Flowchart of dynamic forecasting model

What this also suggests is that the opening bidrsquos impact on pricedoes not remain constant but should be discounted graduallythroughout the auction

One way of discounting the impact of a static variable x is viaits influence on the price evolution That is if x has a strongerinfluence on price at the beginning of the auction then it shouldbe discounted less during that period On the other hand ifx only barely influences price at the end of the auction thenits discounting should be larger at the end of the auction Oneway of measuring the influence of a static variable on the pricecurve is via functional regression analysis as described in Sec-tion 32 Let β(t) denote the slope coefficient from the func-tional regression model y(t) = α(t) + β(t)x + ε similar to (4)thus β(t) quantifies the influence of x on y(t) at any time t Wecombine x and β(t) and compute the influence-weighted ver-sion of the static variable x as

x(t) = xβ(t) (13)

x(t) now carries time-varying information and can conse-quently be included as a time-varying predictor variable

As pointed out earlier our dynamic forecasting model con-sists of two basic parts one part forecasts the price dynamicsand the other part uses these forecasted dynamics as input intothe price forecaster A flowchart of our algorithm is shown inFigure 9

5 EMPIRICAL APPLICATION ANDFORECASTING COMPARISON

We apply the forecasting methodology to our dataset of190 eBay auctions Model fitting and prediction are imple-mented using modules of the R software package We randomlypartition our data into a training set (70 or 130 auctions) anda validation set (30 or 60 auctions) We use the training setto estimate the model and we test the method on the valida-tion set For testing we first remove all price information fromthe last auction day and then compare our results with the trueprice

51 Model Estimation

Estimation of the model is done in two steps We first esti-mate model (7) and then use the forecasted dynamics as inputsin model (6)

511 Modeling Price Dynamics Model (7) is fitted iter-atively This leads to a best fitting model with a quadratic trend(K = 2) and three predictors (P = 3) where x1 x2 and x3 arethe influence-weighted variants of the opening bid the itemvalue and jump bidding respectively The resulting residualsare AR(1) that is R = 1 in (8) Figure 10 shows the signifi-cance of x1 x2 and x3 over the last auction day in the form ofsignificance curves Because we use x1 x2 and x3 to predict

Figure 10 p-value curves for x1 x2 and x3 over the last auctionday Consistent with the three predictors we denote 1 = opening bid2 = item value and 3 = jump bidding The dotted horizontal linemarks the 5 significance level


price dynamics for all time points between days 6 and 7 thesignificance of individual predictors may be different at differ-ent time points Indeed Figure 10 shows that while jump bid-ding (line 3 in the graph) is insignificant during the beginning ofday 6 (with a huge spike around day 63) it becomes significanttoward the end of the auction The opposite is true for the itemvalue which becomes insignificant at the end of the auction Onthe other hand the opening bid remains significant throughoutthe last day This change in significance suggests that the ldquobur-den of predictionrdquo does not remain equally distributed over allthree predictors In fact the burden is heavier on item value atthe beginning of the auction and then shifts to jump bidding atthe end of the auction Meanwhile the opening bid carries thesame prediction burden throughout the last day

Figure 11 illustrates the forecasting performance on the hold-out sample We chose four representative auctions and com-pared the true price velocity over the last day (solid line) withits prediction based on model (7) (broken line) We see that themodel captures the true price dynamics very well

512 Modeling Price We estimate model (5) using thefollowing 11 predictor variables (grouped by their type)

Influence-weighted static predictors Opening bid Reserveprice Seller rating Item condition Item value Earlybidding and Jump bidding

Time-varying predictors Current number of bids and Cur-rent average bidder rating

Price dynamics Price velocityPrice lags Price at time t minus 1

Figure 12 shows the significance curves for all 11 predic-tors Interestingly reserve price seller rating current numberof bids and current average bidder rating are insignificant atthe start of the auction Whereas the significance of the lat-ter two increases toward the end of the auction seller ratingbecomes even more insignificant On the other hand whereasreserve price becomes highly significant at the end of the auc-tion item condition which is significant at the start becomes

(a) (b)

Figure 12 p-value curves for all 11 predictors over the last auc-tion day 1 = opening bid 2 = reserve price 3 = seller rating4 = item condition 5 = item value 6 = early bidding 7 = jump bid-ding 8 = current number of bids 9 = current average bidder rating10 = price velocity and 11 = price at time t minus1 The dotted horizontalline denotes the 5 significance level Panel (b) shows the informationfrom panel (a) ldquozoomed inrdquo for p values between 0 and 08

insignificant at the end All remaining predictors remain at (orbelow) the 5 significance mark throughout the entire auction

52 Price Forecasting

After estimating the model using the training set we apply itto the validation set to obtain forecasts for the price on the lastday Because we removed all price information from the lastauction day we can measure prediction accuracy by comparingthe true price with our forecast

Figure 13 illustrates the forecasting method for four sampleauctions Each of the four graphs in Figure 13 contains threeseparate pieces of information (1) the actual current auctionprice (a step function) (2) the functional price curve and (3) theforecasted price curve The actual current auction price is the

Figure 11 Forecasting performance of model (7) over the last auction day for four sample auctions ( true price velocity forecastprice velocity)


Figure 13 Dynamic forecasting results of last-day price for four sample auctions ( current auction price functional price curveforecast price curve) The x-axis denotes time the y-axis denotes price in $

price observed during the live auction The functional pricecurve is the smoothed functional object based on the observedprices And the forecasted price curve is our forecast based onmodel (5)

