Micro-Demand Systems Analysis of Non-Alcoholic Beverages in the United States: An Application of Econometric Techniques Dealing With Censoring

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1 Mcro-Demand Systems Analyss of Non-Alcoholc Beverages n the Unted States: An Applcaton of Econometrc Technques Dealng Wth Censorng by Pedro A. Alvola IV Program Assocate Department of Agrcultural Economcs and Agrbusness, Unversty of Arkansas, Fayettevlle, AR Oral Capps Jr. Executve Professor Holder of the Southwest Dary Marketng Endowed Char Department of Agrcultural Economcs, Texas A&M Unversty, College Staton, TX and Xmng Wu Assocate Professor, Department of Agrcultural Economcs, Texas A&M Unversty, College Staton, TX Selected Paper prepared for presentaton at the Agrcultural & Appled Economcs Assocaton s 2010 AAEA, CAES & WAEA Jont Annual Meetng, Denver, Colorado, July 25-27, Copyrght 2010 by Pedro A Alvola IV, Oral Capps, Jr and Xmng Wu. All rghts reserved. Readers may make verbatm copes of ths document for non-commercal purposes by any means, provded ths copyrght notce appears on all such copes

2 Mcro-Demand Systems Analyss of Non-Alcoholc Beverages n the Unted States: An Applcaton of Econometrc Technques Dealng Wth Censorng Abstract A censored Almost Ideal Demand System (AIDS) and a Quadratc Almost Ideal Demand System (QUAIDS) were estmated n modelng non-alcoholc beverages. Fve estmaton technques were used, ncludng the conventonal Iterated Seemngly Unrelated Regresson (ITSUR), two-stage methods such as the Heen and Wessells (1990) and the Shonkwler and Yen (1999) approaches, the generalzed maxmum entropy method and the Amemya-Tobn framework of Dong, Gould and Kaser (2004). Our results based on varous specfcatons and estmaton technques are quanttatvely smlar and ndcate that prce elastcty estmates have a greater varablty n more hghly censored nonalcoholc beverage tems such as tea, coffee and bottled water as opposed to less censored non-alcoholc beverage tems such as carbonated softdrnks, mlk and frut uces. Key Words: Censored demand systems, AIDS, QUAIDS, two- step methods, generalzed maxmum entropy, Amemya-Tobn Framework, and non-alcoholc beverages

3 1 The move towards dfferent dets that favor nutrtous foods has n recent years led to the emergence of healther and natural food choces. In partcular, manufacturers and retalers have been responsve n ntroducng new products to the non-alcoholc beverage ndustry, especally uces, energy drnks and others. Ths paper focuses on the nterdependences and demand for certan non-alcoholc beverages, namely; frut uces, tea, coffee, carbonated soft drnks, mlk and bottled water. In the case of the nonalcoholc beverage complex, these products have dfferent levels of market penetraton. Consequently, the consumpton/expendture varables assocated wth these non-alcoholc beverages are censored at zero. That s, certan households have zero expendture, but the correspondng nformaton on household characterstcs, whch forms the bass of the explanatory varables are often readly observed. Several competng estmaton methods have been developed n order to address the censorng ssue n the estmaton of mcrodemand systems. As mcrodata become ncreasngly avalable and more detaled, the estmaton of mcro-demand systems at the household level becomes problematc due to censorng. To our knowledge, no pror research has been done n terms of comparng these respectve approaches wth regard to a partcular data set. In ths study, the demand systems employed were the Quadratc Almost Ideal Demand System (QUAIDS) (Banks, Blundell and Lewbel, 1997) and Almost Ideal Demand System (AIDS) (Deaton and Mulbauer, 1980). The advantages of the QUAIDS model are ts flexblty n ncorporatng nonlnear effects and nteractons of prce and expendtures n the demand relatonshps. Snce the data used are at the household level, censorng typcally s observed as some households report expendtures of a beverage

4 2 product say coffee and none on say bottled water. Thus, n order to model the censorng problem n demand systems, the research utlzed estmaton procedures ncludng twostep estmators (Heen and Wessells, 1990; Shonkwler and Yen, 1999), the maxmum entropy method (Golan, Judge and Mller, 1996) and the maxmum smulated lkelhood estmaton method (Dong, Gould and Kaser, 2004). The terated seemngly unrelated regresson (ITSUR) estmaton wthout adustments for censorng serves as a baselne of comparson for the aforementoned estmaton technques. Fnally, the source of data s the 1999 Nelsen Homescan Panel due to ts vast array of household demographc nformaton. Lterature Revew The Quadratc Almost Ideal Demand System (QUAIDS) commonly s used n appled work. For example, Dhar and Foltz (2005) utlzed a Quadratc AIDS model to estmate values and benefts derved from recombnant bovne somatotropn (rbst)-free mlk, organc mlk and unlabelled mlk. Ther study reled on the use of tme-seres scanner data pertanng to mlk consumpton from 12 key ctes n the Unted States. Ther fndngs ndcate that rbst-free mlk and organc mlk are complements, whle conventonal mlk and rbst-free mlk as well as conventonal mlk and organc mlk are substtutes. The respectve own-prce elastcty estmates were for rbst-free mlk, for organc mlk and for conventonal mlk. Lkewse, a study done by Mutuc, Pan and Reesus (2007) nvestgated household demand for vegetables n the Phlppnes usng the QUAIDS model. Ther fndngs ndcated sgnfcant dfferences n expendture elastctes between rural and urban areas

