This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

Transcription

1 Ths artcle appeared n a journal publshed by Elsever. The attached copy s furnshed to the author for nternal non-commercal research and educaton use, ncludng for nstructon at the authors nsttuton and sharng wth colleagues. Other uses, ncludng reproducton and dstrbuton, or sellng or lcensng copes, or postng to personal, nsttutonal or thrd party webstes are prohbted. In most cases authors are permtted to post ther verson of the artcle (e.g. n Word or Tex form) to ther personal webste or nsttutonal repostory. Authors requrng further nformaton regardng Elsever s archvng and manuscrpt polces are encouraged to vst:

2 Journal of Econometrcs 174 (2013) Contents lsts avalable at ScVerse ScenceDrect Journal of Econometrcs journal homepage: Robust frm prcng wth panel data Benjamn R. Handel a,b, Kanshka Msra c,d, James W. Roberts e,b, a Department of Economcs, Unversty of Calforna at Berkeley, Unted States b NBER, Unted States c London Busness School, Unted Kngdom d The Ross School of Busness, Unversty of Mchgan, Unted States e Department of Economcs, Duke Unversty, Unted States a r t c l e n f o a b s t r a c t Artcle hstory: Receved 20 December 2010 Receved n revsed form 18 December 2012 Accepted 22 February 2013 Avalable onlne 4 March 2013 JEL classfcaton: C14 C44 L11 L13 L15 Frms often have mperfect nformaton about demand for ther products. We develop an ntegrated econometrc and theoretcal framework to model frm demand assessment and subsequent prcng decsons wth lmted nformaton. We ntroduce a panel data dscrete choce model whose realstc assumptons about consumer behavor delver partally dentfed preferences and thus generate ambguty n the frm prcng problem. We use the mnmax-regret crteron as a decson-makng rule for frms facng ths ambguty. We llustrate the framework s benefts relatve to the most common dscrete choce analyss approach through smulatons and emprcal examples wth feld data Elsever B.V. All rghts reserved. Keywords: Frm prcng Mnmax-regret Partal dentfcaton Panel data 1. Introducton Standard approaches for applyng random utlty models to nterpret dscrete choce data mantan assumptons that allow pont dentfcaton of consumer preferences, whch eases counterfactual choce predctons. The pont of ths paper s to relax our modelng assumptons of consumer behavor wthout compromsng our ablty to provde gudance on counterfactual outcomes. To ths end, the frst part of ths paper develops a model that requres only conservatve assumptons about consumer decson-makng processes to partally dentfy preferences and, consequently, counterfactual choces. We focus on settngs wth panel data, and extend pror work by ntegratng conservatve assumptons on nter-temporal decson-makng nto our econometrc framework. The second part of the paper focuses on how frms use the model s output to make strategc decsons. Whle our robust Correspondence to: 213 Socal Scences Buldng, 419 Chapel Drve, Durham, NC 27708, Unted States. Tel.: E-mal addresses: handel@berkeley.edu (B.R. Handel), kmsra@london.edu (K. Msra), j.roberts@duke.edu (J.W. Roberts). modelng assumptons stll allow us to predct counterfactual choces, the analyss of a frm s strategc optmzaton problem s complcated by the fact that those counterfactual demand curves are only partally dentfed. Wth ths nformaton set, how does a frm make a strategc choce lke a prcng decson? When output s pont dentfed, a frm has complete nformaton about the dstrbuton of consumer preferences, and can use ths nformaton to maxmze expected profts. In our settng, and ndeed n any settng where a frm uses partally dentfed parameters as decson-makng nputs, the frm may not be able to construct a pror over the set of feasble preference parameters n order to maxmze expected profts. In ths sense the frm faces Knghtan Uncertanty, or ambguty, about consumer demand. The second prmary contrbuton of ths paper analyses how a frm can use the partally dentfed set of preference dstrbutons arsng from our conservatve econometrc model to choose prces under ambguty. Ths paper thus ntegrates the pror theoretcal work on frm prcng under ambguty wth a novel econometrc framework to () econometrcally model the lack of nformaton nherent n the frm s problem when only conservatve assumptons about consumer decson-makng are made and then to () study how frms wll prce under ambguty f ther nformaton set s /$ see front matter 2013 Elsever B.V. All rghts reserved.

3 166 B.R. Handel et al. / Journal of Econometrcs 174 (2013) consstent wth the output of our econometrc model. We then use ths two part framework to study frm prcng n both smulatons and feld data. We show that there are many cases where, despte ts more conservatve approach, our ntegrated model compares favorably to, and at tmes outperforms, the combned mxed logt and expected proft maxmzaton framework, whch s the workhorse model of the ndustral organzaton lterature. We nvestgate an envronment wth panel data and develop four alternatve models that correspond to dfferent assumptons on how consumer preferences can change over tme. Each model results n a dstnct, partally dentfed set of consumer preferences and, consequently, demand curves and counterfactual choces. Across these models, the prmary parametrc assumpton we mantan s that consumer preferences are a lnear functon of product attrbutes, as n the canoncal dscrete choce framework of McFadden (1974). Under ths assumpton, each alternatve nter-temporal decson-makng framework places restrctons on the range of feasble valuatons for products and ther assocated attrbutes, gven consumers choces. Unlke models wth more powerful statstcal assumptons about the dstrbuton of preferences (e.g. random coeffcents) and the dstrbuton of dosyncratc preference shocks (e.g ndependent and dentcally dstrbuted logt errors), each of our models can be rejected by the data f the underlyng assumptons are volated, ncreasng the credblty of the analyss at the expense of reduced precson. The four frameworks dffer accordng to ther mantaned assumptons on the tme varaton of consumers preferences. 1 In the frst, most basc, setup, consumers have the same exact preferences n each tme perod, wth no dosyncratc component, and we drectly apply the strong axom of revealed preferences to partally dentfy consumer preferences. Ths most basc model lacks flexblty n allowng for wthn-consumer varaton n preferences over tme, and hence wll lkely be rejected by the data. Thus we extend t n three ways to allow for tme varyng consumer utlty. Frst, we study random shocks to each consumer s utlty for each product and tme perod. Unlke prevous models n ths lterature, whch place more structure on the dstrbuton of these random shocks, we only mantan that these errors are bounded n sze by a constant n absolute value and we do not make any ndependence or dstrbutonal assumptons about these error terms. 2 Includng bounded errors allows the model to account for small departures from stable preferences that occur often over the course of multple decsons. We llustrate how the bound () s dentfed and () can be estmated n a frst stage usng only the orgnal panel data set. 3 Our next framework studes data contamnaton, an oft-cted determnant of observed tmevaraton n purchases (see e.g. Keane (1997) or Enav et al. (2010)). Intutvely, ths allows the model to account for large departures from stable preferences that occur rarely. It wll often be the case that for partcular data sets there s exstng knowledge that can be drawn upon to nform the econometrcan about the extent of contamnaton n the data. Our fnal framework combnes our analyss of bounded errors wth our analyss of data contamnaton. Wth these partally dentfed predctons n hand, we then nvestgate the frm prcng problem under ambguty. To our knowledge, ths s the frst work that ntegrates an econometrc 1 As an addtonal extenson to each of these four frameworks, we develop a method to use cross-sectonal varaton n conjuncton wth each of these ntertemporal frameworks to obtan further dentfyng power when the panel s not a representatve sample from the populaton. 2 These assumptons address many of the undesrable features of the standard extreme value random utlty model, as dscussed n Bajar and Benkard (2003). 3 We characterze the dentfed sets n the cases () where the error bound s known by the frm and econometrcan and () where the error bound s unknown by both partes and s estmated. framework that generates partally dentfed demand, due to a frm s lack of nformaton on consumer preferences, wth a model of frm prcng under ambguty. We model frm decson-makng usng the mnmax-regret prcng crtera dscussed elsewhere n a purely theoretcal settng (see e.g. Bergemann and Morrs (2005) Bergemann and Schlag (2007) or Bergemann and Schlag (2008)). Under ths crteron, the frm chooses a prce to mnmze ts maxmum regret over the set of perceved feasble demand curves represented by the partally dentfed output of our econometrc model. Here regret s defned for a gven demand curve n the set of feasble demand curves, and equals the dfference between profts under the optmal prce for that demand curve and the profts under the actual prce. 4 Our analyss consders the cases of monopolstc and duopolstc prcng under ambguty based on the partally dentfed set of demand curves where the latter ncorporates a strategc envronment. 5 Whle we use mnmax-regret as a crteron because t has the desrable property that t trades off potental losses from overprcng (sellng too lttle) versus those from underprcng (not extractng enough consumer value), we note that any crteron for decson-makng under ambguty could be used to make decsons wth our partally dentfed econometrc output. For example, the maxmn crteron (see e.g. Glboa and Schmedler (1989)) s a potental alternatve to mnmax-regret, whch we study brefly n the context of our emprcal examples. We use smulatons to test the performance of our jont econometrc theoretcal framework relatve to two benchmark specfcatons: () the mxed logt wth multvarate mxng and () ex post optmal prcng under perfect nformaton. For plausble underlyng data generatng processes, we analyze how consumer choces translate nto partally dentfed estmates of demand for our dfferent econometrc models. The results show that, n the monopoly settng, the monopolst gets close to ex post effcent prces wth our framework ndependent of the underlyng error data generatng process. On the other hand, the mxed logt performs well f the underlyng data structure has..d. errors but can yeld large dfferences from optmal prcng when ths s volated (such as when there are tme correlated error shocks). In the olgopolstc settng, we analyze mnmax-regret best response curves gven partally dentfed preferences and show that prces under our model are much closer to the ex post effcent prces for many data generatng processes. These results suggest that our ntegrated framework for robust frm prcng provdes a vable alternatve to the canoncal mxed logt model n cases where t s lkely that the frms studed have lmted nformaton. Fnally, we llustrate how our methodology can be appled n an actual emprcal settng n order to recommend an optmal prce when only conservatve assumptons about consumer decsonmakng are made. The settng we consder s retaler mlk prcng. Flud mlk s a frequently purchased non-storable good and s an mportant category for retalers as t has the hghest penetraton of any retal category (Bronnenberg et al. (2008)). It s manly drven by retaler owned, prvate label brands and, mportantly for us, t s a non-storable good. 6,7 By frst estmatng demand 4 Note that ths noton of regret from the statstcal decson lterature (e.g. Savage (1951)) s completely dstnct from the noton of regret dscussed n the psychology and economcs lterature. 5 We developed our framework for strategc frm prcng under ambguty smultaneously and ndependently of recent work by Renou and Schlag (2010) who study foundatons for mnmax-regret strategc prcng equlbrum n a purely theoretcal paper. 6 See Fong et al. (2011) for a revew of marketng papers estmatng prvate label elastcty wth standard models. 7 Ths second observaton s relevant as ths rules out stockplng behavor whch wll generate dynamc choce behavor, the modelng of whch les outsde the scope of ths paper (for papers that do model ths behavor based on more tradtonal demand estmaton technques, see Erdem et al. (2003) or Hendel and Nevo (2006)).