Notice that Figure 13 reveals two levels of ldquotruthrdquo The firstis on the functional level which compares true and forecastedcurves Our forecasting method operates on the functional ob-jects and predicts the price curves In that sense the closerthe forecasted curve is to the functional price curve the betterits functional prediction performance Indeed Figure 13 showsthat the functional and forecasted curves are generally veryclose However the functional price curve is merely an approx-imation of the live auction price Therefore a second level oftruth is revealed by comparing the forecasted price curve withthe actual current auction price On this level the discrepancyis larger which is not surprising The quality of the forecast-ing output is only as good as its input If the quality of the in-put is poor (ie functional objects that do not approximate thecurrent auction price well) then not much can be expected ofthe forecasted output This underlines the importance of gen-erating high-quality functional objects The most reliable wayof checking the quality of the functional objects is via visu-alization Jank Shmueli Plaisant and Shneiderman (in press)proposed several ways of inspecting functional data visuallyAnother way of guaranteeing the quality of the results is viasensitivity studies with respect to the allocation of knots andthe choice of the smoothing parameter (see the App)

521 Forecast Accuracy We measure forecast accuracyon the validation set using the mean absolute percentage error(MAPE) We compute the MAPE in two different ways sim-ilar to Figure 13 once between the forecasted curve and thetrue functional curve (MAPE1) and then between the forecastedcurve and the actual current auction price (MAPE2) The resultis shown in Figure 14

Naturally MAPE2 is higher than MAPE1 because it isharder to reach the second level of ldquotruthrdquo compared with thefirst level MAPE1 is at least on average less than +5 for the

entire prediction period (ie over the last day) implying thatour model has a very high forecasting accuracy MAPE2 is a bitlarger in magnitude because of the inevitable variation in fittingsmoothing splines to the observed data The widths of the con-fidence bounds underline the heterogeneity across all auctionsin our dataset

522 Forecast Accuracy by Auction Characteristics Fo-recast accuracy can lead to new insight on the empiricalregularities of bidding when breaking it down by different auc-tion characteristics We therefore compare forecast accuracyfor different levels of the opening bid secret reserve price itemcondition and value seller reputation bidder experience com-petition and early and jump bidding Table 3 shows the re-sults We find that the error is generally relatively small nolarger than 20 of the true functional price curve and no largerthan 36 of the actual final auction price But there are sub-tle differences across the different variables The error is largerwhen forecasting new items as compared to used items High-value items on the other hand have a smaller error than low-value items which could be attributed to the fact that whenthe stakes are higher bidders spend more time researching the

(a) (b)

Figure 14 Mean absolute percentage errors (MAPEs)MAPE1 (a) is the error between the forecasted price curve andthe true functional price curve MAPE2 (b) is the error between theforecasted price curve and the actual current auction price The dottedlines correspond to the 5th and 95th percentiles


Table 3 Mean absolute percentage errors (MAPEs) broken upby different variables

MAPE1 MAPE2

Variable Case Mean Standard error Mean Standard error

Reserve policy Yes 08 NA 16 NANo 12 02 23 02

Condition New 17 05 31 05Used 09 01 19 02

Item value High 07 01 14 01Low 16 04 31 04

Opening bid High 06 01 14 02Low 20 04 36 04

Seller rating High 14 04 26 04Low 09 02 20 03

Average bidder High 15 04 30 04rating Low 09 02 17 02

Number of bids High 13 03 24 03Low 10 02 22 03

Early bidding Yes 11 02 22 03No 12 03 24 04

Jump Yes 09 01 27 03No 13 03 21 03

NOTE MAPE1 is the error between the forecasted final price and the functional finalprice MAPE2 is the error between the forecasted final price and the actual final price Thestandard error of reserve price is ldquoNArdquo because there is only one auction with a reserveprice in the validation set

item and thus price dispersion is lower Not surprisingly auc-tions with a high opening bid have a smaller forecasting errorbecause when the opening bid is high and the itemrsquos value isrelatively well known as in our situation there is less uncer-tainty about the possible outcomes of the auction Lower sellerreputation results in more accurate forecasts This may be due

to the fact that higher seller ratings often elicit price premi-ums (Lucking-Reiley et al 2000) thereby increasing the pricevariance Bidder experience has a similar impact on forecastingaccuracy As for bidding competition (captured by the numberof bids) higher competition results in larger variation in theforecast errors It is also interesting to note that early biddinghas barely any effect on the predictability of an auction thisagain is different for jump bidding

523 Comparison With Exponential Smoothing Tobenchmark the performance of our method we compare it todouble exponential smoothing Double exponential smoothingis a popular short-term forecasting method that assigns expo-nentially decreasing weights as the observations become lessrecent and also takes into account a possible (changing) trendin the data This method cannot be applied directly to the rawbid data because of their uneven spacing Functional objectsonce again come to the rescue that is we apply double ex-ponential smoothing to a grid of evenly spaced values from thefunctional curve The dashed lines in Figure 15 show the perfor-mance of exponential smoothing for the same four auctions asin Figure 13 We see that the predictions based on exponentialsmoothing are very far from the true auction price and even farfrom the true functional price curve Table 4 compares our fore-casting system with exponential smoothing in terms of MAPEWe find that the forecast error of exponential smoothing is morethan twice the error of our forecasting system