5 3 whereas for the respectve own-prce and cross-prce elastctes, no sgnfcant varatons across rural and urban areas were evdent. Dhar and Foltz (2005) encountered no censorng ssues, and subsequently used the Full Informaton Maxmum Lkelhood (FIML) estmator. In Mutuc, Pan and Reesus (2007) work, censorng problems occurred because of the presence of zero expendtures on some vegetable commodtes consumed by the sample of households. Hence they reled upon the the Shonkwler and Yen (1999) two-step procedure to crcumvent the censorng ssue. The Heen and Wessells (1990) approach mmcs the Heckman two-stage method by frst estmatng probt models to compute nverse Mlls ratos for each commodty. Subsequently, these ratos are ncorporated nto the second-step SUR estmaton of the demand system. On the other hand, Shonkwler and Yen (1999) proposed a consstent estmaton procedure that utlzes a probt estmator n the frst step. Subsequently, the cumulatve dstrbuton functon (cdf) s used to multply the covarates n the demand model and the probablty densty functon (pdf) s ncluded as an ndependent varable n the second step. Both methods fall under the purvew of utlzng two-step estmators. Whle the Shonkwler and Yen approach worked well wth the problem of zero expendtures, Arndt, Lu and Preckel (1999) clamed that t had lmtatons wth respect to dealng wth corner solutons. Several studes ncludng Arndt (1999) and Golan, Perloff and Shen (2001) propose an alternatve maxmum entropy approach to estmate censored demand systems. Ths approach allows for consstent and effcent estmaton of demand systems wthout puttng any restrctons on the error terms. Other researchers

6 4 such as Meyerhoefer, Ranney and Sahn (2005) use the general method of moments (GMM) estmator to address censorng problems n demand systems estmaton. Several studes have crtczed the two-step methods statng that the addng up restrcton n estmatng share equatons n censored demand systems s gnored (Dong, Gould and Kaser, 2004; Yen, Ln and Smallwood, 2003). Together wth Golan, Perloff and Shen (2001), these classes of estmators fall under the Amemya-Tobn framework where the former does not employ maxmum lkelhood estmaton n evaluatng multvarate probablty ntegrals. Dong, Gould and Kaser (2004) and Yen, Ln and Smallwood (2003) utlze numercal methods such as maxmum and quas-maxmum smulated lkelhood estmaton n approxmatng the lkelhood functon. The lterature regardng the use of alternatve estmaton technques such as Bayesan and nonparametrc approaches on mcro-demand system estmaton have been lmted (Tffn and Aquar, 1995). Methodology Almost Ideal Demand System (AIDS) Model Ths research utlzes the AIDS (Deaton and Mulbauer, 1980) n the estmaton of the demand for sx non-alcoholc beverages, namely: frut uces, tea, coffee, carbonated soft drnks, bottled water and mlk. Equaton (1) descrbes the general specfcaton of the AIDS model where p and w are the prce and budget share of the th beverage commodty. The average budget share w s computed as p q /M where M = p q s the total expendture on the sx aforementoned non-alcoholc beverages. The parameters of ths system are α, γ and β, respectvely. One can also ncorporate household

7 5 demographc characterstcs nto the demand system thru the ntercept parameter α. These varables nclude household sze, household ncome, race and regon. Also, a seasonalty component was added. 6 M (1) w = α + γ ln p + β ln + ε a p, -1,2,..6 = 1 ( ) where: 6 o + + = 1 ln a( p) = α α ln p 0.5 γ ln p ln p, and, * α = α + α HHsze + α Inc + α Race + α Season + α Rg In our study, we ncorporate selected demographc varables namely household sze (HHsze), ncome (Inc), race (Race), seasonalty (Season) and regon (Rg) n the analyss. Lkewse, the classcal theoretcal restrctons of addng up, homogenety and symmetry were mposed n the estmaton of the AIDS demand system: n Addng up: α = 1, β = 0, γ = 0 = 1 n = 1 Homogenety: γ = 0 n = 1 Symmetry: γ = γ n = 1 Quadratc Almost Ideal Demand System (QUAIDS) Model The Quadratc Almost Ideal Demand System (QUAIDS) model (Banks, Blundell and Lewbel, 1997) also s utlzed n ths demand analyss. The advantages of usng ths model over competng flexble demand systems s ts unparalleled capablty of ncorporatng non-lnear effects and nteractons of prce and expendtures on the demand

8 6 specfcatons. The mathematcal representaton of the QUAIDS demand system s as follows: 6 M λ M (2) w = α + γ ln p + β ln + + ε a p ln = 1 ( ) b( p) a( p) where: 2, -1,2,.,6 ln a( p) = α + b ( p ) = n = 1 o p β, n = 1 α ln p and γ ln p ln p, * α = α + α HHsze+ α Inc + α Race+ α Season+ α Rg The QUAIDS model s a generalzaton of the AIDS model. Also, f the null hypothess that λ 1 = λ 2 = = λ 6 =0 s reected then the QUAIDS model s a superor model at least statstcally relatve to the AIDS model system. In ths research, the ntercept parameter α ncorporates selected household demographc characterstcs ust as wth the AIDS model. Addng up, homogenety condtons and symmetry condtons also are mposed on the demand system as follows; n Addng up: α = 1, β = 0, γ = 0, λ = 0 = 1 n = 1 Homogenety: γ = 0 = 1 Symmetry: γ = γ n n = 1 n = 1 4 5