4 B.R. Handel et al. / Journal of Econometrcs 174 (2013) and then solvng for mnmax-regret optmal prces, we show that our methodology s applcable n real-world settngs and returns sensble counterfactual recommendatons (the mnmaxregret optmal prce s $2.40/gallon when actual observed prces average about $2.56/gallon). Ths paper helps to advance the twn goals n the broader dscrete choce lterature of () descrbng preferences and () makng counterfactual predctons. Papers that best descrbe preferences n specfc contexts can make conservatve assumptons wth smple decson-theoretc foundatons, but, as a result, are generally not well suted for counterfactual predcton. For example, Samuelson (1938, 1948) study observed consumers choces from dfferent choce sets and prce ncome pars and use ether the weak or strong axom of revealed preference along wth a transtvty assumpton to draw powerful conclusons about preferences for products n the observed envronment. Followng ths lne of work, Varan (1982, 1983) develop an econometrc methodology that () tests f observed choce behavor s consstent wth ratonal choces and () recovers preferences as a functon of prces and budget sets. Whle these approaches nfer preferences under mnmal assumptons, ther emprcal vablty s lmted because they requre very rch choce data, the models can be easly rejected by the data, and they cannot nform predctons n counterfactual settngs. In more recent work, Blundell et al. (2008) use revealed preference restrctons to non-parametrcally dentfy demand responses along Engel curves. 8 Smlar to ths paper, ther objectve s to use theoretcal restrctons to obtan credble preference estmates, gven ndvdual-level data on relatve prces and total expendtures, wthout mposng the usual parametrc and statstcal assumptons that permeate the demand estmaton lterature. Our approaches dffer along multple dmensons, most promnently that Blundell et al. () take a cross-sectonal approach that does not ncorporate tme seres decson restrctons n dentfcaton and () do not ncorporate the noton of products as bundles of attrbutes (or mantan the lnear utlty n attrbutes assumpton that we do). The former mples that our prmary sources of dentfcaton and deal data sets dffer substantally, as we ncorporate theoretcal restrctons on how a gven consumer makes decsons over tme. Whle Blundell et al. study what mnmum level of statstcal perturbaton to ther consumer utlty bounds can justfy a ratonal paradgm gven ther model and choce envronment, our analyss partally dentfes sets of preferences for attrbutes and can be used to make predctons n counterfactual choce settngs where products are composed of those same attrbutes (e.g. wth new products or new choce settngs based on exstng products). Fnally, we ncorporate the possblty of data contamnaton and lnk the output of the econometrc exercse drectly to the frm prcng decson under ambguty. The second, and much more heavly utlzed, branch of the dscrete choce lterature makes stronger assumptons about consumer behavor but s also able to make stronger statements about counterfactual outcomes. These papers assume that consumers have preferences for product attrbutes whch are aggregated to establsh preferences over products (see Lancaster (1966) and McFadden (1974)). The canoncal model assumes that consumer preferences have a specfc parametrc form that maps attrbutespecfc preference parameters, vectors of product attrbutes, and an addtve preference shock known to the frm (but not the researcher) nto product values and preference orderngs. Papers n ths lterature make dfferent assumptons about the dstrbutons 8 In related work Blundell et al. (2003) show how to use non-parametrc methods to detect revealed preference volatons. of determnstc preference heterogenety and the dosyncratc preference shock. Many assume that the error terms are ether ndependent and/or dentcally dstrbuted across consumers, products and tme. 9 Gven the dstrbuton of error shocks and form of utlty, model parameters are dentfed usng observed choce data. 10 Our work uses some of the basc assumptons n ths lterature, such as attrbute based preferences, to mantan the ablty to perform flexble counterfactual analyses, but refrans from makng parametrc assumptons whch are wthout theoretcal foundaton and can be dffcult to nterpret. 11 Of specal note s recent work by Mansk (2007), who studes a sem-parametrc cross-sectonal dscrete choce model wth no assumptons on the dstrbuton of errors. Mansk partally dentfes preferences based on three man sources: () lnear utlty n attrbutes, () the consstency of preference parameters wth observed ratonal behavor for a gven choce set and () cross-sectonal varaton n prces and choce sets. Our paper uses () but has dfferent notons of () and (), prmarly because we study a panel settng where choce consstency over tme must be taken nto. We vew hs paper as complementary to our own from an econometrc perspectve snce t has a smlar underlyng motvaton but apples to a dstnctly dfferent data envronment. Moreover, hs paper does not nvestgate how the output of the model wll be used n decson-makng as we do wth our emphass on frm prcng. The remander of the paper proceeds as follows. Secton 2 sets up the model and derves the dentfcaton regons for each framework. Secton 3 dscusses estmaton. Secton 4 descrbes the frm problem when preferences are partally dentfed. Secton 5 llustrates the methodology through smulatons and n an applcaton to mlk prcng. Secton 6 concludes. 2. Model The problem we consder s one where a frm observes panel data and uses them to make a prcng decson. Our goal s to relax the assumptons underlyng the tradtonal lterature on such behavor by not requrng the frm to know the dstrbuton of demand for ts products. That s, we consder a frm whch seeks 9 It s mportant to note that the mxed-logt class of models does allow for error correlatons (see e.g., McFadden and Tran (2000)). Several other papers provde counterexamples to ths clam and deserve specfc menton. An mportant contrbuton n ths lne of work s Keane (1997) who establshes the presence of state dependent preferences as well as heterogenety n these tastes. Recently, Febg et al. (2010) extend the mxed multnomal logt model to present a generalzed multnomal logt model that allows better modelng of consumers wth extreme and/or random tastes (n the sense that a partcular attrbute of a product drves much of ther decson-makng). In other related work, Geweke (2012) explores recoverng regons of parameters based on observed data but from a Bayesan perspectve. 10 Broadly speakng, these models fall nto four categores (Ben-Avka et al. (1997)): those assumng (1) functonal forms for determnstc utlty (lnear n product attrbutes) and that error terms are..d. accordng to a specfed dstrbuton, such as Type 1 Extreme Value (ths could nclude dynamc structural models of demand, e.g. Erdem and Keane (1996), Erdem et al. (2003) or Hendel and Nevo (2006)); (2) a parametrc functonal form for determnstc utlty (usually lnear n attrbutes) but wth unspecfed error dstrbuton (see Mansk (1975)); (3) a specfc form for the error dstrbuton, but no functonal form assumpton on determnstc utlty (see Haste and Tbshran (1990) and Abe (1995)); and (4) no functonal form for determnstc utlty or the dstrbuton of error terms (see Matzkn (1993)). 11 We vew our work as complementary to the pror lterature that mantans stronger assumptons on the dstrbutons of preferences and preference shocks. If the output from a model wth many mantaned assumptons does not le wthn the bounds our models produce, the researcher should be skeptcal that ther model s correctly specfed. Further, f the pont dentfed output les near one edge of our feasble demand curve set, our model sheds lght on the lkely drecton of any potental model bas. Fnally, f the researcher beleves there are specfc justfcatons for the parametrc assumptons mantaned, ths adds nsght above and beyond our model.

5 168 B.R. Handel et al. / Journal of Econometrcs 174 (2013) to maxmze profts, but cannot necessarly do so n the tradtonal way because t does not know the dstrbuton of types n the populaton. Ths mples that the frm can only partally dentfy demand for each potental prce t consders. In ths secton we show how to non-parametrcally dentfy consumer preferences usng panel data. We consder a varety of models wth ncreasng flexblty to llustrate how one can dentfy demand under specfc ncremental assumptons Base case: tme consstency We begn our analyss wth the base model that assumes that each ndvdual has stable preferences over tme. There are no product or tme specfc preference shocks, whch yelds tght bounds, but a hgh probablty that the model wll be rejected when consumers decsons cannot be ratonalzed wthn the lnear preferences over attrbutes specfcaton Model 1: tme consstent preferences As n Mansk (2007) we examne the dscrete choce problem faced by consumers wthn a treatment response framework. In ths setup, there are A possble dstnct alternatves (products) each unquely characterzed by a K dmensonal attrbute vector x. Each attrbute s assumed to have fnte support, 12 therefore the set A s fnte. We defne the set of possble treatments D as the space of possble choce sets an ndvdual could face, whch n ths settng s the collecton of all non-empty fnte subsets of A. Each ndvdual faces a choce set from D and responds by choosng an element of that set. Formally, there s a populaton of N ndvduals, denoted I, n whch each ndvdual I has a response functon y ( ) : D A mappng choce sets nto unque choces from that set. The probablty dstrbuton P[y( )] of the random functon y( ) : D A descrbes the aggregate choces (product shares) made by the populaton. For example, consder a case where there are three feasble alternatves b, c, and d, so that A = {b, c, d}. Assume that the alternatves are descrbed only by ther dentfed name (b, c or d). In our notaton we would say that K = 3, as each alternatve s descrbed by three ndcators (smlar to fxed effects). Suppose that we observe data from a choce settng where N consumers choose between product b and c, so that D = {b, c}. In our notaton we would say that y ({b, c}) = b for the N b consumers who choose b, y ({b, c}) = c for the N c consumers who choose c and that P[y({b, c}) = b] = N b, P[y({b, c}) = c] = N c. N N Our objectve s to estmate counterfactual choce probabltes. For example, what percentage of consumers would choose d n a choce between all three alternatves? Wthout any assumptons about the underlyng consumer decson-makng process, we cannot say anythng nformatve about ths counterfactual. In our settng we make several conservatve assumptons on ndvdual behavor that wll allow us to make counterfactual predctons. Assumpton 1 (Utlty Maxmzaton). Consumers have well defned preferences and make decsons that maxmze utlty subject to the avalable elements n ther choce set. Under ths assumpton, f consumer faces choce set D we have the followng nformaton about the consumers response functon where u,a s the utlty consumer gets from alternatve a: y (D) = argmax a D u,a. (1) 12 Ths assumpton contrasts wth those n Berry and Hale (2009). Ths assumpton allows us to make nferences about counterfactual consumer choces, and we can classfy the populaton nto types based on ther preferences. A type s defned by preferences over all elements of D. There are D! possble types n the populaton correspondng to dfferent permutatons of the elements of D that could correspond to ratonal preference orderngs. Inference about what type a consumer mght be can be made from observed choce data. In the smple example above, there are 6 types of consumers: 1. b c d, 2. b d c, 3. c b d 4. c d b, 5. d b c, 6. d c b. Observng the fact that N b consumers choose opton b n a choce between {b, c} mples that these consumers are of type 1, 2 or 5. Smlarly, observng N c consumers choose c mples that they are of type 3, 4, or 6. In counterfactual choce settngs, the proporton of consumers who choose c from a choce set {b, c, d} s equvalent to estmatng the proporton of consumers of type 3 or 4. In ths example ths s bounded above at N c. N However, wthout makng more assumptons about the underlyng utlty structure, we cannot estmate the counterfactual choce probablty for choce d snce we never observe d n the consumer choce set. Therefore, we follow the dscrete choce lterature (McFadden (1974)) and consder products to be bundles of attrbutes and assume that consumers utlty functons are lnear n these attrbutes. Assumpton 2 (Lnear Utlty). Indvdual utlty functons are lnear n the K dmensonal attrbute vector x, that descrbes the alternatves n the choce sets. Addtonally, we defne ndvdual specfc K dmensonal parameters ω to descrbe ndvdual s preferences for each attrbute. We defne Ω as the feasble parameter space for these preferences, wth ω Ω. Under these characterzatons, the utlty consumer gets from alternatve a s defned as: u,a = ω x a. (2) Relatng ths to the response functon we now have: y (D) = argmax a D ω x a. (3) In our formulaton of ths utlty model, prce s one attrbute n x a. Thus, we consder one product sold at P dfferent prces as P dstnct feasble alternatves n the set of all alternatves A. Here, a demand curve nvolves constructng a set of counterfactual predctons based on the set of alternatves. Unlke our smple example where we had sx types of consumers n our model wth lnear utlty, we now have a contnuous parameter space wth nfnte consumer types. However, snce we have a space of fnte alternatves, we can represent the contnuous space Ω by a dscrete dstrbuton of types correspondng to the dfferent possble choce functons over A (as n Mansk (2007)). These representatons are equvalent because each of the ω Ω that corresponds to the same preference orderng over all alternatves cannot be dentfed from each other n the data n our model. Formally, let A m, m = 1,..., A! represent the mth permutaton of A. If x m,n s the attrbute bundle of the nth element of A m, then the dscrete type space can be defned: Θ m [ω Ω : ω x m1 > ω x m2 > > ω x m A ]. Let θ l denote a generc element of Θ l whch we can use from ths pont forward to represent that type wthout loss of generalty.