6 CONCLUSION AND FUTURE DIRECTIONS

In this article we propose a dynamic forecasting model forprice in online auctions We set up the forecasting problem inthe context of functional data analysis by treating the price evo-lution in an auction as a functional object This leads to a novel

Figure 15 Comparison of forecasting results of last-day price evolution of individual auctions between using the exponential smoothingmethod and our dynamic forecasting method ( current auction price functional price curve exp smoothing forecast dynamicforecast)


Table 4 Comparison of forecasting accuracy between our dynamicforecasting model and exponential smoothing

MAPE1 MAPE2

Method Mean Standard error Mean Standard error

Dynamic forecasting 12 02 23 02Exponential smoothing 42 03 49 03

NOTE The forecasting accuracy is measured by the mean absolute percentage error(MAPE)

use of FDA for forecasting which has not been considered inthe literature to date It is also new in that it allows dynamicforecasting of an ongoing auction The functional setup allowsus to (1) represent the extremely unevenly spaced series of bidsin a compact form (2) estimate price dynamics via the deriv-atives of the smooth functional objects and integrate this dy-namic information into the forecaster and (3) incorporate bothstatic and time-varying information about the auction into theforecasting system Combining the dynamics with the static andtime-varying information enables forecasting the price in ongo-ing live auctions for different types of products The functionalapproach allows us also to investigate regularities of the biddingdynamics as a function of relevant auction dimensions

We apply our forecasting system to real data from eBay ona diverse set of auctions and find that the combination of sta-tic and time-varying information creates a powerful forecastingsystem The model produces forecasts with low errors and itoutperforms standard forecasting methods such as double ex-ponential smoothing that severely underpredict the price evo-lution This also shows that online auction forecasting is notan easy task Whereas traditional methods are hard to applythey are also inaccurate because they do not take into accountthe dramatic change in auction dynamics Our model on theother hand achieves high forecasting accuracy and accommo-dates the changing price dynamics well

This work can be extended in several ways In this articlewe focus on auctions of the same duration The lessons learnedfrom this work can be used to extend the model to auctionsof different length Combining auctions of different durationsis challenging because it involves registration of misalignedcurves (see eg Ramsay and Silverman 2005 James 2004)However in the auction context the misaligned curves are ofdifferent lengths which poses additional difficulties Anotherextension is to incorporate a concurrency component In onlineauctions bidders have the option to inspect and follow multi-ple auctions at the same time This places new challenges formodeling especially in the functional framework In a relatedseries of articles (see Jank and Shmueli 2007 Hyde Jank andShmueli 2006) we propose some solutions via visualization of

concurrent functional objects and modeling of concurrent finalprices Finally further research is required to better understandthe exact role of price dynamics and their impact on economictheory One possible avenue is the exploration of functional dif-ferential equation models in the auction context (see Jank andShmueli in press)

ACKNOWLEDGMENTS

This research was funded in part by NSF grant DMI-02-05489 The authors gratefully acknowledge support by theCenter of Electronic Markets and Enterprises University ofMaryland

APPENDIX SENSITIVITY ANALYSIS

The choice of our smoothing parameters is governed by rea-sonable fit Because there is a wide range of choices that lead toreasonable curve approximations we investigate the sensitivityof the forecasting accuracy to different choices of the knots andsmoothing parameter λ Table A1 shows the forecasting accu-racy in terms of MAPE1 (between the forecasted price and thefunctional curve) and MAPE2 (between the forecasted curveand the actual current auction price) for three different sets ofknots Similarly Table A2 shows the sensitivity to the choiceof λ In both cases we see that the magnitude of the MAPEvalues remains in the area of 10ndash30 with very little change inthe standard errors

Table A2 Sensitivity analysis of λ selection (knots fixed to ϒ2)

MAPE1 MAPE2

λ Mean Standard error Mean Standard error

1 28 04 32 04

3 23 03 28 03

5 21 03 28 03

7 18 02 26 03

9 16 02 25 021 16 03 27 035 15 03 27 03

10 12 02 24 0215 12 02 23 0220 12 02 23 0225 12 02 23 0230 12 02 23 0240 11 02 23 0250 11 02 23 02

[Received December 2004 Revised May 2006]

Table A1 Sensitivity analysis of knot selection based on different knot scenarios

MAPE1 MAPE2

Set Knots Mean Standard error Mean Standard error

ϒ1 0 1 2 3 4 5 6 625 65 675 68750 7 18 05 29 04ϒ2 0 1 2 3 4 5 6 625 65 675 68125 68750 69375 7 12 02 23 02ϒ3 0 5 1 15 2 3 4 5 6 625 65 675 68125 68750 69375 7 26 04 31 03

NOTE ϒ2 is the one used in this article


REFERENCES

Ba S and Pavlou P A (2002) ldquoEvidence of the Effect of Trust BuildingTechnology in Electronic Markets Price Premiums and Buyer BehaviorrdquoMIS Quarterly 26 269ndash289

Bajari P and Hortacsu A (2003) ldquoThe Winnerrsquos Curse Reserve Prices andEndogenous Entry Empirical Insights From eBay Auctionsrdquo Rand Journalof Economics 3 329ndash355

(2004) ldquoEconomic Insights From Internet Auctionsrdquo Working Pa-per w10076 National Bureau of Economic Research