9 7 Elastcty Estmaton n AIDS and QUAIDS Demand Systems When the demand parameters of the AIDS and QUAIDS demand systems are estmated, the elastcty estmates subsequently, can be calculated. Followng Green and Alston (1990) and Banks, Blundell and Lewbel (1997), the expendture, uncompensated and compensated prce elastctes are gven by the followng formulae; β (3) η = + 1, for the AIDS model w 2λ m β + ln b( p) ( ) (4) a p η = + 1, for the QUAIDS model. w The Marshallan or uncompensated prce elastctes are gven by γ β( α + γ k ln pk ) (5) u k ε = w δk, for the AIDS model, and λβ m γ μ( α + γ k ln pk ) ln u k b( p) a p (6) ε ( ) = δk, for the w QUAIDS model, 2 where 2λ m μ = β + ln ( ) and δ k = the Kronecker delta (1 f =k and b p a( p) 0 otherwse) Fnally, from Slutsky s equaton, the Hcksan or compensated elastctes are calculated c u va the formula; ε = ε + η w, where u ε s the uncompensated prce elastcty of

10 8 beverage wth respect to beverage and η s the budget elastcty of beverage. The term w s the mean budget share of beverage. Estmaton Technques That Address Censorng n a Demand System Two-Step Estmators A class of estmaton technques that deal wth censored systems of equatons s the twostep estmaton procedure. In ths paper we consder two approaches proposed n Heen and Wessells (1990) and Shonkwler and Yen (1999) respectvely. These technques usually consst of estmatng a bnary choce model n the frst step to account for the decson to purchase or not to purchase the partcular beverage. Two mportant by products of the probt estmaton nclude the calculaton of the cumulatve dstrbuton functon (cdf) and probablty densty functon (pdf) from the bnary choce model. In the case of the Heen and Wessells (1990) approach, the calculaton of the nverse Mlls rato (rato of the pdf to the cdf) from the frst step probt estmaton now s ncluded as an added regressor nto the estmaton of the demand system. We note however that for those households that consumed and dd not consume the beverage tem, the formula for the nverse Mlls rato (IMR) s gven as: (7) IMR T φ( W ˆ) η =, for households that consume beverage T Φ( W ˆ) η (8) T φ( W ˆ) η IMR =, for households that dd not consume beverage T 1 Φ( W ˆ) η

11 9 W where, φ( W ηˆ ), Φ ( ηˆ ) and W correspond to the pdf, cdf and vector of socodemographc varables ncludng ncome, race and regon. Thus, the Heen and Wessells (1990) two-step approach of estmatng a demand system can be represented as: n M (9) w = α + γ ln p + β ln + υimr + ε a p, for the AIDS model =1 ( ) n M λ M (10) w = α + γ ln p + β ln + + υimr + ε a p ln, for the = 1 ( ) b( p) a( p) QUAIDS model. On the other hand, the Shonkwler and Yen (1999) consstent two-step approach utlzes the calculated cdf to multply the entre rght hand sde varables of the share equaton and nclude the pdf as an addtonal regressor n the system of budget shares. Ths formulaton can be represented as: n M (11) w = Φ( Wη ) α + γ ln p + β ln + φ( Wη ) + ε, for the AIDS model = 1 a( p) 2 n M λ M (12) w = Φ( Wη ) α + γ ln p + β ln + + φ Wη + ε a p b p ln a p ( ), = 1 ( ) ( ) ( ) for the QUAIDS model. 2 Dong, Gould and Kaser Approach (2004) We also used the Dong, Gould and Kaser (2004) approach, a varant of the Amemya- Tobn model n estmatng a censored AIDS model. In ths approach the AIDS demand model can be wrtten as:

12 10 n M * (13) w = α + γ ln p + β ln + ε, =1 a( p) where w = * p q represents the latent budget share wth p and q correspondng to the prce and quantty of the th beverage. As ponted out by Stockton, Capps and Dong (2007), the censored system wll take nto account the latent budget share f the vector mappng of the latent shares to ts correspondng actual shares addresses the followng condtons concernng the latent share, w. These condtons are ) 0 w 1 and ) * w = 1. Thus, Dong, Gould and Kaser (2004) proposed an approach that addresses both restrctons by applyng the followng mappng condton; (14) w * w = * w εω, f w * > 0 and Ω corresponds to the postve latent share space. w = 0, f w * 0 In ths mappng rule, we fnd that not only s the addng-up condton for latent and observed shares satsfed but because the rule addressed the two constrants mposed on the latent share, non-negatve expendture shares are expected. As for the estmaton procedure, the error structure of the respectve share equaton assumes a multvarate normal dstrbuton, thus the method of maxmum smulated lkelhood was used to evaluate the ntegrals nherent n ths multvarate dstrbuton.

13 11 Generalzed Maxmum Entropy Procedure 1 (GME) The method of maxmum entropy s an nformaton theoretc approach that does not mpose parametrc dstrbutonal assumptons (Golan, Judge and Mller 1996, Golan, Perloff and Shen 2001). Followng the SAS ETS 9.2 ENTROPY Procedure gude (SAS ETS 9.2 User Gude, 2008), the procedure selects the parameter estmates consstent wth the maxmzaton of the entropy dstrbuton. Thus, the entropy metrc for a gven dstrbuton s gven as: (15) max p ln( p ) n s.t. = 1 n p = 1 = 1, where p s the probablty of the th support pont. In a regresson framework, snce ths method assumes no parametrc assumptons, reparameterzatons are used to dentfy the respectve β parameters and the error terms. In a smple two support pont example, the expresson for the reparameterzed coeffcents can be wrtten as β = p h1sh1 + ph2sh2 where p h1 and p h2 represent the probabltes and s h1 and s h2 are the upper and lower bounds values based on pror nformaton on β. Lkewese, the reparametrzed error term can be wrtten as ε = r e + r e where r z1 and r z2 are assocated weghts of the error term s upper and z1 z1 z2 z2 lower bound values of e z1 and e z2 (SAS ETS 9.2 User Gude, 2008). From ths reparameterzaton, the GME maxmzaton problem can be notatonally wrtten as: (16) max G( p, r) = p'ln( p) r'ln( r) 1 The SAS ETS 9.2 Entropy procedure gude provdes excellent dscusson on how to use the SAS expermental procedure, The ENTROPY Procedure. The ensung dscusson follows theoretcal exposton of the general maxmum entropy estmaton prncple gven n the SAS procedure.