6 B.R. Handel et al. / Journal of Econometrcs 174 (2013) We have now mapped our parameter space from a contnuous set Ω to a set of dscrete types {θ 1,..., θ A! }. 13 We partally dentfy the dstrbuton of types wth panel data by usng ths model to dentfy the feasble range of preferences for each ndvdual based on ther choces and then aggregatng these to form an aggregate bound on the dstrbuton of types. In our panel data, for each ndvdual I we observe choces a t from dstnct choce sets over tme (e.g., choce made every week) d t for t = 1,..., T. 14 The man advantage of these data are that we can use all T observatons for the ndvdual to gan more nformaton about a gven ndvdual and aggregate feasble types. However, the panel framework also presents addtonal complcatons snce t s possble for data on ndvdual decsons to be nconsstent wth a constant preference parameter over tme. Below we present models that allow for the most commonly gven explanatons for such apparent nconsstences: tme varyng preferences and data contamnaton. 15 We begn, however, wth a base model that assumes that consumers have stable preferences over tme and that there s no data contamnaton. It s the strctest of the models we present snce t s the least flexble n terms of how t can ratonalze a sequence of observed choces. Defnton 1 (Tme-Consstent Utlty). An ndvdual n the populaton s tme-consstent f they always make decsons accordng to a fxed θ. Ths defnton mples that a consumer s utlty n each purchase occason s descrbed completely by observable (to the researcher) attrbutes. An ndvdual can be tme-consstent f: Θ 1 θ : θ T {θ : θ x a t > θ x a, a t d t } t where a t s s purchase decson at tme t and a t s an element not chosen from that set. Here, Θ 1 denotes the set of feasble θ for ndvdual gven the decsons we observe over tme. Under the tme-consstency assumpton, the partally dentfed probablty of an ndvdual beng of type θ s: H[Pr(θ = θ)] = [Pr(Θ 1 = {θ}), Pr(θ Θ 1 )]. (4) Ths says that the lower bound of a specfc consumer beng a certan type s the probablty that the dentfcaton regon for an ndvdual ncludes only that type, whle the upper bound s the probablty that a gven type s ncluded n an dentfcaton regon. If an ndvdual s not tme-consstent so that Θ 1 =, we conclude that ndvdual s not n the A! ratonal types and les nstead n the larger collecton of types that s descrbed by all permutatons of possble choces across feasble choce sets. In our base model, we consder the sample that we analyze to be the populaton of nterest. Ths smplfes exposton of our model n that t allows us to focus on dentfcaton nstead of samplng propertes at ths pont. The set of feasble dstrbutons for θ n the populaton comes drectly from dentfyng the feasble types 13 As shown n Mansk (2007) the lnear utlty specfcaton has some dentfyng power as we reduce the number of feasble choce functons before gong to the data. In the smulatons that we study n Secton 5.1, the dmensonalty of feasble dscrete types s reduced approxmately by a factor of ten when we mpose the lnear model, mplyng that the number of ponts n the dstrbuton that we are estmatng s also reduced by a factor of ten. 14 For smplcty, we wll consder models where consumers make only one dscrete choce at each pont n tme, though nothng about our setup precludes us from observng multple choces at multple ponts n tme for a gven ndvdual, where lnkng contemporaneous decsons would also add dentfyng power. 15 Two other potental explanatons for such nconsstences are decson-makng errors and non-lnear utlty. To focus on the core ssue of robust frm prcng, we leave the exploraton of these types of models to future work. at the ndvdual level (as descrbed above) and then determnng all combnatons of these types aggregated to the populaton level. Then, all such feasble aggregatons descrbe the partally dentfed set of type dstrbutons. More formally, the set of feasble dstrbutons satsfes: H[F(θ)] F(θ) f (θ) = 1 I[θ = θ ], θ Θ 1. (5) N I To understand the set defnton, consder the example from above where there are sx consumer types. Suppose that there are two consumers for whom, based on ther sequence of purchases, we have determned that the frst can be type 1 or 2 and the second can be type 2, 3 or 4. Then there are sx possble dstrbutons and H[F(θ)] ncludes all sx possbltes: 1. f (1) = 1/2, f (2) = 1/2; 2. f (1) = 1/2, f (2) = 1/2 3. f (1) = 1/2, f (4) = 1/2; 4. f (2) = 1 5. f (2) = 1/2, f (3) = 1/2; 6. f (2) = 1/2, f (4) = 1/2. The knowledge of the partally dentfed dstrbuton of preferences allows us to study counterfactual choce settngs and, hence, counterfactual demand. At the ndvdual-level, a gven consumer ether could or could not choose product a from counterfactual choce set D. Ths bnary possblty depends drectly on whether the partally dentfed preference set for that ndvdual contans at least one preference profle where the ndvdual would choose a from D. Gven ths, we defne bounds on demand for product a when the populaton faces choce set D relatve to H[F(θ)], the set of feasble preference dstrbutons at the populaton level. Mnmum demand for product a comes from the feasble dstrbuton n H[F(θ)] where the fewest consumers would purchase a (and vce-versa for the maxmum): H[P(y(D)) = a] = max F( ) H[F( )] mn F( ) H[F( )] 1[y(D) = a]f (θ), θ 1[y(D) = a]f (θ). (6) θ 2.2. Relaxng tme consstency: bounded preference shocks and data contamnaton The base model above s predcated on consumers havng constant preferences over tme. Gven ths nflexblty, t s possble that an ndvdual s observed purchase decsons cannot be ratonalzed by such a model. Therefore, we now present two extensons to make the model more realstc. The frst of these allows for ndvdual-tme-product specfc preference shocks as most dscrete choce models do, but assumes no structure on the populaton dstrbuton of the shocks except that they are bounded. Ths approach s motvated by Bajar and Benkard (2003) who llustrate that canoncal dscrete choce models wth unbounded errors have some notable undesrable propertes. Frst, as the number of products n the choce set becomes large, the standard approach mples that all consumer decsons are drven by unobserved error shocks. Ths mples that n settngs wth large choce sets researchers cannot learn about underlyng consumer preferences. Second, n any choce settng, the standard approach mples that every product has a non-zero probablty of beng chosen by a gven consumer, regardless of underlyng preferences. In ths secton, our model wth bounded errors allows researchers to learn about preferences over tme even wth a large number of products and, addtonally, allows for a zero probablty of choosng a domnated product. The second of these extensons allows for the possblty that the data may be contamnated and thus observed purchases do

7 170 B.R. Handel et al. / Journal of Econometrcs 174 (2013) not reflect actual choces. There are numerous reasons that data may be contamnated ncludng, but not lmted to, recordng errors, non-response, or nterpolaton/extrapolaton. Heurstcally, bounded errors allow the model to explan consumers makng frequent, but small departures from stable preferences. Data contamnaton, on the other hand, permts less frequent, though larger fluctuatons n mpled preferences Model 2: random utlty Most attempts to estmate demand wth panel data employ a utlty model of the form: u,a = θ x a + ε,a. (7) A consumer s utlty n each purchase occason s descrbed by observable (to the researcher) attrbutes and unobserved error shocks. Here we make the followng assumpton on ε: Assumpton 3 (Random Utlty Model wth Bounded Errors). An ndvdual n the populaton receves random utlty shocks ε,a,t for each, a, and t. The only assumpton about these shocks are that they are (strctly) bounded wthn some range [ δ, +δ]. Therefore, at each pont n tme, and for any product, an ndvdual receves a shock to hs utlty of magntude no greater than δ. Further, as opposed to what s done n the lterature, we make no dstrbutonal or ndependence assumptons about ths shock, apart from settng ths bound. 16 In theory, the frm could know δ or calbrate t from a varety of data sources. In Secton 3 we present one methodology for how a frm may non-parametrcally select an approprate δ n a gven emprcal settng. Wthout any assumptons about the dstrbuton of ε, all we know s that a 1, a 2 D, 2δ ε,a1,t ε,a2,t 2δ. Therefore the dentfcaton regon for ths model s gven: Θ 2 {θ : θ T {θ : θ x a θ x t a 2δ, a t d t }}. (8) t As n the tme-consstency case, the partally dentfed probablty of an ndvdual beng of type θ and the feasble populaton dstrbutons of types are gven by Eqs. (4) and (5), respectvely, wth Θ 2 replacng Θ 1. As n the base model, knowledge of the partally dentfed dstrbuton of preferences allows us to study counterfactual choce settngs. We can derve the probablty that alternatve a s chosen from a choce set D as was done n Eq. (6) Model 3: data contamnaton Whle bounded errors provde flexblty n terms of descrbng consumer decsons that depart by a small magntude from ther stable preferences, n some cases there may be large fluctuatons n mpled preferences for an ndvdual as a functon of observed choces. Here, we expand the base model by allowng for a proporton of the data to be contamnated. An alternatve nterpretaton of large devatons from apparently stable preferences that s based more on behavoral foundatons, s that consumers occasonally make sub-optmal decsons. 17 We prefer 16 If there are dstnct consumer types n the data whch are observable, then δ could vary across consumer types. For smplcty we assume no such dstnctons exst as the extenson s straghtforward. 17 Ths may occur for a varety of reasons. It may be, for nstance, that consumers do not have full nformaton about the optons avalable and make decsons based on some ndvdual heurstc. For example, there s evdence showng that tmeconstraned consumers are more lkely to purchase tems from the mddle of store shelves (Dreze et al. (1994)). As researchers, we do not know when ndvduals use such heurstcs, nor whch heurstcs they use. the data contamnaton explanaton and so proceed under that nterpretaton. 18,19 Assumpton 4 (Data Contamnaton Model wth Tme-Consstent Utlty). Indvduals have tme-consstent utlty but up to φ percentage of ther decsons could be recorded wth error. There are numerous reasons that data may be contamnated and multple papers have explored the extent of data contamnaton and ssues relatng to dentfcaton and estmaton n ts presence (see for example Horowtz and Mansk (1995), Keane (1997), Erdem et al. (1999) or Enav et al. (2010)). Defne T φ as the set of observatons for that are not contamnated, wth T φ T and T φ (1 φ) T. Gven T φ the partally dentfed set of preferences for each consumer are defned 20 : Θ 3 (T φ ) θ : θ φ T {θ : θ x a θ x t a a t d t }. (9) t For a gven bound φ, there are many possble canddates for T φ snce the researcher cannot observe whch data are actually contamnated. As a result, all feasble T φ must be consdered to determne the true partally dentfed set Θ 3 : Θ 3 T φ Θ 3 (T φ ). (10) Here, T φ s the set of all feasble T φ. The unon of these sets s the correct metrc, because each potental T φ could reflect the set of all actual purchases. It s mportant to note that, from an emprcal standpont, the only potental T φ the researcher needs to consder are those that have exactly 1 φ observatons ( T φ = (1 φ) T ). Ths s true because the set of potental preferences can only expand as restrctons from choces are removed, so those feasble data sets wth the fewest choce restrctons (largest possble proporton of msclassfed data) wll be the most nclusve. As n the tme-consstency case, the partally dentfed probablty of an ndvdual beng of type θ and the feasble populaton dstrbutons of types are gven by Eqs. (4) and (5), respectvely, wth Θ 3 replacng Θ 1.21 Counterfactually, once we determne the partally dentfed dstrbuton of preferences we can derve the probablty that alternatve a s chosen from a choce set D as was done n Eq. (6). 18 There s an added beneft of avodng a model of sub-optmal decson-makng. If consumer decsons are actually random some fracton of the tme, a frm may wsh to set an nfnte prce. As we study frm prcng below, t would be dffcult to ratonalze ths strategy wth emprcal evdence that prces are rarely nfnte. We thank the referees for pontng ths out. 19 An alternatve nterpretaton of φ could be that wth a small probablty, a partcular consumer s observaton s recorded wth error. We do not adopt ths approach because f φ were the probablty that an ndvdual observaton was msclassfed, then t s possble, albet unlkely, that all observatons n any one data set would be contamnated. Ths would be equvalent to havng our nterpretaton of φ as a proporton and that proporton beng 1. φ can be thought of as a value that wll be above the proporton of msclassfed data for almost every ndvdual. 20 A related process s dscussed n Keane and Sauer (2009, 2010), but n a dfferent context where the authors consder the case of employment status msclassfcaton when modelng female labor supply. 21 We assume that at most φ percent of observatons are msclassfed/ contamnated for each ndvdual. If the true data generatng process (DGP) s bnomal where, wth some probablty p, an observaton s recorded wth error, any bound less than φ = 1 could n theory be volated. In the context of our assumed bound, f a consumer actually has a proporton of msclassfed data greater than φ, the model wll be rejected by the data f the msclassfcaton, treated as an actual choce, leads to mpled preferences that are not consstent across purchases. In ths case, the dentfed set of parameters wll be empty. Our model already conservatvely accounts for the case where actual msclassfcatons are less than the assumed bound φ.

