Modeling zero response data from willingness to pay surveys A semi-parametric estimation

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1 Economcs Letters 71 (001) locte/ econbse Modelng zero response dt from wllngness to py surveys A sem-prmetrc estmton Seung-Hoon Yoo *, T-Yoo Km, J-K Lee, b c Insttute of Economc Reserch, Kore Unversty, 5-1 Anm-Dong, Sungbuk-Ku, Seoul, , South Kore b Techno-Economcs nd Polcy Progrm, Seoul Ntonl Unversty, 56-1 Sn, Shnrm-Dong, Kwnk-Ku, Seoul, , South Kore c Deprtment of Interntonl Are Studes, Hoseo Unversty, 9-1 Sechul-R, Bebng-Myun, Asn Chungnm, , South Kore Receved 19 Aprl 000; ccepted 11 October 000 Abstrct Ths pper models zero response dt from wllngness to py surveys by employng prmetrc nd sem-prmetrc estmton methods. The result of the specfcton test ndctes the sem-prmetrc estmton outperforms the prmetrc estmton sgnfcntly. 001 Publshed by Elsever Scence B.V. Keywords: Zero response; Wllngness to py; Symmetrclly trmmed lest squres JEL clssfcton: C4 1. Introducton A typcl chrcterstc of wllngness to py (WTP) for n tem s tht mny respondents would not be wllng to py nythng for t. In our dt on household WTP for the eductonl ner vdeo-on-demnd (ENVOD) servce, ths s the cse for 6.% of ll observtons. In such cses, lest squres estmtes wll be nconsstent. To ccount for ths, one type of lmted-dependent vrble model, the fmlr Tobt model (Tobn, 1958), hs been wdely used. The Tobt estmtor, bsed on the mxmum lkelhood estmton (MLE) method, ssumes the homoscedstcty nd normlty on the dstrbuton of the error term. If the ssumptons re not stsfed, t s gn nconsstent (e.g. see Robnson (198)). Snce economc theory generlly yelds no restrctons concernng the form of the error dstrbuton or homoscedstcty of the resduls, the *Correspondng uthor. Tel.: ; fx: E-ml ddress: shyoo01@ntzen.com (S.-H. Yoo) / 01/ $ see front mtter 001 Publshed by Elsever Scence B.V. PII: S (01)

2 19 S.-H. Yoo et l. / Economcs Letters 71 (001) senstvty of lkelhood-bsed procedures to such ssumptons s serous concern. The test results, explned n Secton 3, mply tht the hypotheses of homoscedstcty nd normlty re clerly rejected t the 1% level. Recently, the econometrc theory of sem-prmetrc estmton methods for the Tobt model hs gned much ttenton. However, emprcl pplctons of the methods remn lckng. Ths pper, therefore, hs two mjor gols. The frst s to nlyze some determnnts of WTP for the ENVOD servce. The second gol s to explore the use of consstent nd robust estmtor when estmtng WTP equton usng zero response dt from the WTP survey. To ths end, we propose systemtc pproch of testng the prmetrc ssumptons on the error term dstrbuton of the Tobt model, sem-prmetrc re-estmton of the model, nd specfcton test of the prmetrc estmton vs. the sem-prmetrc estmton.. A model of WTP An ndvdul s optml WTP cn be derved wthn the constrned utlty mxmzton frmework. Assumng the utlty functon s contnuous nd qus-concve, then the optml WTP cn be expressed s functon of the respondent s tstes or personl chrcterstcs. Denote these determnnts of WTP s vector x nd ssume lner functonl form for the WTP equton. Then, for ndvdul 5 1,..., T, the optml WTP, y *, cn be wrtten s: y* 5 x9b 1 u where b s vector of prmeters, nd u s rndom error. In relty, n ndvdul s choce s subject to non-negtvty constrnts, nd, therefore, corner soluton could result. One representtve wy to ccommodte corner solutons s to use the Tobt model, n whch cse observed WTP, denoted y, reltes to the ltent WTP y* such tht: y 5 mxhy *,0j where u s ssumed to be dstrbuted s norml wth men zero nd stndrd devton s. (1) () 3. Dt nd pretests The dt on household WTP for the ENVOD servce, nd chrcterstcs used n ths nlyss come from 1998 survey of 47 households n Pusn, Kore. The smple s censored, wth 11 households (6.%) reportng zero WTP. The vrbles n the model re descrbed n Tble 1. We test for the exstence of heteroscedstcty nd non-normlty n the error term of the Tobt model. Frstly, we consder the heteroscedstc Tobt model n whch we specfy tht s 5 s exp(x9 ). The null hypothess of homoscedstcty s 5 0. The Lgrnge multpler (LM) sttstc ws clculted s Ths s symptotclly dstrbuted s ch-squred wth 10 degrees of freedom 0.01 under the null hypothess. Gven tht x (10) 5 3.1, the hypothess cn be rejected t the 1% level. Secondly, we test for censored normlty n the Tobt model, usng n LM test devsed by Chesher