14 12 s.t. q = X S p + E r 1 H = (I H Θ 1 ' L ) p 1 Z = (I Z Θ 1 ' L ) r where q s the vector of response varable, X s the matrx of ndependent covarate observatons. S and p denote the vectors of support ponts and ther assocated probabltes, whle r s a weght vector assocated wth the support ponts contaned n E. And fnally I H and I Z are dentty matrces. The symbol Θ s the Kronecker product. However for ths exercse, we deal wth censored shares n a demand system. As such that we make modfcatons n solvng the prmal problem of the entropy procedure found n equaton (16). For example, gven that q = w s the share n the AIDS model, w n M = α + γ ln p + β ln + ε, we apply the followng condtons: = 1 a( p) w * 6 α + = γ = 1 M ln p + β ln + ε : w a( p) lowerbound : w 0 > 0 Thus for ths case, the prmal optmzaton problem can be wrtten as (17) max G( p, r) = p'ln( p) r'ln( r) s.t n M w = α + γ ln p + β ln Sp + Er =1 a( p) LB n LB M LB w α + γ ln p + β ln S p + =1 a( p) E LB r LB 1 H = (I H Θ 1 ' L ) p

15 13 1 Z = (I Z Θ 1 ' L ) r A smlar constructon can be done n the QUAIDS model. Estmaton Issues The estmaton of the AIDS and QUAIDS specfcaton usng the maxmum entropy technque was done usng the expermental SAS procedure called PROC ENTROPY. However, ths expermental procedure at present s only lmted to estmaton of systems of lnear relatonshps. Thus, attempts were made to lnearze the demand system by usng the startng values generated from the ITSUR specfcaton and smplfyng through the use of mean values of the non-lnear components such as the nonlnear prce ndex ln(a(p)) and Cobb Douglas prce aggregator b(p) nto constants n both the AIDS and QUAIDS model. Thus, n ths case, the lnearzed AIDS and QUAIDS model can be represented as: n (18) w = α + γ ln p + βδ + ε, for the AIDS model =1 where M Δ = ln and ln C s a calculated constant of ln a(p) C n 2 (19) w = α + γ ln p + βδ + λγ + ε, for the QUAIDS model = 1 where ln M C D 2 2 Γ = wth lnc as the calculated constant of ln a(p) and D s the constant representng the Cobb-Douglas prce aggregator b(p).

16 14 The mposton of classcal restrctons of symmetry and homogenety was not done n the maxmum entropy estmaton of the demand system. Dffcultes were encountered n dentfyng the values of support ponts of those coeffcents beng restrcted. And wth so many restrctons beng mposed, the dentfcaton of problematc constrants was a maor problem. Thus, n usng the maxmum entropy estmaton procedure, the estmaton of the AIDS and QUAIDS models were done wthout the usual mposton of the classcal theoretcal constrants. The use of the Dong, Gould and Kaser (2004) technque was only performed n the AIDS model. We dd not attempt to use ths procedure n the QUAIDS model specfcaton. Agan ths acton was necessary due prmarly to the hghly non-lnear nature of the QUAIDS model. Convergence assocated wth ths procedure was dffcult to acheve. Data The data used n the study s the 1999 AC Nelsen HomeScan Panel where the data set s a complaton of household purchase transactons of ths calendar year. In ths data set, the transacton records of each household relate to total expendtures and quanttes of commodtes purchased prmarly n retal groceres, ncludng the use of dscounts coupons. The number of households n the sample s 7, 195 and because quarterly observatons are used for each household, the total sample sze comes to 28,780. Ths sample sze can be thought of a natonally representatve sample of the purchases made by U.S. households from retal grocery stores or mass merchandsers for the calendar year 1999.

17 15 Insert Table 1 In ths study, the selected soco-demographc varables used were household ncome, household sze, race, regon and seasonalty. From Table 1, we fnd the mean household ncome s $51,740 and the domnant household sze for the sample s those wth two members (38%). As for race, approxmately 94 percent are whte and black households and for regons, 34 percent come from the South whle the rest has the followng breakdown: East (20%), Central (25%) and West (20%). Another feature of the data set s that commodty prces are not readly avalable. Instead one uses the dervaton of expendtures over quanttes of the purchased tem, called unt values and these unt values serve as proxes for the prce varables. If both the expendtures and quanttes were zero, then ths study utlzed a smple prce mputaton procedure restng on the use of ncome, race and regonal dummy varables. If p = 0 for a partcular household, then Pfrutuce = (hnc* ) + (whte* ) + (black* ) + (orental* ) + (central* ) + (south* ) + (west* ); Ptea = (hnc* ) + (whte* ) + (black* ) + (orental* ) + (central* ) + (south* ) + (west* ); Pcoffee = (hnc* ) + (whte* ) + (black* ) + (orental* )+ (central* ) + (south* ) + (west* ); Pcsd = (hnc* ) + (whte* ) + (black* ) + (orental* ) + (central* ) + (south* )+ (west* ); Pwater = (hnc* ) + (whte* ) + (black* ) + (orental* ) + (central* ) + (south* ) + (west* ); Pmlk = (hnc* ) + (whte* ) + (black* ) + (orental* ) + (central* ) + (south* ) + (west* );