Bapna R Goes P Gupta A and Jin Y (2004) ldquoUser Heterogeneity and ItsImpact on Electronic Auction Market Design An Empirical ExplorationrdquoMIS Quarterly 28 1

Daniel K D and Hirshleifer D (1998) ldquoA Theory of Costly Sequential Bid-dingrdquo technical report Northwestern University Kellogg Graduate Schoolof Management

Dellarocas C (2003) ldquoThe Digitization of Word-of-Mouth Promise andChallenges of Online Reputation Mechanismsrdquo Management Science 491407ndash1424

Easley R F and Tenorio R (2004) ldquoJump Bidding Strategies in Internet Auc-tionsrdquo Management Science 50 1407ndash1419

Engle R F and Russel J R (1998) ldquoAutoregressive Conditional DurationA New Model for Irregularly Spaced Transaction Datardquo Econometrica 661127ndash1162

Escabias M Aguilera A M and Valderrama M J (2004) ldquoModelling En-vironmental Data by Functional Principal Component Logistic RegressionrdquoEnvironmetrics 16 95ndash107

Farawary J J (1997) ldquoRegression Analysis for a Functional Responserdquo Tech-nometrics 39 254ndash261

Ghani R and Simmons H (2004) ldquoPredicting the End-Price of OnlineAuctionsrdquo in Proceedings of the International Workshop on Data Miningand Adaptive Modelling Methods for Economics and Management heldin conjunction with the 15th European Conference on Machine Learning(ECMLPKDDD) [on-line] available at httpwwwaccenturecomxdocenservicestechnologypublicationspricepredictionpdf

Hortacsu A and Cabral L (2005) ldquoDynamics of Seller Reputation Theoryand Evidence From eBayrdquo working paper University of Chicago

Hyde V Jank W and Shmueli G (2006) ldquoInvestigating Concurrency inOnline Auctions Through Visualizationrdquo The American Statistician 60241ndash250

Isaac M Salmon T C and Zillante A (2002) ldquoA Theory of Jump Biddingin Ascending Auctionsrdquo Journal of Economic Behavior and Organization62 144ndash164

James G M (2002) ldquoGeneralized Linear Models With Functional PredictorsrdquoJournal of the Royal Statistical Society Ser B 64 411ndash432

(2004) ldquoCurve Alignment by Momentsrdquo unpublished manuscriptJank W and Shmueli G (in press) ldquoStudying Heterogeneity of Price Evolu-

tion in eBay Auctions via Functional Clusteringrdquo submitted to Handbook onInformation Series Business Computing eds G Adomavicius and A GuptaElsevier

(2006) ldquoFunctional Data Analysis in Electronic Commerce ResearchrdquoStatistical Science 21 155ndash166

(2007) ldquoModeling Concurrency of Events in Online Auctions viaSpatio-Temporal Semiparametric Modelsrdquo Journal of Royal Statistical So-ciety Ser C 60 1ndash27

Jank W Shmueli G Plaisant C and Shneiderman B (in press) ldquoVi-sualizing Functional Data With an Application to eBayrsquos Online Auc-tionsrdquo in Handbook on Computational Statistics on Data Visualization edsC-H Chen W Haumlrdle and A Unwin Heidelberg Springer-Verlag

Krishna V (2002) Auction Theory San Diego Academic PressLucking-Reiley D Bryan D Prasad N and Reeves D (2000) ldquoPennies

From eBay The Determinants of Price in Online Auctionsrdquo technical reportUniversity of Arizona

Ramsay J O (2000) ldquoFunction Components of Variation in HandwritingrdquoJournal of the American Statistical Association 95 9ndash15

Ramsay J O and Silverman B W (2000) Functional Data Analysis (1st ed)New York Springer-Verlag

(2005) Functional Data Analysis (2nd ed) New York Springer-Verlag

Roth A E and Ockenfels A (2002) ldquoLast-Minute Bidding and the Rulesfor Ending Second-Price Auctions Evidence From eBay and Amazon on theInternetrdquo American Economic Review 92 1093ndash1103

Shmueli G and Jank W (2005) ldquoVisualizing Online Auctionsrdquo Journal ofComputational and Graphical Statistics 14 299ndash319

Shmueli G Jank W Aris A Plaisant C and Shneiderman B (2006) ldquoEx-ploring Auction Databases Through Interactive Visualizationrdquo Decision Sup-port Systems 42 1521ndash1538

Shmueli G Russo R P and Jank W (in press) ldquoThe Barista A Model forBid Arrivals in Online Auctionsrdquo Annals of Applied Statistics

Simonoff J S (1996) Smoothing Methods in Statistics New York Springer-Verlag


state-of-the-art functional data methodology for directly model-ing temporal bidding information and its dynamic change Forexample Jank and Shmueli (in press) use functional data analy-sis to find that the price dynamics change quite sharply over thecourse of an auction even for auctions for the same item Infact functional data analysis proves to be a very suitable toolfor the analysis of online auction data and more discussion onthe versatility of this toolset in the broader context of electroniccommerce research can be found in Jank and Shmueli (2006)We build upon the general versatility of functional data analysisin this work