8 B.R. Handel et al. / Journal of Econometrcs 174 (2013) Model 4: random utlty wth data contamnaton Assumpton 5 (Data Contamnaton Model wth Random Utlty). Indvduals have random utlty wth bounded errors and up to φ percentage of ther decsons could be recorded wth error. Ths model s a combnaton of models 2 and 3 dscussed above. There are two parameters: δ, the bound for the random utlty shocks and φ, the bound for the frequency of data contamnaton. The combnaton of these factors s attractve n stuatons where there s a large probablty that ndvdual preferences change by small amounts over dfferent choce settngs and a small probablty that an ndvdual appears to makes a decson that departs completely from our descrpton of ther preferences. To estmate consumer preferences wth these assumptons gven a feasble T φ, we defne: Θ 4 (T φ ) θ : θ φ T {θ : θ x a θ x t a 2δ a t d t } t (11) where T φ s defned as above. Then, as n model 3, the full partally dentfed set of consumers preferences s defned: Θ 4 T φ Θ 4 (T φ ). (12) As n the tme-consstency case, the partally dentfed probablty of an ndvdual beng of type θ and the feasble populaton dstrbutons of types are gven by Eqs. (4) and (5), respectvely, wth Θ 4 replacng Θ 1.22 Counterfactually, once we determne the partally dentfed dstrbuton of preferences we can derve the probablty that alternatve a s chosen from a choce set D as was done n Eq. (6). As wth model 2, the frm could know δ or calbrate t from a varety of data sources. Secton 3 presents one methodology for how a frm may non-parametrcally select an approprate δ condtonal on φ. In our emprcal exercse n Secton 5, we calbrate φ usng pror work on data contamnaton, and dscuss several alternatve approaches. As an addtonal extenson to each of these four frameworks, n Appendx A we develop a method to use cross-sectonal varaton n conjuncton wth each of these nter-temporal frameworks to obtan further dentfyng power when the panel s not a representatve sample from the populaton Prce endogenety and bounded errors Whle most dscrete choce models rely on exogenous varaton of the ndependent varables, such as prce, our model makes no explct ndependence assumptons and therefore does not requre exogenous varaton. In general, endogenety s a much bgger concern wth aggregate purchase data (see e.g. Berry et al. (1995)) than when the researcher has panel scanner data n our settng (see e.g. Erdem et al. (2003)). Whle endogenety s thus not lkely to be a major concern n our context, we note as a robustness pont that the partally dentfed estmates of consumer preference dstrbutons would not be based even f frms had any addtonal amount of nformaton that they were ncorporatng nto prces. To see ths consder the utlty functon as n Berry et al. (1995) a canoncal model n ndustral organzaton, where u jt = 22 In the case that the true DGP s one where there s some probablty p an observaton s msclassfed, t s possble that a partcular ndvdual has more than φ percent of observatons msclassfed. In that event, our estmate of δ (as descrbed n Secton 3) would ncrease to account for apparent larger devatons n stable preferences from msclassfed observatons not accounted for wth φ. Ths agan would lead to conservatve estmates, as our partally dentfed parameters sets would reflect the larger δ values that would be necessary to ratonalze these purchases. β X jt + ξ jt + ε jt where ξ jt s a common shock that mpacts all consumers. The common assumpton s that the frm observes ξ jt and therefore sets a prce P jt that ncludes ths nformaton. In our specfcatons wth bounded errors, we consder ξ jt to be a part of ε jt, whch, n most standard dscrete choce models wth ndependent and..d. errors would cause prce endogenety as the error term would be correlated wth the ndependent varables. However, n our model we make no ndependence or dentcal dstrbuton assumptons and therefore, wth the caveat that the error term must le wthn the bounds, the model s stll estmated consstently wth endogenous ndependent varables. To see ths explctly, say ξ jt s a negatve shock. In our model ths wll mply all ε jt wll smultaneously receve negatve shocks. However, as long as the shocks le wthn the assumed bound δ, the partally dentfed set wll stll be consstent and contan the true demand dstrbuton. Ths feature remans true allowng for some proporton of the data to be contamnated, as we do n our settng. As wth the bounded error models, so long as the proporton of assumed data contamnaton falls underneath our assumed upper bound for such decsons (we dscuss how ths can be determned systematcally n the next secton), the set of estmated preference dstrbutons s stll consstent and the true determnstc demand curve s contaned wthn ths set. Thus, our model wll be robust to endogenety concerns wth respect to prce or other features of the envronment such as advertsng or marketng Smulaton In order to llustrate our methodology, we study a smulated market where the frm or frms have nformaton on consumer purchase behavor that they use to estmate a demand curve. The smulaton gves us the ablty to study the degree to whch the partally dentfed demand output from the varous proposed models lnks to underlyng preferences. In our smulaton we wll have two products and an outsde opton and n each tme perod consumers decde whch product to purchase (f any) gven the specfed prce. We smulate the preferences of 300 ndvduals from the populaton who obey the utlty specfcaton: U jt = α j + β p jt. Here, α are product fxed effects whch we use to aggregate preferences for all attrbutes except for prce, as well as any other brand specfc utlty component. Ths s wthout loss of generalty for our prcng problem snce we assume that frms n ths market do not change product attrbutes over tme, except for prce. Addtonal nformaton about product attrbutes can only help refne the model further. Consumer chooses product k at tme t gven the decson set d t based on the decson rules n the four models just descrbed. We set the utlty of the outsde opton for each person and tme perod to 0 and normalze the value of α 1 = 1 n order to obtan dentfcaton. Throughout the analyss there are two possble products, so there are two free preference parameters for each ndvdual n the populaton. For ths populaton, we draw α 2 from a unform dstrbuton on [0.5, 1.5] and β ndependently from a unform dstrbuton on the range [ 3.75, 1.75]. We then smulate 208 tme perods of choces (correspondng to four years of weekly data 23 ) of decsons for each ndvdual. In order to do so, we assume that both products are offered n every perod and 23 Most panel data sets avalable to researchers have 4 years of weekly purchase data. We have also expermented wth fewer perods. If we have 50 weeks our overall conclusons do not change.

9 172 B.R. Handel et al. / Journal of Econometrcs 174 (2013) Fg. 1. Sample dentfcaton sets for a consumer. The black dot represents the true preference parameters, and the blue regon represents the partally dentfed regon. (For nterpretaton of the references to colour n ths fgure legend, the reader s referred to the web verson of ths artcle.) that ther prces are drawn ndependently and unformly from the range [0.1, 0.7]. We defne the feasble dentfcaton regon for (α 2, β ) to be [0, 5] [ 5, 0] so that the feasble dentfcaton regon covers a large regon of reasonable relatve preferences. We smulate data for each of the four choce models descrbed above. For model 1 we smulate the data as just descrbed. For the models wth random utlty shocks (models 2 and 4) we set δ = Ths mples that random utlty shocks here are at most 10% of the base value of the preference for product 1. We allow there to be three types of consumers. The frst 100 have..d. errors. The second 100 have errors that are correlated across products. Ths correlaton s generated n each perod by frst drawng ε 1t unformly from the range [ δ, δ], resultng n ε 1t. We then draw ε 2t unformly from [ε 1t δ, δ] f ε 1t > 0 and from [ δ, ε 1t + δ] f ε 1t < 0. The last 100 have errors that are correlated over tme. Ths correlaton s generated for each product by frst drawng ε j1 unformly from the range [ δ, δ], resultng n ε j1. We then draw ε j2 unformly from [ ε j1 δ, δ] f ε j1 > 0 and from [ δ, ε j1 + δ] f ε j1 < 0. We then repeat for perod three (and so on) replacng ε j1 wth ε j2. 24 For models 3 and 4 we set the φ parameter to Ths 24 The average (over consumers) correlaton n the errors across brands s and the average correlaton n the errors between tmes t and t + k s 0.483, 0.223, 0.108, and for k = 1, 2, 3, 4 and 5, respectvely. mples that 10% of the data are contamnated. We also dvde the consumers nto three types: 100 have (up to) 10% of ther decsons ms-recorded (randomly), 100 choose brand 1 ndependent of preferences (up to) 10% of the tme and the other 90% of the tme they make utlty based decsons and the last 100 choose not to buy (up to) 10% of the tme. However, these 10% are more lkely (100 tmes) to occur after a purchase of brand 1 n tme t 1. These rules are ntended to smulate data contamnaton. Gven each ndvdual s choce data, we can partally dentfy her true parameters wthn ths feasble set. The sze of the partally dentfed set vares based on the observed choce behavor and the choce. Below, we gve examples of partally dentfed sets for two consumers, based on the four choce models presented. The frst ndvdual (see Fg. 1) s an example of a consumer who purchases both products (and the outsde opton) at some pont n tme n our generated data for all four choce models. Ths allows us to refne the partally dentfed set of alternatves to a relatvely small regon for all four models. All the estmated dentfcaton regons contan the true parameters. The largest dentfcaton regon wll, by defnton, occur for model 4 snce the data generatng process ncorporates both data classfcaton errors and utlty shocks, mplyng less nformaton about consumers actual preferences than the other specfcatons. The second example, depcted n Fg. 2, s that of a consumer who never purchases product 2 n the

10 B.R. Handel et al. / Journal of Econometrcs 174 (2013) Fg. 2. Sample dentfcaton sets for a consumer. The black dot represents the true preference parameters, and the blue regon represents the partally dentfed regon. (For nterpretaton of the references to colour n ths fgure legend, the reader s referred to the web verson of ths artcle.) entre data sample. Gven these data we cannot acheve such tght bounds. After we have found the partally dentfed set of preferences for all 300 consumers, we aggregate these regons n the manner descrbed for each model to obtan a partally dentfed set for the jont dstrbuton of tastes n the populaton. Ths dstrbuton s generated n a two dmensonal space (α 2, β space) and s used to estmate demand. The demand estmates are shown n two dfferent formats. Frst, we consder a demand curve for each product f t were sold n solaton (wthout the other product) and then estmate demand wth both products sellng. From Fg. 3 we obtan several nsghts. We correctly capture the true demand n the bounds for all of our models. The bounds are tghtest n the model wth the tme-consstency assumpton and are wdest n model 4. Overall, the bounds are tght and can be nformatve for manageral decson-makng. One way of thnkng of ths n terms of parametrc dscrete choce modelng s that any correctly specfed model that descrbes these data must predct demand to be wthn the bounds specfed. Fnally, n Fg. 4 we consder demand n a settng where both products are sold. These representatons show that we do capture large parts of the demand curve wth tght bounds across all models. Once agan, the smallest bounds are wth model 1 and the largest bounds are wth model Estmaton The econometrc framework just descrbed assumes that the researcher and frm know the bound on the ndvdual consumerlevel preference shock δ as well as the maxmum extent of data contamnaton φ. For cases where the frm and researcher know, or assume, δ and φ, there s a drect lnk from the data observed to the partally dentfed preference sets that are our prmary econometrc output, as descrbed n Secton 2. It s mportant to note that our model can be rejected by the data f the values of δ and φ mposed by the econometrcan cannot explan the observed varaton n the data. In most emprcal settngs, researchers wll have lmted nformaton about these parameters. In ths secton we propose a smple method to dentfy and estmate δ n a frst stage that precedes the mplementaton of the framework set out n Secton 2. To do ths, we frst resolve the ssue of where the data contamnaton parameter φ comes from n an emprcal settng, and then dscuss the frst stage estmaton of δ condtonal on that value of φ Data contamnaton There are numerous reasons that data may be contamnated ncludng, but not lmted to, recordng errors, non-response,

11 174 B.R. Handel et al. / Journal of Econometrcs 174 (2013) Fg. 3. Partally dentfed demand curves f products were sold n solaton. The rows represent the four models presented n ths paper and the columns represent the demand curves for product 1 and product 2, respectvely.