18 16 Insert Tables 2, 3 and 4 The coeffcents were derved by regressng the prce of each non-alcoholc beverage tem wth household ncome (hnc), race (whte,black and orental) and regons (central, south and west). Tables 2, 3 and 4 present the mean total expendtures, quantty purchased and prces for the sx non-alcoholc beverages consdered. In ths case we fnd that the top household purchases wth respect to non-alcoholc beverages were carbonated soft drnks, frut uces, mlk and coffee. The mean prces are as follows: frut uces ($4.71/gal), tea ($2.06/gal), coffee ($1.41/gal), carbonated soft drnks ($2.48/gal), bottled water ($2.06/gal) and mlk ($3.08/gal). On the other hand, Table 5 presents the mean budget shares of the beverage tems. For the perod 1999, approxmately 81 percent of total expendtures for non alcoholc beverages are captured by carbonated soft drnks, frut uces and mlk. The remanng 19 percent are devoted to tea (4.7 %), coffee (11%) and bottled water (3.8 %). Insert Table 5 Table 6 descrbes the degree of censorng assocated wth each type of nonalcoholc beverages for each household on a quarterly bass. From the table, tems wth mnmal to medum censorng are mlk (6.77%), carbonated soft drnks (8.84 %) and frut uces (23.09 %). On the other hand, the remanng hghly censored non-alcoholc beverage tems are tea (54.88 %), coffee (42.77 %) and bottled water (60.65 %). Insert Table 6

19 17 Emprcal Results Estmated Demand Parameters 2 Almost all of the soco-demographc coeffcents n both specfcatons and across all estmaton technques are statstcally sgnfcant. Also, almost all of the parameters n both AIDS and QUAIDS and across estmaton technques are relatvely close to one another and the same can be sad for the AIDS and QUAIDS unrestrcted cases. Thus t can be postulated that because of a relatvely large sample sze, the varous estmaton procedures converged to yeldng relatvely close parameter estmates. Also, the parameters assocated wth the quadratc term n the QUAIDS specfcaton are hghly sgnfcant, suggestng n part a bas towards the QUAIDS specfcaton over the AIDS model across the varous estmaton procedures, wth or wthout ncorporatng demand restrctons. In Table 7, we fnd that the symmetry, homogenety and the combnaton of both restrctons are reected n both the AIDS and QUAIDS models. Insert Table 7 Expendture and Compensated Elastctes In Tables 8 to 15, we present the calculated expendture and compensated elastctes of non-alcoholc beverages across model specfcatons, estmaton technques and mposton of theoretcal restrctons. From the tables, we fnd that both expendture elastctes and own-prce elastctes were generally smlar across model specfcatons, estmaton technques and whether or not the theoretcal restrctons were mposed. All of 2 Due to space constrants, the estmated parameters are not ncluded n the text, but are avalable upon request.

20 18 the expendture elastctes are postve ndcatng that all non-alcoholc beverages are normal goods. Also, f we look at the compensated cross-prce elastctes across model specfcatons, estmaton technques and wth or wthout theoretcal restrctons, we fnd that almost all of them are postve ndcatng that the set of non-alcoholc beverages are net substtutes. Smlarly, the maor substtutes for frut uce and tea are coffee, carbonated soft drnk and mlk. On the other hand the maor substtutes for coffee are frut uce, carbonated soft drnks and mlk. For carbonated soft drnks the maor substtutes are coffee and mlk. Coffee, carbonated soft drnks and mlk represent the maor non-alcoholc beverage substtutes for bottled water. Fnally, the maor commodty substtutes for mlk are frut uce, coffee and carbonated soft drnks. Insert Tables 8 to 15 Elastcty Comparsons across Censored Estmaton Technques of Non-Alcoholc Beverages In Table 9, we present the AIDS compensated or Hcksan prce elastcty matrx of nonalcoholc beverages. We note more varablty of cross-prce elastctes estmates of nonalcoholc beverage that are hghly censored, that s for tea, coffee and bottled water. On the other hand, relatvely less varable cross-prce elastcty estmates were observed for commodtes wth relatvely fewer censorng ssues. For example, n mlk, the cross-prce elastcty estmates of mlk wth respect to frut uce ranged from to The cross-prce elastcty values for bottled water wth respect to frut uce ranged from to Also note that assocated p-values for all prce elastctes are mostly sgnfcant. For the QUAIDS specfcaton, we note the same clam that the greater number of

21 19 censored observatons the commodty, the more varable the respectve own- and crossprce elastctes are. For mlk the compensated prce elastctes wth respect to frut uce ranged from to 0.283, whle for the bottled water, the compensated prce elastctes ranged from to (Table 11). On the other hand, the same observaton can be made for the AIDS and QUAIDS unrestrcted cases. For example the cross-prce elastcty of mlk wth respect frut uce ranged from to for AIDS (Table 13) and to for QUAIDS (Table 15), whle the cross-prce elastcty of bottled water wth respect to frut uce ranged from to for the AIDS specfcaton and to for the QUAIDS model. Elastcty Comparsons across Model Specfcatons (AIDS vs. QUAIDS) The compensated own- and cross-prce elastcty matrces of non-alcoholc beverages of both the AIDS and QUAIDS models are presented n Tables 9 and 11. We note relatvely smlar prce elastcty estmates especally wth respect to the own-prce elastcty values of both models. For mlk, the range of the AIDS own prce elastctes were from to , whereas for the QUAIDS model, the values ranged from to Also f we look at a hghly censored commodty such as bottled water, the cross-prce elastcty of bottled water wth respect to tea ranged from to for the AIDS model and to n the QUAIDS specfcaton. The same fndngs also were observed for the unrestrcted cases of AIDS and QUAIDS where the calculated compensated prce elastctes were remarkably smlar.