In this article we develop a dynamic forecasting model to pre-dict price in online auctions By dynamic we mean a model thatoperates during the live auction and forecasts price at a futuretime point of the ongoing auction and as a by-product also atthe auction end This is in contrast to static forecasting modelsthat predict only the final price and that take into considerationonly information available at the start of the auction Such infor-mation may involve the length of the auction its opening priceproduct characteristics or the sellerrsquos reputation and may bemodeled using standard least squares regression analysis How-ever a static approach cannot account for information that be-comes available after the start of the auction (eg the amountof competition or current price level) and it cannot incorporatesuch information ldquoon the flyrdquo As we explain throughout thisarticle we find functional data analysis a very suitable tool fordeveloping dynamic price predictions

Forecasting price in online auctions can have benefits to dif-ferent auction parties For instance price forecasts can be usedto dynamically score auctions for the same (or similar) itemby their predicted price On any given day there are severalhundreds or even thousands of open auctions available espe-cially for very popular items such as Apple iPods or MicrosoftXboxes Dynamic price scoring can lead to a ranking of auc-tions with the lowest expected price Such a ranking could helpbidders focus their time and energy on only a few select auc-tions that is those that promise the lowest price Auction fore-casting can also be beneficial to the seller or the auction houseFor instance the auction house can use price forecasts to of-fer insurance to the seller This is related to the idea by Ghaniand Simmons (2004) who suggested offering seller insurancethat guarantees a minimum selling price In order to do sohowever it is important to correctly forecast the price at leaston average Whereas Ghani and Simmonsrsquo method is static innature our dynamic forecasting approach could potentially al-low more flexible features such as an ldquoInsure-It-Nowrdquo optionwhich would allow sellers to purchase insurance either at thebeginning of the auction or during the live auction (with a time-varying premium) Price forecasts can also be used by eBay-driven businesses that provide services to buyers or sellers Re-cently the authors were contacted by a company that providesbrokerage services for eBay sellers about using the dynamicforecasting system to create a secondary market for eBay-basedderivatives

While there has been some work related to price forecastingin online auctions our approach is novel particularly becauseof its dynamic nature (see also Hortacsu and Cabral 2005 forthe dynamics of seller reputation) As pointed out earlier Ghaniand Simmons (2004) using data-mining methods also predict

the end-price of online auctions however that method is staticand cannot account for newly arriving information in the liveauction Structural models to recover the bid distribution (Bajariand Hortacsu 2003) although able to more explicitly accountfor mechanism design are also focused on the final price Thedynamic nature of our forecasting approach is founded withinthe framework of functional data analysis (FDA) In FDA thecenter of interest is a set of curves shapes or objects or moregenerally a set of functional observations This is in contrastto classical statistics where the interest centers around a set ofdata vectors Although the concept of FDA has been around fora longer time the field has gained fast momentum lately dueto the work of Ramsay and Silverman (2000) There is quite abit of interest in the statistics literature to generalize classicalmodels and methods to the functional context Examples areregression analysis for a functional response (Farawary 1997)functional logistic regression (Escabias Aguilera and Valder-rama 2004) and generalized linear models for functional data(James 2002)

Our forecasting approach presents several methodologicaladditions to this stream of literature First to the best of ourknowledge forecasting functional data is a topic that has notbeen sufficiently addressed in the FDA literature to date Infact the use of functional data analysis presents several prac-tical and conceptual advantages for online auction data Tradi-tional methods for forecasting time series such as exponentialsmoothing or moving averages cannot be applied in the auc-tion context at least not directly because of the special datastructure Traditional forecasting methods assume that data ar-rive in evenly spaced time intervals such as every quarter orevery month In such a setting one trains the model on dataup to the current time period t and then uses this model topredict at time t + 1 Implied in this process is the importantassumption that the distance between two adjacent time peri-ods is equal which is the case for quarterly or monthly dataNow consider the case of online auctions Bids arrive in veryunevenly spaced time intervals determined by the bidders andtheir bidding strategies and the number of bids within a shortperiod of time can sometimes be very sparse and other times beextremely dense In this setting the interval between bids cansometimes be more than a day and at other times only secondsSecond online auctions even for the same product can expe-rience price paths with very heterogeneous price dynamics Byprice dynamics we mean the speed at which price travels dur-ing the auction and the rate at which this speed changes Tradi-tional models do not account for instantaneous change and itseffect on the price forecast This calls for new methods that canmeasure and incorporate this important dynamic information

Unevenly spaced data have always been very challenging fortraditional forecasting methods Some of the more recent andinfluential work in this area includes Engle and Russel (1998)who propose the use of the so-called autoregressive conditionalduration model for unevenly spaced transaction data such ashigh-frequency asset price data Similarly Shmueli et al (inpress) develop a class of three-stage nonhomogeneous Pois-son processes the BARISTA to model the changing bid arrivalprocess In both of these cases the underlying assumption isthat the discrete observations are manifestations of a continu-ous process Following this idea we take a functional data ap-proach Specifically we assume an underlying price curve that






























(a) (b)




























y( j) = (y( j)

1 y( j)n

) (1)




(a)

(b)



(a) (b) (c)




l=1




PENSS( j)λm =

nsum

i=1

(y( j)


m (3)






(a) (b)

(c) (d)








y(t) = XTβ(t) + ε(t) (4)



































y(t) = α +Qsum

i=1

βixi(t) +Jsum

j=1

γjD( j)y(t) +

Lsum

l=1

ηly(t minus l) (5)



i=1


j=1

γjD( j)y(T + h|T)