12 B.R. Handel et al. / Journal of Econometrcs 174 (2013) Fg. 4. Partally dentfed demand curves f both products are sold. The rows represent the four models presented n ths paper and the columns represent the demand curves for product 1 and product 2, respectvely.

13 176 B.R. Handel et al. / Journal of Econometrcs 174 (2013) or nterpolaton/extrapolaton. Numerous authors have explored the extent, and ssues relatng to dentfcaton and estmaton n the presence of data contamnaton across myrad emprcal settngs (see for example Horowtz and Mansk (1995), Keane (1997), Erdem et al. (1999) or Enav et al. (2010)). In our setup, we cannot separately dentfy both δ and φ usng just the panel data because a large δ can be used to emprcally justfy the large departure from stable preferences represented by data contamnaton. 25 In our approach, we lean on the ablty to dentfy the extent of data contamnaton from past studes or a smple emprcal nvestgaton set up by the researcher or frm. We then use the panel data we observe to estmate δ condtonal on φ. 26 Prevous work, such as Erdem et al. (1999) or Enav et al. (2010) provde excellent sources for researchers to learn about the extent of data contamnaton. Addtonally, n prncple t s feasble n most emprcal contexts for the frm or researcher to perform a smlar type of valdaton study to get ˆφ Estmatng the error bound Once we have a conjecture ˆφ, we can estmate δ n a frst stage that uses the same panel data used throughout the rest of the analyss. Defne δ as the true value of δ we are tryng to recover. We construct an estmator for δ that leverages both the tmeseres and cross-sectonal varaton across consumers n the panel data that we observe. As n Secton 2, we llustrate the estmaton wth homogeneous δ. In practce we can condton estmaton of δ on observable demographcs wthout alterng the methods descrbed here. 27 Defne δ ˆφ T as the lowest δ that can ratonalze consumer s decsons when there are T tme perods observed and ˆφ s the extent of data contamnaton. Ths s the mnmum value of δ that results n the dentfed set for consumer to be a non-empty set condtonal on ˆφ. Explctly from the methodology n Secton 2, ths mples that the δ ˆφ T s the lowest value of δ, such that the set θ : θ ˆφ {θ : θ x a θ x T t a 2δ a t d t } t s non-empty for some feasble T ˆφ, wth the set of feasble T ˆφ as defned n Secton In addton to the assumpton that δ s homogeneous condtonal on observable demographcs, we assume that the bound s tght for the potental purchase data we observe. That s, as we observe nfnte data, the most extreme values of δ wll be realzed for each consumer. Formally, we have lm T P(δ ˆφ = T δ ) = 1. Gven these assumptons, we proceed as follows. In any sample we can defne ˆφ T ordered so that δ ˆφ 1T = {δ ˆφ T, I} and assume that the sample s δ ˆφ IT. A smple estmator of δ s ˆδ = max( ˆφ T ). Ths wll be based downwards relatve to δ. Though ˆδ would be consstent for δ, we follow the econometrc lterature on estmatng boundares and correct for ths bas n our estmator (see the summary of current methods n Karunamun and Alberts (2005)). Denote ths bas as γ. Whle the lterature on estmatng boundares has a varety of sophstcated methods for estmatng the edge of a dstrbuton, Hall and Park (2002) dscuss a smple m-out-of-n subsamplng bootstrap method n whch the estmate of the bas s γ HP m = (δ n +1 δ n )K(/m) K(/m), where K( ) s a nonnegatve functon, whch can be nterpreted as a kernel weghted average of the dfference between consecutve values of δ. We adapt ther approach, whch s for contnuous bounded random varables (as s generally the case n ths lterature) to our settng n whch all admssble values of δ are from a dscrete set (as we have a dscrete set of types) n our model. 29 Despte ths dfference, we show n Monte Carlo exercses n Appendx B that our procedure effectvely corrects for the bas n ˆδ. Defne ˆf (δ ˆφ T ˆφ ) as the emprcal dstrbuton of δt across I for fxed T, whch wll be a dscrete dstrbuton n our settng. Our estmator for γ s: γˆ T = δ ˆφ ˆφ T T (ˆδ δ ˆφ T )ˆf ˆφ (δ T ). (13) For ths estmator to be consstent, we need lm T γˆ T = 0. Ths s true as lm T f (δ ˆφ ˆφ T ) = 0, δ ˆ T δ by our assumpton of a common δ for the populaton n queston. In other words both the smple estmator ˆδ and the bas-corrected estmator ˆδ + ˆγ are consstent for δ, but the latter s more conservatve n the sense that t s less lkely to underestmate δ. 30 The extent to whch γˆ T + δˆ T s upward or downward based n a gven applcaton depends on the nteracton between the assumed form of the bas correcton n Eq. (13) and the data. To evaluate the performance of ths estmator we run a seres of Monte Carlo smulatons n Appendx B. Our smulatons reveal that wth more than 50 consumers and 100 tme perods (reasonable values n the context of exstng panel data sets) our estmator provdes a relable and conservatve estmate for the true δ. Under a varety of data generatng processes, the bas-corrected estmator has very few nstances where estmated ˆδ < δ and s usually qute close to δ comng from above. It s worth notng that, even when the bas correcton we mplement leads to an estmator that slghtly overstates the true δ n the fnte 25 In our model as δ, all data can be ratonalzed as unbounded large random shocks and as φ 1, all data can be ratonalzed as completely contamnated data. 26 We note that, n prncple, we could estmate the model by usng an estmate of δ from outsde the panel data we observe and then estmate φ wth our data condtonal on that value for δ. However, snce t s easy to thnk of how one would construct data valdaton studes outsde of the panel data, but dffcult to thnk about studes that would nform the extent of preference shocks, we beleve that the approach outlned n ths secton s more practcal for most emprcal settngs. 27 It s mportant to pont out that as T becomes large and we observe more purchase data per consumer, t becomes more attractve to try and estmate unobserved heterogenety n δ, condtonal on a demographc profle. In the lmt as T (and there s enough prce varaton), we dentfy the true δ for each ndvdual. The estmaton here recognzes that we do not observe nfnte data n realty, and uses homogenety condtonal on demographcs to help dentfy δ n the relevant group of nterest. 28 ˆφ If a set of preferences s feasble any gven T then t s feasble n general. 29 The lterature on boundary estmaton consders a setup where the econometrcan observes N draws from a contnuous unvarate dstrbuton F wth a unknown and fnte upper boundary. Our estmator below s a dscrete analog to these methods, snce the dstrbuton of ˆδ T s dscrete. As n Sddqu (1960) and Bloch and Gastwrth (2002) we take the dfference between the mth hghest draw (of the N draws) from F and the hghest draw, and weght by 1. As n the Hall and Park (2002) M estmator dscussed n the text, we consder a kernel weghted average of dfferent values of m rangng from 2 to M out of the N total draws. Ths allows us to use the range of draws whle not havng the worry about the fact that, n dscrete data, the dfference between the j and j + 1 hghest draws could often be zero, as one would have to f usng Hall and Park (2002) drectly. 30 Estmatng the choce model defned n ths paper wth a value of δ less than the true δ could result n based estmates of ndvdual types (.e. partally dentfed sets that do not contan true preferences). On the other hand, estmatng the choce model wth a value of δ greater than the true δ wll lead to a loss of effcency (the partally dentfed sets wll be larger), but not a bas. Ths s one reason that we feel t s mportant to correct for the bas n ˆδ wth ˆγ.

14 B.R. Handel et al. / Journal of Econometrcs 174 (2013) sample analyss, n our framework ths leads to more conservatve estmates n the form of larger partally dentfed preference sets. It s mportant to note that, wth the assumptons mantaned n ths secton, δ for the populaton (condtonng on any observable demographcs) s dentfed by the purchases of a gven consumer as T and there s suffcent varaton n prces. Wth fnte data on T, a larger number of consumers I mproves the precson of ˆδ and ˆγ. Also, we note that whle the model can be rejected by the data for the general case where φ and δ are mposed by the econometrcan, for the estmaton methodology proposed here the model wll not be rejected by the data snce δ s estmated as the mnmum ratonalzable value. 4. The frm problem We now demonstrate how a frm whch has partally dentfed demand n the manner descrbed thus far can make strategc decsons n the face of the resultng ambguty about consumer preferences. We focus on perhaps the quntessental frm problem: how to set prces. We examne the cases of a monopoly and a duopoly Monopoly A monopolst observes a panel of ndvdual decsons and makes a decson on what prce to charge to the same target populaton that composes the sample. Gven the framework above, the monopolst observes a range of feasble type dstrbutons that characterze the populaton and uses that nformaton to arrve at a prcng decson. The monopolst wshes to maxmze profts, but cannot necessarly do so n the tradtonal way because he does not know the dstrbuton of types n the populaton and thus can only partally dentfy demand for each potental prce. Gven that the frm does not know the dstrbuton of types, expected proft maxmzaton s not possble. Therefore, we must take a stand on how the frm makes ts prcng decson. As n Bergemann and Schlag (2007), we examne the monopolst s problem usng the mnmax-regret crteron, whch does not ncorporate subjectve belefs on the state space by the decsonmaker. Instead, ths decson-makng crteron s to mnmze the largest possble dstance from what the actual best choce would have been, were t to know the true state of the world ex post. It s conservatve n the sense that t analyses the maxmum regret (that s, the maxmum dstance from the deal value over all possble states), but less conservatve than a pure maxmn crteron. Ths s because the mnmax-regret crteron accounts for devatons from possbly very good outcomes as well as just consderng the worst case scenaro (as would the maxmn crteron). We assume that the frm solves a constraned mnmax-regret problem where the vector of possble prces chosen s fxed, nstead of allowng for random prcng or menu prcng. The monopolst n our settng has data on past purchase decsons by the populaton of consumers and seeks to maxmze profts n a counterfactual settng n whch prces can be set at levels not yet observed n the data for a gven set of products. In our settng, the fundamental state s the dstrbuton F(θ) descrbng the populaton of dscrete types. If the monopolst knew ths state exactly, t could easly construct a demand functon for ts product. We wll denote demand as D(F(θ), p). Here, p s a prce vector and D(F(θ), p) s the demand vector, where both quanttes are vectors because we assume that the monopolst can sell multple brands. A monopolst s regret s a mappng from any chosen prce and gven dstrbuton of preferences nto a scalar whch measures how far the profts resultng from the chosen prce are from the profts that would result from the optmal prce f the canddate dstrbuton of preferences were the true dstrbuton. Maxmum regret for the monopolst, gven a choce of a prce vector p and the potental dstrbutons of types H[F(θ)], s defned as: R(p, H[F(θ)]) = max F(θ) H[F(θ)] p (F(θ))D(F(θ), p ) pd(f(θ), p). (14) The frst term n Eq. (14) denotes the optmal profts for the monopolst f t knew that the true dstrbuton of types was F(θ). Here, p s the prce vector that mplements ths deal proft level. From now on, we wll denote the deal proft level gven a state F(θ) as π (F(θ)). In addton, we wll smplfy notaton by alludng to the quantty pd(f(θ), p) as π(p, F(θ)), the profts earned by the monopolst n state F(θ) gven some chosen prce vector p. The key emprcal challenge s estmatng π(p, F(θ)) for every possble dstrbuton of types. For every potental prce, we calculate demand for each dstrbuton of preferences. We fnd the optmal prce as the prce wth the hghest demand for any gven dstrbuton and regret for any other prce s the dfference between the proft under that prce and the optmal prce. Once we have calculated the regret for every potental prce and dstrbuton combnaton, we can easly calculate the maxmum regret for any prce (maxmum over dstrbutons). Then, we choose the prce whch mnmzes the maxmum regret. To be clear, we can defne the monopolst s mnmax-regret, gven the dentfcaton regon H[F(θ)], as: MMR(H[F(θ)]) = mn p max π (F(θ)) π(p, F(θ)). (15) F(θ) H[F(θ)] For any combnaton of (p, F(θ)), a monopolst s regret wll stem from ether overprcng or underprcng based on whether p s greater than or less than p, respectvely. In the Bayesan setup, ths overprcng and underprcng for each (p, F(θ)) par s weghted by a subjectve Bayesan pror over H[F( )] and regret mnmzaton wth respect to ths weghtng s equvalent to expected proft maxmzaton. We denote the mnmax-regret soluton as p MMR. Snce the mnmax-regret state space s drectly defned by the econometrc exercse, the state space s complex n the sense that t s mpossble to obtan an analytcal soluton to ths problem. Ths s an mportant way n whch the current paper dffers from that of Bergemann and Schlag (2007), snce ther model reles on the set noton of an ε neghborhood, gven some sze ε, and fnds solutons analytcally. In practce, mplementng the mnmaxregret soluton requres a mult-stage algorthm gven H[F(θ)]. Frst, for each dstrbuton F(θ) and each feasble prce vector p, we compute the demand vector D(F(θ), p). Then, we compute the deal proft gven for each F(θ). Afterward, we compute the maxmum regret for each prce vector over the dentfcaton regon for feasble dstrbutons. Fnally, we mnmze these maxmum regrets over all possble prce vectors. Before we move on to the olgopoly problem, we present a stylzed example meant to llustrate the way one can thnk about the monopolst s mnmax-regret problem Stylzed example Ths secton presents a smple example of the monopolst mnmax-regret problem for one good. We assume that there s one preference parameter whch translates drectly nto demand wth feasble values n [1, 2] gven our econometrc nput. Ths s a much smplfed verson of our model, n whch we must frst translate partally dentfed dstrbutons of types nto demand n a non-trval way. The beneft s that t provdes ntuton for mnmax-regret n a very smple framework. Suppose, that gven the possble dstrbutons n H[F(θ)], the monopolst knows that, for a gven p, the range of demand s [1 2p, 1 p], where ths range comes from the mappng D(F(θ), p), taken over H[F(θ)] for