22 20 Elastcty Comparsons across Imposton of Theoretcal Restrctons In Tables 9 and 13, we show the compensated own- and cross-prce elastcty matrces of the AIDS restrcted and unrestrcted cases. Two notable results were observed; own-prce elastcty estmates (absolute values) were larger n the restrcted case vs-à-vs the unrestrcted case. On the other hand compensated cross-prce elastctes were generally larger n absolute terms n the unrestrcted case relatve to the values generated n the restrcted case. The same result also was observed for the QUAIDS restrcted and unrestrcted models (Tables 11 & 15). Ft Comparsons across Econometrc Technques Table 16 present the R-square values of the budget share equatons from dfferent censorng econometrc technques across demand system specfcaton and mposton of theoretcal restrctons. From the estmates, we fnd that across model specfcatons and theoretcal restrctons, the Heen and Wessells approach had the hghest R-square values n ts budget share equatons. On the other hand, R-square values generated by the Shonkwler and Yen technque regstered second f theoretcal restrctons are relaxed. Lkewse, the ITSUR technque placed last across demand model specfcatons and theoretcal mpostons n terms of goodness of ft. Insert Table 16 Conclusons We fnd that the prce elastctes especally the compensated prce elastctes were robust and relatvely smlar and statstcally sgnfcant across model specfcatons, estmaton technques and restrcton mpostons. The sgns of the compensated cross-

23 21 prce elastctes across the board were generally postve ndcatng that the respectve non-alcoholc beverages are net substtutes. Comparatve analyss show that across estmaton technques, greater varablty of compensated cross-prce elastcty estmates were observed n hghly censored non-alcoholc beverages such as tea, coffee and bottled water. As for the comparson between model specfcatons (AIDS versus QUAIDS), the compensated prce estmates were remarkably smlar especally for the own-prce elastcty values. Fnally, the estmates for unrestrcted compensated cross-prce elastctes were generally greater vs-à-vs the restrcted cases. The reverse s generally true wth regard to the compensated own-prce elastcty estmates. The robustness of both the parameter estmates and the calculated expendture and prce elastctes may be explaned n part to the avalablty of hgh number of observatons (n~30,000). However, snce most censored data sets do not usually have ths partcular characterstc, then studes that smulate the effect of sample sze wll be benefcal on determnng whether robustness wll stll be observed for parameter estmates and prce and expendtures elastctes n the presence of dfferng sample szes.

24 References Arndt, C An Entropy Approach to Treatng Bndng Non-Negatvty Constrants n Demand Systems Estmaton and an Applcaton to Herbcde Demand n Corn. Journal of Agrcultural and Resource Economcs 24(1): Arndt, C., S. Lu and P.V. Preckel On Dual Approaches to Demand Systems Estmaton n the Presence of Bndng Quantty Constrants. Appled Economcs 31(8), Banks, J., R. Blundell and A. Lewbel Quadratc Engel Curves and Consumer Demand. The Revew of Economc and Statstcs 79(4): Deaton, A., and J. Muelbauer An Almost Ideal Demand System. Amercan Economc Revew 70(3): Dhar, T and J.D. Foltz Mlk by Any Other Name Consumer Benefts from Labeled Mlk. Amercan Journal Agrcultural Economcs 87(1): Dong, D., B.W. Gould and H.M. Kaser Food Demand n Mexco: An Applcaton of the Amemya-Tobn Approach to the Estmaton of a Censored Food System. Amercan Journal Agrcultural Economcs 86(4): Golan, A., J.M Perloff and E.Z. Shen Estmatng a Demand System wth Nonnegatvty Constrants: Mexcan Meat Demand. The Revew of Economcs and Statstcs 83(3): Golan, A., G. G. Judge, and D. Mller Maxmum Entropy Econometrcs: Robust Estmaton wth Lmted Data, John Wley & Sons.

25 Green, R., and J. Alston Elastctes n the AIDS Models. Amercan Journal of Agrcultural Economcs 72(2): Greene, W.T Econometrc Analyss. 6 th Edton. Upper Saddle Rver, New Jersey: Prentce Hall. Heen, D. and C.R. Wessells Demand Systems Estmaton wth Mcrodata: A Censored Regresson Approach. Journal of Busness and Economc Statstcs 8(3): Meyerhoefer, C., C.K. Ranney and D.E. Sahn Consstent Estmaton of Censored Demand Systems usng Panel Data. Amercan Journal Agrcultural Economcs 87(3): Mutuc, M.E., S. Pan, and R.M. Reesus Household Vegetable Demand n the Phlppnes: Is There an Urban-Rural Dvde. Agrbusness: An Internatonal Journal 23(4): SAS Insttute Inc SAS/ETS 9.2 User s Gude. Cary, North Carolna: SAS Insttute Inc. Shonkwler, J.S. and S.T. Yen Two-Step Estmaton of a Censored System of Equatons. Amercan Journal of Agrcultural Economcs 81(4): Stockton, M.C., O. Capps, Jr and D. Dong A Mcro Level Demand Analyss of Selected Non-Alcoholc Beverages wth Emphass on Contaner Sze. Department of Agrcultural Economcs, Texas A&M Unversty, Research Report.