+Lsum

l=1







D( j)y(t) =Ksum

k=0

aktk +Psum

i=1

bixi(t) + u(t) t = 1 T (7)


u(t) =Rsum

i=1



u(T + 1|T) =Rsum

i=1



D( j)y(T + 1|T)

=Ksum

k=0

ak(T + 1)k +Psum

i=1

bixi(T + 1|T) + u(T + 1|T) (10)


D( j)y(T + h|T)

=Ksum

k=0

ak(T + h)k +Psum

i=1

bixi(T + h|T) + u(T + h|T) (11)



y(t) = α + βx (12)







x(t) = xβ(t) (13)





51 Model Estimation












(a) (b)















(a) (b)




MAPE1 MAPE2










Jump Yes 09 01 27 03No 13 03 21 03










MAPE1 MAPE2








ACKNOWLEDGMENTS





MAPE1 MAPE2


1 28 04 32 04

3 23 03 28 03

5 21 03 28 03

7 18 02 26 03

9 16 02 25 021 16 03 27 035 15 03 27 03

10 12 02 24 0215 12 02 23 0220 12 02 23 0225 12 02 23 0230 12 02 23 0240 11 02 23 0250 11 02 23 02



MAPE1 MAPE2


ϒ1 0 1 2 3 4 5 6 625 65 675 68750 7 18 05 29 04ϒ2 0 1 2 3 4 5 6 625 65 675 68125 68750 69375 7 12 02 23 02ϒ3 0 5 1 15 2 3 4 5 6 625 65 675 68125 68750 69375 7 26 04 31 03



REFERENCES




























































(a) (b)




























y( j) = (y( j)

1 y( j)n

) (1)




(a)

(b)



(a) (b) (c)




l=1




PENSS( j)λm =

nsum

i=1

(y( j)


m (3)






(a) (b)

(c) (d)








y(t) = XTβ(t) + ε(t) (4)



































y(t) = α +Qsum

i=1

βixi(t) +Jsum

j=1

γjD( j)y(t) +

Lsum

l=1

ηly(t minus l) (5)



i=1


j=1

γjD( j)y(T + h|T)

+Lsum

l=1







D( j)y(t) =Ksum

k=0

aktk +Psum

i=1

bixi(t) + u(t) t = 1 T (7)


u(t) =Rsum

i=1



u(T + 1|T) =Rsum

i=1



D( j)y(T + 1|T)

=Ksum

k=0

ak(T + 1)k +Psum

i=1

bixi(T + 1|T) + u(T + 1|T) (10)


D( j)y(T + h|T)

=Ksum

k=0

ak(T + h)k +Psum

i=1

bixi(T + h|T) + u(T + h|T) (11)



y(t) = α + βx (12)







x(t) = xβ(t) (13)





51 Model Estimation












(a) (b)















(a) (b)




MAPE1 MAPE2










Jump Yes 09 01 27 03No 13 03 21 03










MAPE1 MAPE2








ACKNOWLEDGMENTS





MAPE1 MAPE2


1 28 04 32 04

3 23 03 28 03

5 21 03 28 03

7 18 02 26 03

9 16 02 25 021 16 03 27 035 15 03 27 03

10 12 02 24 0215 12 02 23 0220 12 02 23 0225 12 02 23 0230 12 02 23 0240 11 02 23 0250 11 02 23 02



MAPE1 MAPE2


ϒ1 0 1 2 3 4 5 6 625 65 675 68750 7 18 05 29 04ϒ2 0 1 2 3 4 5 6 625 65 675 68125 68750 69375 7 12 02 23 02ϒ3 0 5 1 15 2 3 4 5 6 625 65 675 68125 68750 69375 7 26 04 31 03



REFERENCES

























































y( j) = (y( j)

1 y( j)n

) (1)




(a)

(b)



(a) (b) (c)




l=1




PENSS( j)λm =

nsum

i=1

(y( j)


m (3)






(a) (b)

(c) (d)








y(t) = XTβ(t) + ε(t) (4)



































y(t) = α +Qsum

i=1

βixi(t) +Jsum

j=1

γjD( j)y(t) +

Lsum

l=1

ηly(t minus l) (5)



i=1


j=1

γjD( j)y(T + h|T)

+Lsum

l=1







D( j)y(t) =Ksum

k=0

aktk +Psum

i=1

bixi(t) + u(t) t = 1 T (7)


u(t) =Rsum

i=1



u(T + 1|T) =Rsum

i=1



D( j)y(T + 1|T)

=Ksum

k=0

ak(T + 1)k +Psum

i=1

bixi(T + 1|T) + u(T + 1|T) (10)


D( j)y(T + h|T)

=Ksum

k=0

ak(T + h)k +Psum

i=1

bixi(T + h|T) + u(T + h|T) (11)



y(t) = α + βx (12)







x(t) = xβ(t) (13)





51 Model Estimation












(a) (b)















(a) (b)




MAPE1 MAPE2










Jump Yes 09 01 27 03No 13 03 21 03










MAPE1 MAPE2








ACKNOWLEDGMENTS





MAPE1 MAPE2


1 28 04 32 04

3 23 03 28 03

5 21 03 28 03

7 18 02 26 03

9 16 02 25 021 16 03 27 035 15 03 27 03

10 12 02 24 0215 12 02 23 0220 12 02 23 0225 12 02 23 0230 12 02 23 0240 11 02 23 0250 11 02 23 02