15 178 B.R. Handel et al. / Journal of Econometrcs 174 (2013) each p. We wll assume the margnal cost equals zero for smplcty. The monopolst s mnmax-regret problem s: mn p max π (F(θ)) π(p, F(θ)) F(θ) H[F(θ)] mn p max β [1,2] 1 4β p + βp2. Now, when the monopolst solves for hs maxmum regret over β gven hs choce of prce, he only has to consder two states, β {1, 2}. Ths s because, gven the optmal prce p (F(θ)), the proft functon s monotoncally decreasng on R + gong n both drectons from that optmum snce proft s a quadratc functon of prce. Ths mples that for any gven prce, the maxmum regret wll be one of the endponts of the range of β, snce the optmal prce gven β s monotoncally decreasng n β. Thus, we can map the range of β drectly nto a range of optmal prces, p (β), and for any gven p, maxmum regret wll occur at the maxmum possble dstance from a feasble p (β), whch wll always correspond to an extreme value of β. In our example, the regret functons for p gven β {1, 2} are: R(β = 1, p) = 1 4 p + p2 R(β = 2, p) = 1 8 p + 2p2. The frst functon s mnmzed wth zero regret at p = 1 2, whle the second s mnmzed at p = 1. Each functon s monotoncally 4 ncreasng n both drectons from ts respectve mnmum, so we know that the mnmax-regret must occur n the range [ 1, 1 ] at the 4 2 pont where both of these functons have dentcal regret values gven p. Ths occurs when: 1 4 p + p2 = 1 8 p + 2p2 p MMR = 1. 8 The soluton s easy to verfy. If p s ncreased or decreased from p MMR, the maxmum regret ncreases because one of the two regret functons correspondng to β {1, 2} must ncrease. Ths provdes some nsght nto the MMR soluton that we wll derve n our prcng experments n the next secton. For a gven set of dstrbutons H[F(θ)], there wll be an extreme dstrbuton that corresponds to the mnmum and maxmum demand for a gven prce. For that prce, t wll then be possble to compute the maxmum regret, whch wll then be compared over all prces to derve the fnal soluton, whch wll balance the potental losses from prcng low n a low elastcty state and prcng hgh n a hgh elastcty state Olgopoly In addton to the monopolst s problem specfed above, we analyze a statc olgopoly game. In ths settng, every frm shares the same nformaton set and evaluates payoffs accordng to mnmax-regret over H[F(θ)], gven the other frms prces p. Ths extends the assumpton that the frm and the researcher have the same nformaton set to one where both frms and the researcher have the same nformaton set, whch we beleve s more reasonable n stuatons where frms observe smlar lmted data to base ther prcng decsons on. 31 A further more detaled model where frms have only partal nformaton about the other frms nformaton sets (or ther perceptons of the dstrbuton of types) would be nterestng, but for now we stck to ths base case and leave ths extenson to future work. Let frms be ndexed by j correspondng to J dfferent sets of brands. The frms play a game where each evaluates outcomes by mnmzng maxmum regret over possble prce vectors gven 31 Our emprcal example s one stuaton where ths may be reasonable. ther opponents prces. The frms evaluate maxmum regret for p j gven the opponents prce vector p j as follows: R(p j, p j, F(θ)) = max F(θ) H[F(θ)] p (F(θ), j p j)d(f(θ), p, j p j) p j D(F(θ), p j, p j ). (16) Here, the frm evaluates regret at a gven state of nature condtonal on hs opponents prces. Hs mnmax-regret gven p j s: MMR(H[F(θ)], p j ) = mn p j max π (F(θ), p j ) F(θ) H[F(θ)] π(p j, p j, F(θ)) (17) π (F(θ), p j ) s the deal proft for frm j, gven a specfc dstrbuton of types drawn from the dentfcaton set and p j. π(p j, p j, F(θ)) s the proft for frm j gven the type dstrbuton and opponent s prce. In the game that the frms play, the acton space s the set of feasble non-negatve prces, and we restrct each frm to the use of pure strateges. The game s one of complete nformaton between players n the sense that each frm knows the uncertanty faced by the other wth respect to the dstrbuton of types. We assume that both frms have common knowledge and look for a pure strategy Nash equlbrum n prce vectors. We say that frm prces p NE are a Nash equlbrum f the followng best response condtons are smultaneously satsfed: p NE j arg mn p j max π (F(θ), p NE ) j π(p j, p NE, F(θ)), j F(θ) H[F(θ)] j J. (18) In our smulaton descrbed n the next secton we fnd a pure strategy equlbrum usng the best response curves of each frm to hs opponent s prce, gven the dentfcaton regon H[F(θ)]. 5. Emprcal analyss The purposes of ths secton are () to compare the results of our model to those of the most commonly used dscrete choce models and () to show an example of how our method can be appled to data. We wll start wth a smulaton experment n Secton 5.1 where we show that when the underlyng data volate the..d. assumpton of the standard dscrete choce models, our model provdes a more relable and robust prcng recommendaton. We then consder panel data from mlk purchases from two competng retalers n Secton 5.2. We show that our methodology s applcable n ths real-world settng and that t returns sensble counterfactual recommendatons Smulaton experment of the frm problem In order to llustrate our methodology, we study a smulated market where the frm or frms have nformaton on consumer purchase behavor that they use to determne how prces should be set. The smulaton gves us the ablty to study how solvng the frm problem wth our method compares to what the frm would do f t knew the true dstrbuton of preferences n the populaton. In addton, t allows us to study the predctons of our model compared to more famlar models, such as a mxed logt model, when estmated wth the same data. We smulate the preferences of 100 ndvduals wth utlty u jt = α j + β p jt, as was done n Secton We assume that consumer chooses product k at tme t gven the decson set d t 32 For clarty, as was done n Secton 2.3, we set the utlty of the outsde opton for each person and tme perod to 0 (locaton nvarance) and normalze the value of α 1 = 1 (scale nvarance) for dentfcaton. Throughout the analyss

16 B.R. Handel et al. / Journal of Econometrcs 174 (2013) Table 1 Results of monopoly prcng from smulaton. See text for detals on smulaton desgn. Consumer type Prcng model Prce 1 Prce 2 % of optmal profts..d. shocks ex post effcent Mxed logt Mnmax-regret Brand correlated shocks ex post effcent Mxed logt Mnmax-regret Tme correlated shocks ex post effcent Mxed logt Mnmax-regret based on the decson rules n model 2 descrbed n Secton 2. We set δ = 0.20 (for ths exercse, φ = 0,.e. there s no data msclassfcaton). Gven our parameterzaton, ths mples that random utlty shocks here are at most 20% of the base value of the preference for product 1. We smulate three types of consumers. The frst type have..d. errors drawn from a unform dstrbuton between [ δ, δ ]. The second type have errors that are correlated across products. Ths correlaton s generated n each perod by frst drawng ε 1t unformly from the range [ δ, δ], resultng n ε 1t. We then draw ε 2t unformly from [ε 1t δ, δ] f ε 1t > 0 and from [ δ, ε 1t + δ] f ε 1t < 0. The thrd type have errors that are correlated over tme. Ths correlaton s generated for each product by frst drawng ε j1 unformly from the range [ δ, δ], resultng n ε j1. We then draw ε j1 unformly from [ ε j1 δ, δ] f ε j1 > 0 and from [ δ, ε j1 + δ] f ε j1 < 0. We then repeat for perod three (and so on) replacng ε j1 wth ε j2. We proceed frst by partally dentfyng each consumer s preferences as was llustrated n Secton 2.3. Gven each ndvdual s choce data, we can partally dentfy true parameters as beng wthn ths feasble set. Next we turn to the ndustry prcng problem. To evaluate our model we compare t wth two benchmarks: (1) ex post effcent prces based on true parameters and (2) a mxed logt wth multvarate normal mxng. 33 Monopoly We begn by consderng a mult-product monopolst possessng the purchasng decsons of each set of 100 consumers over 100 perods. The frm must now set prces for each of ts goods. In Table 1 we show three optons for the optmal prces: (1) ex post effcent prces, from the smulated values of each ndvdual; (2) optmal prces from the mxed logt model where we consder the prces that maxmze expected profts; and (3) optmal prces from the mnmax-regret model as descrbed n Secton 4. there are two possble products, so there are two free preference parameters for each ndvdual n the populaton. For ths populaton, we draw α 2 from a unform dstrbuton on [0.5, 1.5] and β ndependently from a unform dstrbuton on the range [ 3.75, 1.75]. We then smulate 100 tme perods of choces (correspondng to about two years of weekly data) of decsons for each ndvdual. In order to do so, we assume that both products are offered n every perod and that ther prces are drawn ndependently and unformly from the range [0.1, 1.0]. We defne the feasble dentfcaton regon for (α 2, β ) to be [0, 5] [ 5, 0] so that the feasble dentfcaton regon covers a large regon of reasonable relatve preferences. 33 Our mxed logt model s specfed as u,j,t = α,j + β P j,t + ε,j,t for j = 1, 2 and u,0,t = ε,0,t. We assume the ε are dstrbuted..d. Type 1 Extreme Value. For heterogenety we assume (α,1, α,2, β ) N((α 1, α 2, β), Σ). We use smulated maxmum lkelhood wth 100 draws to estmate α 1, α 2, β and the Cholesky decomposton of Σ (we follow the estmaton procedure n Revelt and Tran (1998)). Fg. 5. Best response curves n duopoly smulaton. See text for detals on smulaton desgn. Note that for the mxed logt and the mnmax-regret model we consder prces n 0.09 ncrements between 0.09 and The frst set of results n Table 1 are for the case where consumers are drawn wth..d. errors. In ths case the mnmaxregret model estmates prces that are close to ex post, wth frms earnng nearly 100% of potental ex post profts. The mxed logt optmal prces are also close to the ex post effcent ones, and recover nearly 90% of the optmal ex post profts. Ths suggests that both models recommend nearly ex post effcent prces when consumers have..d. error draws. The second and thrd sets of results n Table 1, where consumers have ether brand correlated error shocks or tme correlated error shocks, are notceably dfferent. Here whle the mnmax-regret model sll recommends prces close to ex post effcent prces, the optmal prces from the mxed logt are far too hgh. Ths s partcularly evdent for consumer type 3 wth tme correlated shocks. Duopoly We now turn to the case of multple sngle product frms n a dfferentated goods ndustry. We focus on the duopoly case, pool all 300 smulated consumers from the monopoly experment (100 consumers of each type) and now assume that the two products are sold by two dfferent frms. Each frm must now choose the prce for ts good takng nto account what the other frm wll do. We solve for the equlbrum of the prcng game by fndng the ntersecton of the frms best response curves, depcted n Fg. 5. We also show the true ex post effcent response curves for the frms n ths fgure. Here the mnmax-regret model estmates best response curves close to the ex post effcent best response curve. The model recommends duopoly prces of 0.18 for each product. In comparson, the mxed logt model here would recommend duopoly prces of 0.90 for each product. Ths occurs because, as before, consumers n ths smulaton have non-..d. error draws Usng a fner grd dd not materally affect the qualtatve fndngs n ths secton. 35 As a pont of further comparson, we have estmated the optmal prces wth a dfferent non-pror based decson rule: maxmn. Under maxmn preferences, the