26 Tffn, R., and M. Aquar Bayesan Estmaton of an Almost Ideal Demand System for Fresh Frut n Portugal. European Revew of Agrculture Economcs 22(4): Yen, S.T., B. Ln, and D.M. Smallwood "Quas and Smulated Lkelhood Approaches to Censored Demand Systems: Food Consumpton by Food Stamp Recpents n the Unted States." Amercan Journal of Agrcultural Economcs 85(2):

27 Table 1. Descrptve Statstcs of Relevant Household Demographc Varables. Varables Mean Std. Devaton Mn Max Household Income($) 51,740 26,254 5, ,000 Household Sze (%) One member Two members Three members Four members Fve members Race (%) Whte Black Orental Other Regon (%) East Central South West Observatons 28,780

28 Table 2. Descrptve Statstcs for Total Expendture for Each Non- Alcoholc Beverage Item (n=28,780). Mean ($) Std. Devaton ($) Mn ($) Max ($) Frut Juces Tea Coffee Carbonated Soft Drnks Bottled Water Mlk

29 Table 3. Descrptve Statstcs for Quanttes for Each Non Alcoholc Beverage Item (n=28,780). Mean (gallons) Std. Devaton (gallons) Mn (gallons) Max (gallons) Frut Juces Tea Coffee Carbonated Soft Drnks Bottled Water Mlk

30 Table 4. Descrptve Statstcs for Prces 1 for Each Non-Alcoholc Beverage Item (n=28,780). Mean ($/gallon) Std. Devaton ($/gallon) Mn ($/gallon) Max ($/gallon) Frut Juces Tea Coffee Carbonated Soft Drnks Bottled Water Mlk When expendture and quanttes are equal to zero, prce mputaton was used P =f (ncome, race and regon).

31 Table 5. Descrptve Statstcs for Budget Shares for Each Beverage Item for Calendar Year Beverage Product Average Budget Share Std. Devaton Mn Max Frut Juces Tea Coffee Carbonated Soft Drnks Bottled Water Mlk

32 Table 6. Number of Censored Responses for Each Beverage Item. Number of Observatons Percentage Frut Juces 6, Tea 15, Coffee 12, Carbonated Soft Drnks 2, Bottled Water 17, Mlk 1,

33 Table 7. Tests of Symmetry, Homogenety and Combnaton of Symmetry and Homogenety Restrcton Based on Wald Tests. A. AIDS model Symmetry χ 2 - Statstc p-value Homogenety χ 2 - Statstc p-value Symmetry and Homogenety χ 2 - Statstc p-value ITSUR < < <.0001 Heen & Wessells < < <.0001 Shonkwler & Yen < < <.0001 B. QUAIDS model ITSUR < < <.0001 Heen & Wessells < < <.0001 Shonkwler & Yen < < <.0001

34 Table 8. Expendture Elastctes 1 of Non-Alcoholc Beverages Usng the AIDS System and 1999 ACNelsen Homescan Data ITSUR Heen & Wessells Shonkwler & Yen GME Dong et al. Dong et al. Mean Standard Item Estmate Estmate Estmate Estmate Actual Estmates Latent Estmates Devaton Frut Juce (0.000) (0.000) (0.000) (0.000) (0.005) Tea (0.000) (0.000) (0.000) (0.000) (0.000) Coffee (0.000) (0.000) (0.000) (0.000) (0.089) Carbonated Soft drnks (0.000) (0.000) (0.000) (0.000) (0.000) Bottled Water (0.000) (0.000) (0.000) (0.000) (0.000) Mlk (0.000) (0.000) (0.000) (0.000) (0.000) Note: p-values are n brackets 1 Calculated usng sample means.

35 Table 9. Compensated Own- and Cross-Prce Elastcty Matrx 1 of Non-Alcoholc Beverages Usng the AIDS and the 1999 ACNelsen Homescan Data Frut Carbonated Bottled Juce Tea Coffee Soft Drnks Water Mlk Frut Juce ITSUR [.0001] [.0001] [.0001] [.0001] [.0108] [.0001] Heen &Wessells [.0001] [.0001] [.0001] [.0001] [.0149] [.0001] Shonkwler & Yen [.0001] [.0245] [.0001] [.0001] [.9528] [.0001] Dong et. al (actual) [.0001] [0.0001] [.0001] [.0001] [.1923] [.0001] Dong e.t al (latent) GME(unrestrcted) Mean Std. Devaton Tea ITSUR [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [.0001] [.0001] [.5905] [.0001] [.2609] [.0001] Shonkwler & Yen [.0073] [.0001] [.0001] [.0001] [.0001] [.0001] Dong et al (actual) [.0001] [0.0001] [.0001] [.0001] [.0001] [.0001] Dong et al (latent) GME (unrestrcted) Mean Std. Devaton Coffee ITSUR [0.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [0.0001] [.3918] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [0.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Dong et. al (actual) [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Dong et. al (latent) GME (unrestrcted) Mean Std. Devaton

36 Table 9. Contnued Frut Carbonated Bottled Juce Tea Coffee Soft Drnks Water Mlk Carbonated Soft drnks ITSUR [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Dong et al (actual) [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Dong et al (latent) [.0001] GME (unrestrcted) Mean Std. Devaton Bottled Water ITSUR [.0089] [.0001] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [.0090] [.8978] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [.8326] [.0001] [.0001] [.0001] [.0001] [.0001] Dong et al (actual) [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Dong et al (latent) GME (unrestrcted) Mean Std. Devaton Mlk ITSUR [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Dong et al (actual) [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Dong et al (latent) GME (unrestrcted) Mean Std. Devaton Note: p-values are n brackets 1 Calculated usng sample means.