3 Tble 1 Descrpton of vrbles n model Vrbles WTP INTEREST SATISFACTION NEED HELP EXTRA AGE CATV EBS EDUCATION INCOME Defnton S.-H. Yoo et l. / Economcs Letters 71 (001) Wllngness to py for eductonl ner vdeo-ondemnd servce (Unt: Koren won ) Degree of respondent s nterest n eductng hs/her chld (From 15very lttle to 55very much) Degree of how much respondent s stsfed wth current educton n school (From 15very lttle to 55very much) Opnon bout how necessry extrcurrculr work s for respondent s chld (From 15very lttle to 55very much) Opnon bout how helpful the proposed eductonl vdeo-on-demnd s for respondent s chld (From 15very lttle to 55very much) Dummy for respondent s chld beng engged n extrcurrculr work (05no; 15yes) Age of the respondent Dummy for respondent s wtchng cble televson (05no; 15yes) Dummy for respondent s chld wtchng the Eductonl Brodcstng System (05no; 15yes) Educton level (from 15lowest to 95hghest) Monthly household totl ncome fter tx deducton (Unt: Koren won ) At the tme of the survey, US$1 ws pproxmtely equl to 1400 Koren won. nd Irsh (1987). The test s equvlent to testng the null hypothess of g15 g5 0 n modfcton 3 of the norml cumultve dstrbuton functon (cdf), Pr(u t, t) 5 F(t) 5 F(t 1 g01 g1t 1 gt ), where 1 F(? ) s the unvrte stndrd norml cdf. The LM sttstc follows symptotclly x dstrbuton wth two degrees of freedom under the null hypothess of censored normlty. The test sttstc ws computed s 41.89, whch s lrge enough to reject the hypothess t the 1% level, gven tht x 0.01() Thus, t ppers cler from the test results gven bove tht the ssumptons requred to use the Tobt model re too strong to be stsfed. 1 If the model contns constnt term, g wll not be dentfed, so for the cse we re consderng here, the normlzton 0 g 5 0 would be mposed. 0

4 194 S.-H. Yoo et l. / Economcs Letters 71 (001) A sem-prmetrc estmton One could estmte the heteroscedstc Tobt models nd use n lterntve dstrbuton to rectfy heteroscedstcty nd non-normlty of the error term n the Tobt MLE model. However, ths method does not necessrly solve the problem, nd my mke t worse. Accordngly, devsng new estmtor tht s robust to these two problems requrng fewer ssumptons s more ppelng. As n lterntve to the Tobt model, we use symmetrclly trmmed lest squres (STLS) estmtor proposed by Powell (1986), whch s consstent nd symptotclly norml for wde clss of error dstrbutons wth heteroscedstcty of unknown form nd censored dependent vrble. For smple of sze T, the STLS estmtor, ˆ b, s defned s: STLS T ˆ bstls 5rg mnoi(x b. 0) fmn( y,x b ) x b g (3) b [B 51 where B denotes the relevnt prmeter spce for b, nd the ndctor functon, I(? ), tkes the vlue of one f the rgument s true, nd zero otherwse. See Powell (1986) for n tertve procedure to obtn ˆ b, proof of consstency of ˆ STLS b STLS, nd ts symptotc covrnce. Rther thn estmte the symptotc covrnce for the STLS estmtor, we use bootstrp estmtor of the covrnce, V, ˆ gven by R 1 j j ] R STLS STLS STLS STLS j51 Vˆ 5 O(bˆ b )(bˆ b )9 (4) R j where b 5 (1/R)o bˆ STLS j51 STLS nd R s the number of bootstrp replctons. Ths bootstrp procedure results n consstent estmtor of the covrnce of ˆ b STLS, whch s robust to voltons of the ssumpton tht the resduls re dentclly dstrbuted. 5. Estmton results nd specfcton test To obtn the STLS estmtor, convergence occurs when the mxmum chnge n ny prmeter 5 estmte s less thn 10 n two consecutve tertons. The STLS estmtor converged n 48 tertons. The vlue of R s set to Tble shows the coeffcents of our bsc equton estmted by Tobt MLE nd STLS, respectvely. To compre the prmetrc nd sem-prmetrc estmtons more generlly, we conduct Husmn s (1978) type specfcton test gven n Melenberg nd vn Soest (1996). The test requres n estmtor tht s consstent nd effcent under the null hypothess but nconsstent under the lterntve (the Tobt estmtor), nd n estmtor tht s consstent under both hypotheses, but neffcent under the null hypothess (the STLS estmtor). We cn construct the Wld sttstc s: Alterntvely, Yoo et l. (000) ppled lest bsolute devton estmtor developed by Powell (1984) to del wth the zero WTP dt.