17 180 B.R. Handel et al. / Journal of Econometrcs 174 (2013) Fg. 6. Observed prces for prvate label, 1 gallon mlk for two competng stores n Pttsfeld, MA. Table 2 Smulaton results for monopoly and duopoly prcng when δ s estmated. Actual δ used to generate data s See text for more detals on smulaton desgn. δ Monopoly Duopoly Prce 1 Prce 2 Prce 1 Prce 2 Assumed to be Estmated to be Assumed to be Smulaton results wth estmatng δ The smulaton results for the mnmax-regret model for both the monopoly and duopoly results assume that we know the true value of δ. In ths secton estmate a value of δ, as descrbed n Secton 3.2, and then determne optmal prces from the mnmaxregret model. We use the same 300 consumers from above and reestmate the monopoly and duopoly prces wth an estmated value of δ. Recall that the true value of δ used to generate the data was For these data, the lowest value of δ that can ratonalze all observed decsons s Wth the bas adjustment, the estmated value of δ s 0.285, whch s a conservatve estmate of We report the recommended prces when we assume we know the true δ and when we estmate t to be n Table 2. For both the monopoly and duopoly cases, the suggested prces based on the estmated δ are the same as when we know δ s true value. Ths need not be the case. For example, f we assume δ = 0.5 and agan make optmal recommended prces, they wll dffer substantally as the fnal row n the table llustrates Feld data analyss The purpose of ths secton s to provde an llustratve example to show how our method can be used to study prcng n an emprcal context wth feld data. We apply our model to sx years of IRI panel data from the flud mlk category n Pttsfeld, MA. Mlk s a frequently purchased, non-storable, product category n whch the top sellng UPCs are prvate label brands. One reason for consderng mlk s that t s non-storable and s thus unlkely to be stockpled. Ths s mportant because stockplng, n addton to creatng complex error structures that could nvaldate standard logt model assumptons, wll generate a dynamc choce process, the modelng of whch s beyond the scope of ths paper (for papers that do model ths behavor see Erdem et al. (2003) or Hendel and Nevo (2006)). We choose to study the prcng decsons of two neghborng retalers that have the hghest unt sales n the IRI panel data n Pttsfeld, MA (see Bronnenberg et al. (2008) for a detaled dscusson of these data). As s common for retal mlk prces, both stores charge the same prce for all prvate label, one gallon mlk products ndependent of fat content (Khan et al. (2012)). For these two stores we observe nearly 6 years (297 weeks) of panel data for 396 panelsts. 36 Fg. 6 dsplays the average weekly prces for each store over the sample. The medan prces n the data for the two stores over the 6 years are $3.20 and $3.22 per gallon, respectvely. The correlaton n prces across stores s On average, 9% of panelsts buy from store 1 and 17% of panelsts buy from store 2 n a gven week. We fnd 55% of the panelsts make at least one purchase n each store over the sx years. To compute optmal prces we must have a measure of storelevel margnal costs. The majorty of ths cost s lkely to be the wholesale prce the stores pay for mlk. As a proxy for wholesale prces, we collect average monthly Co-op prces 37 n Massachusetts for 2008 (data from USDA s Agrcultural Marketng Servces). Note ths tme perod overlaps wth the last 6 months of our panel data. In the overlappng tme perod we fnd the average wholesale prce s $0.60 per gallon and the average retal prces at the two stores are $2.68 and $2.45 per gallon, respectvely. frm chooses the prce that maxmzes ts profts gven the realzaton of worstcase demand for that prce, selected from the set of feasble demand curves (see e.g. Glboa and Schmedler (1989)). We fnd that, consstent wth the arguments dscussed n Bergemann and Schlag (2008) and Mansk (2005), the maxmn crteron tends to prescrbe overly conservatve decsons. For example, n the duopoly case we fnd that the maxmn duopoly prces are 0.09 and 0.09 and the model recovers only about 50% of the ex post optmal profts. 36 We consder panelsts who make at least 50 mlk purchases from ths store n sx years. 37 We wll assume that ths s each frms margnal costs. It s realstc to assume that each pays the same wholesale prce, but unrealstc to assume that ths wholesale prce s the entrety of each frm s margnal cost, and therefore we vew our optmal prces lkely as beng lower bounds on the true optmal prces for each store.

18 B.R. Handel et al. / Journal of Econometrcs 174 (2013) To apply our model n ths settng we frst need to calbrate a value of φ. Enav et al. (2010) study a panel scanner data settng smlar to our own, and report two estmates for the level of data contamnaton. Frst, they report that about 20% of purchase trps n the panel scanner data that they observe are ncorrectly recorded. Second, they report that approxmately 50% of actual purchase trps are omtted n the scanner data. In our data, consumers purchase from ether store n 26.5% of weeks. From ths analyss, we construct two alternatve measures of the extent of data contamnaton that mght be present n our data, φ, defned as the maxmum proporton of purchase recordng errors per ndvdual. Our frst measure assumes that only the frst type of error (observed purchases that are ncorrect) occurs. Ths suggests that 20% of the 26.5% of purchases made every week are contamnated, or that 5.3% of all purchase/no-purchase decsons are contamnated. Our second measure allows for data contamnaton stemmng from purchases that are not recorded, suggestng that 10.6% (50% of 26.5% 5.3%) purchases were never recorded n the scanner data n our context. As there s some room for nterpretaton n the way the Enav et al. (2010) results lnk to our settng, we estmate δ and study frm prcng n the two alternatve cases where φ = and φ = Whle the Enav et al. (2010) study s the best source for calbratng a value of φ n our context, t s not deal. Ths s because they focus on Nelsen Homescan data n whch consumers record ther purchases at home. Although some panelsts n the IRI data we use also record ther purchases at home, many scan them n the store (see SymphonyIRI (2012) or Bronnenberg et al. (2008) for more nformaton on the IRI method of data collecton). 39 It s lkely that the extent of data recordng error s greater for those scannng at home as opposed to at the store. 40 Therefore, for nternal consstency and robustness we also nvestgate frm prcng for smaller φ equal to 0.026, or half the lowest value calbrated from the Nelsen Homescan data. 41 For φ = 0.053, we fnd that the lowest value of δ that can ratonalze all observed decsons n the data s Usng the bas correcton methodology dscussed n Secton 3.2, the estmated value of δ as Usng δ = and φ = we estmate preferences and compute the mnmax-regret best response curves for each store (assumng a wholesale prce of $0.60 per gallon). Our estmates yeld a unque pure strategy equlbrum where one store charges $2.40/gallon and the other charges $2.35/gallon for prvate label mlk. 42 We beleve that these are reasonable estmates for ths market snce they are smlar to the observed prce levels n the raw data, where the prces between the stores are often close together (the medan weekly prce dfference between the two stores s zero). Repeatng the analyss for φ = yelds 38 If we assume that the probablty a gven observaton s msclassfed equals 0.053, as opposed to the upper bound on the proporton of msclassfed observatons per ndvdual, a bnomal calculaton reveals that, wth 100 purchases per ndvdual n the panel, 99.3% of ndvduals wll have 11 msclassfed observatons or less, so only very few ndvduals would volate the hgher bound φ = The frm prcng results are qute robust to doublng φ to 0.106, as shown below. 39 As dscussed n Bronnenberg et al. (2008), of the panelsts n our data, 47% only scan purchases n the store and 53% scan ther purchases both n the store and at home. 40 Addtonally, the type of data recordng error may dffer across the two data collecton methods as well. Snce only the extent of data recordng error matters for our model, we gnore ths addtonal dfference n data recordng errors n the current paper. 41 In 2009, IRI and Nelsen merged ther panel data collecton operatons and, from that pont on, these data sets () reflect the same panel of consumers and () come exclusvely from home scanned data (see Nelsen (2009)). 42 The bootstrap standard errors, based on 5000 teratons, for the prces are 0.02 and 0.05, respectvely. equlbrum prce estmates of $2.50/gallon and $2.35/gallon for stores one and two respectvely. For the lower value of φ, 0.026, estmated equlbrum prces are $2.40/gallon and $2.30/gallon. Ths suggests that our prce estmates are relatvely robust to the possblty that the IRI data that we perform the analyss wth has a lower amount of data recordng errors than the Nelsen Homescan data that we use to calbrate φ. 6. Concluson Ths paper presents an econometrc framework that partally dentfes consumer preferences and market demand under weak assumptons n a settng wth panel data. The dentfcaton restrctons we make combne non-parametrc methods from the Samuelsonan revealed preference tradton wth an attrbute based product representaton commonly used n the dscrete choce lterature. Overall, the cost of mantanng very weak assumptons on the structure of consumer utltes s that we produce bounds on the dstrbuton of preferences nstead of a pont estmate, as almost all work n ths area does. However, the predctons that we can make wth our methodology are more credble than those made under the tradtonally strong assumptons found n ths lterature. We vew ths work as complementary to past work snce the results from an approach wth more (correct) assumptons should fall wthn the bounds that our model provdes, whle the placement of pont dentfed results relatve to our bounds shed lght on the potental drecton of bas n these results. The purpose of developng and analyzng our econometrc framework s to better model and understand how frms prce n an envronment wth lmted nformaton leadng to a hgh level of uncertanty. We characterze ths uncertanty here wth the noton of ambguty though ths s just one possblty for how frms wth lmted nformaton could prce (relyng more on smple heurstcs s another possblty). We model the frm prcng problem under ambguty, borrowng from the theory lterature, and lnk ths to the econometrc framework by assumng that the ambguty the frm faces s descrbed by the partally dentfed dstrbuton of the demand curve. We nvestgate ths jont framework n both monopoly and olgopoly settngs to ncrease the scope of the framework. Perhaps the most substantal contrbuton of ths paper s to develop a jont theoretcal and emprcal framework that s a credble alternatve to the full nformaton mxed logt and expected proft maxmzaton workhorse model used to analyze frm decson-makng. Through smulatons we show how our framework performs relatve to ths standard approach, concludng that a robust prcng framework can perform as well or better n terms of predctng what prces frms actually choose. We also llustrate how the methodology can be appled n an actual emprcal settng. In stuatons where frms have lmted nformaton and the combned logt-expected proft maxmzaton approach seems mplausble, we provde a credble framework that can be used to study standard ndustral organzaton problems wthout assumng such a hgh bar for frm knowledge and decsonmakng. Acknowledgments We especally thank Charles Mansk. We also wsh to acknowledge helpful comments from Erc Anderson, Karsten Hansen, Igal Hendel, Al Hortacsu, Carl Mela, Avv Nevo and Mchael Whnston. We also thank semnar partcpants at Northwestern, London Busness School, UC Berkeley, Stanford and the 2011 QME conference for useful feedback and comments. Fnally, we would lke to thank the extremely helpful comments of two anonymous referees that have substantally mproved the paper.