MAPE1 MAPE2


ϒ1 0 1 2 3 4 5 6 625 65 675 68750 7 18 05 29 04ϒ2 0 1 2 3 4 5 6 625 65 675 68125 68750 69375 7 12 02 23 02ϒ3 0 5 1 15 2 3 4 5 6 625 65 675 68125 68750 69375 7 26 04 31 03



REFERENCES
































(a) (b) (c)




l=1




PENSS( j)λm =

nsum

i=1

(y( j)


m (3)






(a) (b)

(c) (d)








y(t) = XTβ(t) + ε(t) (4)



































y(t) = α +Qsum

i=1

βixi(t) +Jsum

j=1

γjD( j)y(t) +

Lsum

l=1

ηly(t minus l) (5)



i=1


j=1

γjD( j)y(T + h|T)

+Lsum

l=1







D( j)y(t) =Ksum

k=0

aktk +Psum

i=1

bixi(t) + u(t) t = 1 T (7)


u(t) =Rsum

i=1



u(T + 1|T) =Rsum

i=1



D( j)y(T + 1|T)

=Ksum

k=0

ak(T + 1)k +Psum

i=1

bixi(T + 1|T) + u(T + 1|T) (10)


D( j)y(T + h|T)

=Ksum

k=0

ak(T + h)k +Psum

i=1

bixi(T + h|T) + u(T + h|T) (11)



y(t) = α + βx (12)







x(t) = xβ(t) (13)





51 Model Estimation












(a) (b)















(a) (b)




MAPE1 MAPE2










Jump Yes 09 01 27 03No 13 03 21 03










MAPE1 MAPE2








ACKNOWLEDGMENTS





MAPE1 MAPE2


1 28 04 32 04

3 23 03 28 03

5 21 03 28 03

7 18 02 26 03

9 16 02 25 021 16 03 27 035 15 03 27 03

10 12 02 24 0215 12 02 23 0220 12 02 23 0225 12 02 23 0230 12 02 23 0240 11 02 23 0250 11 02 23 02



MAPE1 MAPE2


ϒ1 0 1 2 3 4 5 6 625 65 675 68750 7 18 05 29 04ϒ2 0 1 2 3 4 5 6 625 65 675 68125 68750 69375 7 12 02 23 02ϒ3 0 5 1 15 2 3 4 5 6 625 65 675 68125 68750 69375 7 26 04 31 03



REFERENCES





































y(t) = XTβ(t) + ε(t) (4)



































y(t) = α +Qsum

i=1

βixi(t) +Jsum

j=1

γjD( j)y(t) +

Lsum

l=1

ηly(t minus l) (5)



i=1


j=1

γjD( j)y(T + h|T)

+Lsum

l=1







D( j)y(t) =Ksum

k=0

aktk +Psum

i=1

bixi(t) + u(t) t = 1 T (7)


u(t) =Rsum

i=1



u(T + 1|T) =Rsum

i=1



D( j)y(T + 1|T)

=Ksum

k=0

ak(T + 1)k +Psum

i=1

bixi(T + 1|T) + u(T + 1|T) (10)


D( j)y(T + h|T)

=Ksum

k=0

ak(T + h)k +Psum

i=1

bixi(T + h|T) + u(T + h|T) (11)



y(t) = α + βx (12)







x(t) = xβ(t) (13)





51 Model Estimation












(a) (b)















(a) (b)




MAPE1 MAPE2










Jump Yes 09 01 27 03No 13 03 21 03










MAPE1 MAPE2








ACKNOWLEDGMENTS





MAPE1 MAPE2


1 28 04 32 04

3 23 03 28 03

5 21 03 28 03

7 18 02 26 03

9 16 02 25 021 16 03 27 035 15 03 27 03

10 12 02 24 0215 12 02 23 0220 12 02 23 0225 12 02 23 0230 12 02 23 0240 11 02 23 0250 11 02 23 02



MAPE1 MAPE2


ϒ1 0 1 2 3 4 5 6 625 65 675 68750 7 18 05 29 04ϒ2 0 1 2 3 4 5 6 625 65 675 68125 68750 69375 7 12 02 23 02ϒ3 0 5 1 15 2 3 4 5 6 625 65 675 68125 68750 69375 7 26 04 31 03



REFERENCES























































y(t) = α +Qsum

i=1

βixi(t) +Jsum

j=1

γjD( j)y(t) +

Lsum

l=1

ηly(t minus l) (5)



i=1


j=1

γjD( j)y(T + h|T)

+Lsum

l=1







D( j)y(t) =Ksum

k=0

aktk +Psum

i=1

bixi(t) + u(t) t = 1 T (7)


u(t) =Rsum

i=1



u(T + 1|T) =Rsum

i=1



D( j)y(T + 1|T)

=Ksum

k=0

ak(T + 1)k +Psum

i=1

bixi(T + 1|T) + u(T + 1|T) (10)


D( j)y(T + h|T)

=Ksum

k=0

ak(T + h)k +Psum

i=1

bixi(T + h|T) + u(T + h|T) (11)



y(t) = α + βx (12)







x(t) = xβ(t) (13)





51 Model Estimation












(a) (b)















(a) (b)




MAPE1 MAPE2










Jump Yes 09 01 27 03No 13 03 21 03










MAPE1 MAPE2








ACKNOWLEDGMENTS





MAPE1 MAPE2


1 28 04 32 04

3 23 03 28 03

5 21 03 28 03

7 18 02 26 03

9 16 02 25 021 16 03 27 035 15 03 27 03

10 12 02 24 0215 12 02 23 0220 12 02 23 0225 12 02 23 0230 12 02 23 0240 11 02 23 0250 11 02 23 02