5 Tble Estmton results of WTP equtons Vrbles Tobt STLS Constnt (.8199) (7.0510) INTEREST (0.59) (0.3708) SATISFACTION (0.61) (0.336) NEED (0.35) (0.948) HELP (0.3155) (1.54) EXTRA (0.3697) (0.5167) AGE (0.048) (0.0860) CATV (0.4769) (0.8583) EBS (0.6157) (1.9899) EDUCATION (0.143) (0.741) INCOME (0.0007) (0.007) x S.-H. Yoo et l. / Economcs Letters 71 (001) The vrbles re defned n Tble 1. The numbers n prentheses below the estmtes re stndrd errors. The stndrd errors of Tobt estmtes re computed usng the nlytc second dervtves of the log-lkelhood. Those of STLS estmtes re clculted by the use of the bootstrp method wth 5000 replctons. ˆ ˆ 1 ˆ ˆ Tobt STLS Tobt STLS W 5 (b b )9L (b b ) (5) ˆ ˆ 3 where L s the covrnce mtrx of (btobt b STLS). Ths s x sttstc wth K degrees of freedom under the null hypothess, where K s rnk of L. Thus, the null hypothess for W s tht the prmetrc Tobt model estmtes would be consstent. The lterntve hypothess s tht the model s sem-prmetrc. The clculted sttstc for the STLS estmtor s 40.. Gven tht x (11) 5 4.7, we cn reject the null hypothess t the 1% level The smple verson of L cn be estmted by Vr(b ˆ ) Vr(b ˆ ) under the null hypothess. CLAD Tobt

6 196 S.-H. Yoo et l. / Economcs Letters 71 (001) Ths s rther drmtc evdence of the mpct of chngng estmtor n stuton of heteroscedstcty or the flure of the normlly dstrbuted error term s relted to censorng. 6. Concludng remrks The prmetrc estmton technque for delng wth zero WTP dt, the MLE of the Tobt model, s vulnerble to exstence of heteroscedstcty nd non-norml error structures. The STLS estmtor s robust under those stresses nd cn del wth zero response dt. In the pplcton reported here, the STLS estmton outperformed the prmetrc estmton sgnfcntly, reducng the mplct restrctons nvolved n the prmetrc model. Even f ths sem-prmetrc estmton method cn be esly clculted wth ny commercl computer pckges such s GAUSS, t hs not been populr becuse of the dffculty of the lterture to people ccustomed to MLE. However, judgng from our pplcton t s prctcl nd theoretclly promsng wy of modelng zero WTP dt. References Chesher, A., Irsh, M., Resdul nlyss n the grouped dt nd censored norml lner model. Journl of Econometrcs 34, Husmn, J., Specfcton test n econometrcs. Econometrc 46, Melenberg, B., vn Soest, A., Prmetrc nd sem-prmetrc modelng of vcton expendtures. Journl of Appled Econometrcs 11, Powell, J.L., Lest bsolute devtons for censored regresson model. Journl of Econometrcs 5, Powell, J.L., Symmetrclly trmmed lest squres estmton for tobt models. Econometrc 54, Robnson, P.M., 198. On the symptotc propertes of estmtors of models contnng lmted dependent vrbles. Econometrc 50, Tobn, J., Estmton of reltonshps for lmted dependent vrbles. Econometrc 6, Yoo, S.-H., Kwk, S.-J., Km, T.-Y., 000. Delng wth zero response dt from contngent vluton surveys: pplcton of lest bsolute devtons estmtor. Appled Economcs Letters 7,