37 35 Table 10. Expendture Elastctes 1 of Non-Alcoholc Beverages Usng the QUAIDS System and 1999 ACNelsen Homescan Data ITSUR Heen & Wessells Shonkwler & Yen GME Mean Standard Item Estmate Estmate Estmate Estmate Devaton Frut Juce (0.000) (0.000) (0.000) Tea (0.000) (0.000) (0.000) Coffee (0.000) (0.000) (0.000) Carbonated Soft drnks (0.000) (0.000) (0.000) Bottled Water (0.000) (0.000) (0.000) Mlk (0.000) (0.000) (0.000) Note: p-values are n brackets 1 Calculated usng sample means

38 36 Table 11. Compensated Own- and Cross-Prce Elastcty Matrx 1 of Non-Alcoholc Beverages Usng the QUAIDS and the 1999 ACNelsen Homescan Data Carbonated Bottled Frut Juce Tea Coffee Soft Drnks Water Mlk Frut Juce ITSUR [.0001] [.0001] [.0001] [.0001] [.0108] [.0001] Heen &Wessells [.0001] [.0001] [.0001] [.0001] [.0214] [.0001] Shonkwler & Yen [.0001] [.2032] [.0001] [.0001] [.4151] [.0001] GME (unrestrcted) Mean Std. Devaton Tea ITSUR [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [.0001] [.0001] [.9184] [.0001] [.4564] [.0001] Shonkwler & Yen [.0772] [.0001] [.0001] [.0001] [.0001] [.0001] GME (unrestrcted) Mean Std. Devaton Coffee ITSUR [0.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [0.0001] [.5303] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [0.0001] [.0001] [.0001] [.0001] [.0001] [.0001] GME (unrestrcted) Mean Std. Devaton Carbonated ITSUR [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Soft drnks Heen &Wessells [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] GME (unrestrcted) Mean Std. Devaton

39 37 Table 11. Contnued Carbonated Bottled Frut Juce Tea Coffee Soft Drnks Water Mlk Bottled Water ITSUR [.0693] [.0001] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [.0140] [.8310] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [.5349] [.0001] [.0001] [.0001] [.0001] [.0001] GME (unrestrcted) Mean Std. Devaton Mlk ITSUR [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] GME (unrestrcted) Mean Std. Devaton Note: p-values are n brackets Calculated usng sample means.

40 38 Table 12. Expendture Elastctes 1 of Non-Alcoholc Beverages Usng the AIDS System and 1999 ACNelsen Homesan Data (Unrestrcted) ITSUR Heen & Wessells Shonkwler & Yen GME Mean Standard Item Estmate Estmate Estmate Estmate Devaton Frut Juce (0.000) (0.000) (0.000) Tea (0.000) (0.000) (0.000) Coffee (0.000) (0.000) (0.000) Carbonated Soft Drnks (0.000) (0.000) (0.000) Bottled Water (0.000) (0.000) (0.000) Mlk (0.000) (0.000) (0.000) p-values are n parenthess 1 Calculated usng sample means

41 39 Table 13. Compensated Own- and Cross-Prce Elastcty Matrx 1 of Non-Alcoholc Beverages Usng the AIDS and 1999 ACNelsen Homescan Data (Unrestrcted) Carbonated Bottled Frut Juce Tea Coffee Soft Drnks Water Mlk Frut Juce ITSUR [.0001] [.0001] [.0001] [.0001] [.9024] [.0001] Heen &Wessells [.0001] [.0001] [.0001] [.0001] [.7842] [.0001] Shonkwler & Yen [.0001] [.0002] [.0001] [.0001] [.2473] [.0001] GME Mean Std. Devaton Tea ITSUR [.0001] [.1954] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [.0001] [.4191] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [.0001] [.1649] [.0001] [.0001] [.0001] [.0001] GME Mean Std. Devaton Coffee ITSUR [0.0001] [.0003] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [0.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [0.0002] [.0001] [.0001] [.0001] [.0001] [.0001] GME Mean Std. Devaton

42 40 Table 13. Contnued Carbonated Bottled Frut Juce Tea Coffee Soft Drnks Water Mlk Carbonated ITSUR [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Soft drnks Heen &Wessells [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] GME Mean Std. Devaton Bottled Water ITSUR [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] GME Mean Std. Devaton Mlk ITSUR [.0001] [.0009] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] GME Mean Std. Devaton Note: p-values are n brackets 1 Calculated usng sample means

43 41 Table 14. Expendture Elastctes 1 of Non-Alcoholc Beverages Usng the QUAIDS System and 1999 ACNelsen Homescan Data (Unrestrcted) ITSUR Heen & Wessells Shonkwler & Yen GME Mean Standard Item Estmate Estmate Estmate Estmate Devaton Frut Juce (0.000) (0.000) (0.000) Tea (0.000) (0.000) (0.000) Coffee (0.000) (0.000) (0.000) Carbonated Soft Drnks (0.000) (0.000) (0.000) Bottled Water (0.000) (0.000) (0.000) Mlk Note: p-values are n brackets 1 Calculated usng sample means (0.000) (0.000) (0.000)

44 42 Table 15. Compensated Own- and Cross-Prce Elastcty Matrx 1 of Non-Alcoholc Beverages Usng the QUAIDS and the 1999 ACNelsen Homescan Data (Unrestrcted) Carbonated Bottled Frut Juce Tea Coffee Soft Drnks Water Mlk Frut Juce ITSUR [.0001] [.0001] [.0001] [.0001] [.8061] [.0001] Heen &Wessells [.0001] [.3264] [.0001] [.0001] [.8260] [.0001] Shonkwler & Yen [.0001] [.0001] [.0001] [.0001] [.8804] [.0001] GME Mean Std. Devaton Tea ITSUR [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] GME Mean Std. Devaton Coffee ITSUR [0.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [0.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [0.0001] [.1667] [.0001] [.0001] [.2921] [.0001] GME Mean Std. Devaton Carbonated Soft drnks ITSUR [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Heen &Wessells [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] Shonkwler & Yen [.0001] [.0001] [.0001] [.0001] [.0001] [.0001] GME Mean Std. Devaton