MAPE1 MAPE2


ϒ1 0 1 2 3 4 5 6 625 65 675 68750 7 18 05 29 04ϒ2 0 1 2 3 4 5 6 625 65 675 68125 68750 69375 7 12 02 23 02ϒ3 0 5 1 15 2 3 4 5 6 625 65 675 68125 68750 69375 7 26 04 31 03



REFERENCES



































x(t) = xβ(t) (13)





51 Model Estimation












(a) (b)















(a) (b)




MAPE1 MAPE2










Jump Yes 09 01 27 03No 13 03 21 03










MAPE1 MAPE2








ACKNOWLEDGMENTS





MAPE1 MAPE2


1 28 04 32 04

3 23 03 28 03

5 21 03 28 03

7 18 02 26 03

9 16 02 25 021 16 03 27 035 15 03 27 03

10 12 02 24 0215 12 02 23 0220 12 02 23 0225 12 02 23 0230 12 02 23 0240 11 02 23 0250 11 02 23 02



MAPE1 MAPE2


ϒ1 0 1 2 3 4 5 6 625 65 675 68750 7 18 05 29 04ϒ2 0 1 2 3 4 5 6 625 65 675 68125 68750 69375 7 12 02 23 02ϒ3 0 5 1 15 2 3 4 5 6 625 65 675 68125 68750 69375 7 26 04 31 03



REFERENCES







































(a) (b)















(a) (b)




MAPE1 MAPE2










Jump Yes 09 01 27 03No 13 03 21 03










MAPE1 MAPE2








ACKNOWLEDGMENTS





MAPE1 MAPE2


1 28 04 32 04

3 23 03 28 03

5 21 03 28 03

7 18 02 26 03

9 16 02 25 021 16 03 27 035 15 03 27 03

10 12 02 24 0215 12 02 23 0220 12 02 23 0225 12 02 23 0230 12 02 23 0240 11 02 23 0250 11 02 23 02



MAPE1 MAPE2


ϒ1 0 1 2 3 4 5 6 625 65 675 68750 7 18 05 29 04ϒ2 0 1 2 3 4 5 6 625 65 675 68125 68750 69375 7 12 02 23 02ϒ3 0 5 1 15 2 3 4 5 6 625 65 675 68125 68750 69375 7 26 04 31 03



REFERENCES







































(a) (b)




MAPE1 MAPE2










Jump Yes 09 01 27 03No 13 03 21 03










MAPE1 MAPE2








ACKNOWLEDGMENTS





MAPE1 MAPE2


1 28 04 32 04

3 23 03 28 03

5 21 03 28 03

7 18 02 26 03

9 16 02 25 021 16 03 27 035 15 03 27 03

10 12 02 24 0215 12 02 23 0220 12 02 23 0225 12 02 23 0230 12 02 23 0240 11 02 23 0250 11 02 23 02



MAPE1 MAPE2


ϒ1 0 1 2 3 4 5 6 625 65 675 68750 7 18 05 29 04ϒ2 0 1 2 3 4 5 6 625 65 675 68125 68750 69375 7 12 02 23 02ϒ3 0 5 1 15 2 3 4 5 6 625 65 675 68125 68750 69375 7 26 04 31 03



REFERENCES

































MAPE1 MAPE2










Jump Yes 09 01 27 03No 13 03 21 03










MAPE1 MAPE2








ACKNOWLEDGMENTS





MAPE1 MAPE2


1 28 04 32 04

3 23 03 28 03

5 21 03 28 03

7 18 02 26 03

9 16 02 25 021 16 03 27 035 15 03 27 03

10 12 02 24 0215 12 02 23 0220 12 02 23 0225 12 02 23 0230 12 02 23 0240 11 02 23 0250 11 02 23 02



MAPE1 MAPE2


ϒ1 0 1 2 3 4 5 6 625 65 675 68750 7 18 05 29 04ϒ2 0 1 2 3 4 5 6 625 65 675 68125 68750 69375 7 12 02 23 02ϒ3 0 5 1 15 2 3 4 5 6 625 65 675 68125 68750 69375 7 26 04 31 03



REFERENCES

































MAPE1 MAPE2








ACKNOWLEDGMENTS





MAPE1 MAPE2


1 28 04 32 04

3 23 03 28 03

5 21 03 28 03

7 18 02 26 03

9 16 02 25 021 16 03 27 035 15 03 27 03

10 12 02 24 0215 12 02 23 0220 12 02 23 0225 12 02 23 0230 12 02 23 0240 11 02 23 0250 11 02 23 02



MAPE1 MAPE2


ϒ1 0 1 2 3 4 5 6 625 65 675 68750 7 18 05 29 04ϒ2 0 1 2 3 4 5 6 625 65 675 68125 68750 69375 7 12 02 23 02ϒ3 0 5 1 15 2 3 4 5 6 625 65 675 68125 68750 69375 7 26 04 31 03



REFERENCES
































REFERENCES































Explaining and Forecasting Online Auction Prices and … 2008... · Explaining and Forecasting...

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Transcript of Explaining and Forecasting Online Auction Prices and … 2008... · Explaining and Forecasting...