19 182 B.R. Handel et al. / Journal of Econometrcs 174 (2013) Appendx A. Cross-sectonal dentfcaton: for onlne publcaton In many cases a panel sample may not be representatve of the entre market. In ths appendx we present a framework to use cross-sectonal dentfcaton n conjuncton wth panel dentfcaton to form bounds on demand parameters when the panel s not representatve of the populaton. We add two elements to the models outlned n Secton 2. Frst, we post that n addton to observng the panel across each of T tme perods, we now also observe aggregate purchase data for each tme perod. Second, we assume that the panel accurately represents q% of the populaton. For example, f the projected market sze were 100,000 and the panel sze were 2000, the assumpton that the panel represents 80% of the populaton mples that the demand estmates from the panel represent the quantty demanded by 80,000 of the 100,000 n the populaton. We ncorporate cross-sectonal dentfcaton to develop bounds on the populaton dstrbuton of preferences usng the aggregate data and panel data together n each tme perod. Snce the panel represents the same q% of the populaton n each tme perod, once we account for the nformaton learned from the panel n the aggregate data, the remanng (100 q)% of the populaton s the same over each tme perod. Partton the overall populaton Ψ nto two sets: Γ, the porton represented by the panel, and Υ, the porton not represented by the panel (Γ Υ Ψ ). Our dentfcaton proceeds n two steps. Frst, we use one of the four models developed n Secton 2 to partally dentfy the dstrbuton of preferences n the panel sample. Next, we represent the purchases made by Γ wth the data from the panel, proportonally scaled up, and construct an observaton each perod for aggregate purchases made by the part of the populaton not represented by the panel. If Q t (Ψ ) s the vector of purchases n perod t for the entre populaton and Q t (Γ ) s the vector of purchases for the populaton represented by the panel, the aggregate purchase observaton for Υ n each t s: Q t (Υ ) = Q t (Ψ ) Q t (Γ ). (19) To place bounds on demand parameters for the entre populaton, we combne the bounds on preferences derved from the panel data wth a bound on the aggregate preferences for the remanng populaton derved from observng Q t (Υ ) over all tme perods. Once we construct the resdual observaton Q t (Υ ) from aggregate and panel data, dentfcaton of the preferences of consumers n Υ s ndependent of the panel preference dentfcaton. It s mportant to note that the assumpton that the panel represents q% of the populaton s testable wthn our framework. If the bounds on the preference parameters of Υ are the empty set, then as long as we accept the assumptons on nter-temporal varatons n preferences from the panel model n Secton 2 that we are usng, then q% s assumed to be too large. For example, ths would be the case f Q t (Υ ) ever has any negatve entres. Ths s a one-sded test snce the data wll never reveal that q s too low. To partally dentfy the dstrbuton of preferences n Υ we construct tghtest bounds from the seres of observatons (Q t (Υ ), p t ) where p t s the prce vector for each t. We use two theoretcal restrctons that must be satsfed by the populaton Υ. The frst concerns domnated prce movements. Here x denotes a specfc product and x denotes all other products. Condton 1 (Purchase Consstency I). t t, x, Q xt Q xt f p xt p xt and p x t p x t. (20) Ths condton says that f the prce of one product goes down and the prces of all other products go up, then we must see a hgher aggregate purchase level for x n Υ. The second condton concerns purchase behavor of the outsde opton x 0 relatve to certan types of prce changes. Condton 2 (Purchase Consstency II). t t Q x0 t Q x0 t f p t p t. (21) Ths condton states that f the prces of all goods go weakly up or down, then the amount of ndvduals not purchasng also moves weakly up or down. We further defne the followng two objects: Φ x (p c ) p : p x p xc, p x p x c x (p c ) p : p x p xc, p x p x c. For any counterfactual prce vector p c, Φ x (p c ) s the set of feasble prce vectors such that the prce of product x s weakly greater than p xc (the prce of product x n the vector p c ) and the prce of every other product s weakly smaller than ts prce n p c. x (p c ) s the converse, where p x s weakly lower than p xc and all other products are weakly more expensve. We can now defne, gven purchase consstency condtons I and II, the bounds on demand for product x D under counterfactual prce vector p c for Υ : H[P(y(D) = x) Υ ] = [max p1,...,p T Φ x (p c )Q xt, 1 max p1,...,p T x (p c )Q x t ]. (22) The bounds on preferences and counterfactual demand for the resdual populaton Υ can then be combned wth those from the populaton represented by the panel, Γ, to fnd the bounds for preferences and counterfactual demand for the entre populaton Ψ : H[P(y(D) = x) p c ] = qh[p(y(d) = x) Γ, p c ] + (1 q)h[p(y(d) = x) Υ, p c ]. (23) The smulaton n the next secton reveals that both the panel and cross-sectonal components of ths model add sgnfcant predctve power by tghtenng the bounds on preferences and counterfactual demand. Smulaton wth cross-sectonal and panel data To smulate the scenaro where we have cross-sectonal and panel data, we smulate two sets of ndvduals. The frst set represents the panelsts and second set represents the consumers that are not represented n the panel. For the analyss we wll observe (a) all decsons made n every tme perod by the panelsts and (b) aggregate cross-sectonal decsons (across both groups) for every tme perod. In ths settng we consder the preferences of the panelsts to represent 80% of the entre populaton. The utlty formulaton for the panelsts s exactly as n Secton 2.3. For the consumers not represented by the panel, we draw the α 2 parameter from a unform dstrbuton on [0.0, 2.0] and β ndependently from a unform dstrbuton on the range [ 2.5, 1.0]. Therefore, these consumers are on average less prce senstve and have more vared tastes for product 2 than the panelsts. We smulate 200 tme perods of data and smulate each panelst s ndvdual decsons and the aggregate decson across all consumers. We estmate the demand curves n four steps. Frst, we estmate the dentfcaton regon for each panelst based on ther purchase data. Second, we bound the counterfactual demand for each panelst. Thrd, we estmate the counterfactual bounds for the aggregate consumers not represented n the panel. Fourth, we estmate the populaton s demand curve bounds by addng the panel and the aggregate estmates.

20 B.R. Handel et al. / Journal of Econometrcs 174 (2013) Fg. 7. Partally dentfed demand curves f products were sold n solaton. The rows represent the dfferent levels of data and the columns represent the demand curves for product 1 and product 2, respectvely. Fg. 7 below dsplays three sets of bounds for each product s demand curve. The frst row represents the demand curves usng both the panel and the cross-sectonal data. The second row represents the demand curve usng just the panel data and the thrd row represents the demand curve usng the cross-sectonal data for the consumers not represented by the panel. Observe that the estmated bounds wth only the crosssectonal data are qute wde. Ths suggests that whle these data do provde some nformaton, we cannot tghtly bound the counterfactual demand. On the other hand, as we have seen above, we can tghtly bound the counterfactual demand wth panel data. The dfference between these two charts shows the addtonal beneft of panel data n estmatng tght counterfactual bounds. We can now combne both peces of nformaton to create a bound for the entre populaton (top row of Fg. 7). In these charts observe that we get qute tght bounds n the mddle of the demand curve

21 184 B.R. Handel et al. / Journal of Econometrcs 174 (2013) Fg. 8. Partally dentfed demand curves f both products were sold. The panels represent the demand curves for product 1 and product 2, respectvely. Table 3 Table represents the percentage of estmated δ that are below the true value 0.10 by changng the number of consumers and the number of tme perods. See text for more detals on smulaton desgn. Number of consumers Number of tme perods 50 (%) 100 (%) 500 (%) 1000 (%) Fg. 9. Box plot of the estmated maxmum of a dstrbuton based on a small sample. Here we consder 100 experments wth 20 ndvduals. The experments dffer n the number of observatons per consumer. The dark black lnes represent the mean. The box represents the nner quartle range. The whskers extend to the most extreme data pont whch s no more than 1.5 tmes the length of the box away from the box. The dots represent outlers that le outsde the whskers. (For nterpretaton of the references to colour n ths fgure legend, the reader s referred to the web verson of ths artcle.) for prces between 0.35 and However the bounds are qute wde for hgher prce ponts. Ths s manly drven by the fact that we estmate wde bounds for the cross-sectonal data at hgh prce ponts. Overall ths demand curve can be nformatve and can be used for frm decson-makng. In Fg. 8 we dsplay the jont demand curves when both products are sold. Appendx B. Monte Carlo studes of the procedure to estmate δ: for onlne publcaton In ths secton we study the performance of our estmator of δ, the bound on consumers utlty shocks. We do ths through two Monte Carlo studes. In the frst study, we abstract from any choce model and study our estmator of the upper bound of the support from whch a random varable s drawn. The study s desgned as follows. Fx some number of tme perods T and set the number of consumers to be 20. For each of these 20 consumers, draw T realzatons of δ unformly from [0, 0.10] and compute the maxmum of the T draws for each of the 20 consumers, yeldng 20 maxmum values of δ. Then use these 20 maxmum values of δ to estmate the upper bound of the nterval from whch δ was drawn (the true value s 0.10) as descrbed n Secton 3.2. We then repeat ths estmaton procedure 100 tmes for each nteger value of T between 1 and 100. The results are reported n Fg. 9. These suggest that we have a conservatve estmate of the true maxmum and as the number of realzatons ncrease, we asymptote to the correct value. The second Monte Carlo study we perform nvolves an actual choce model. Each study nvolves N consumers and T tme perods over whch choces are made. Wthn a consumer, a tme perod s dstngushed by the value of δ nt for that consumer n durng a tme perod t, whch s drawn unformly from [0, 0.10]. We use the same smulaton as n Secton 2.1. Each tme perod, a consumer makes a choce accordng to buy one of the two goods n the market or the outsde opton. Then we choose the lowest value of δ for each consumer that can ratonalze the choces he made durng the T tme perods. Then we collect these N-lowest-values-of-δ and estmate δ accordng to the method descrbed n Secton 3.2. We search over a grd of possble δs wth spacng. Fnally, we repeat 40 tmes for each combnaton of N and T. The mportant statstc here s to understand how often we estmate a value of δ that s less than 0.10 as ths can lead to a bas n our dscrete choce estmates. The results are reported n the Table 3, whch shows that for reasonable numbers of consumers or tme perods our method performs well. In Fg. 10 we report the box plot for the estmates of δ from observng 50 consumers. Once agan these do suggest that we have a conservatve estmate of the true maxmum and as the number of tme perods ncrease we observe less varance across smulatons.