A Problem in Small Area Estimation: Cash Rental Rates

Transcription

1 A Problem n Small Area Esmaon: Cash Renal Raes Wllam Cecere 1, Emly Berg 1, Malay Ghosh 1 Naonal Agrculural Sascs Servce USDA, Unversy of Flora Absrac The USDA Naonal Agrculural Sascs Servce (NASS conucs an annual Cash Rens survey o esmae couny level cash renal raes for hree lan use caegores (non-rrgae croplan, rrgae croplan, an pasure. A cash ren s lan rene on a per acre bass for cash only. Esmaes of cash renal raes are useful o farmers n eermnng renal agreemens, economss n suyng research quesons, an polcy maers n compung paymen raes for he Conservaon Reserve Program. A major ssue s ha survey ncaons for counes are unsable ue o small realze sample szes. We nvesgae he use of mxe moels o oban relable esmaes of average cash renal raes a he couny level. Key Wors: mxe moels, small area esmaon, emprcal bayes, benchmarng 1. Inroucon The Naonal Agrculural Sascs Servce (NASS conucs hunres of surveys each year o oban esmaes relae o verse aspecs of US agrculure. NASS s large scale surveys prouce relable esmaes of agrculural characerscs a naonal an sae levels. Esmaon for small omans, such as counes, s more ffcul ue o small realze sample szes. The focus of our research s esmaon of average cash renal raes for non-rrgae croplan, rrgae croplan, an pasurelan a he couny level. In a cash renal agreemen, a enan rens croplan or pasurelan from a lanowner n uns of ollars per acre. Cash renal agreemens ffer from anoher common seup calle share-renal agreemens, whch nvolve paymens n erms of a share of he prouce goos. The enan n a cash renal agreemen s ypcally responsble for all managemen ecsons, acqures all of he prouce goos, an assumes all rs n proucng hose goos. NASS esmaes of couny-level cash renal raes serve many purposes. Farmers use he esmaes of local cash renal raes for guance n eermnng approprae renal agreemens (Dhuyveer an Kasens, 009. Agronomss use he esmaes o suy research quesons relae o he nerplay beween cash renal raes an oher economc characerscs such as commoy prces an fuel coss (Wooar e al., 010. NASS publshe couny-level cash ren esmaes have mmeae mplcaons for he Conservaon Reserve Program, a polcy ha ams o proec naural resources by provng renal paymens o agrculural lanowners who choose o conserve her lan. Because of he role of cash renal raes n he Conservaon Reserve Program, he 008 Farm Bll requre NASS o conuc an annual survey of cash renal raes for nonrrgae croplan, rrgae croplan, an permanen pasure. To sasfy he requremens of he 008 Farm Bll, NASS mplemene an annual Cash Ren Survey. A concern s ha rec esmaors of average couny renal raes from he Cash Ren Surveys are ofen unsable ue o small realze sample szes. Ulmaely, esmaes are esre for he hree lan use caegores (non-rrgae croplan, rrgae croplan, an pasurelan for counes wh a leas 0,000 acres of croplan or pasurelan. Esmaes are publshe a hree levels of geographc eal: sae,

2 agrculural sascs src, an couny. We nvesgae he use of mxe moels (Rao, 003 o sablze he esmaors of couny-level cash renal raes. Moel-base proceures were evelope usng he 009 an 010 Cash Ren Survey aa, an he mehos were laer apple o he aa from he 011 Cash Ren Survey. NASS esmaes sae-level cash renal raes usng a combnaon of aa from a naonal survey calle he June Area Survey an he Cash Ren Survey. The sae-level cash ren esmaes are consere o be more relable an are publshe before couny level esmaon from he Cash Ren Survey s complee. To manan nernal conssency, s mporan ha appropraely weghe sums of couny precors are equal o he prevously publshe sae esmaes. We use he benchmarng proceure propose by Ghosh an Seors (011 o ensure ha he couny precors preserve he prevously publshe sae esmaes.. Daa for Moelng Cash Renal Raes.1 NASS Cash Ren Survey The man source of aa for esmang couny-level cash renal raes s he annual NASS Cash Ren Survey. The frs secon of he quesonnare ass wheher or no he operaon owne, rene or lease from ohers, or rene or lease o ohers. The secon secon ass f he operaon rene croplan or pasurelan for cash. The operaons ha rene croplan or pasurelan for cash are ase o repor he acres rene an he cash renal rae or he oal ollars rene for each of he hree caegores: nonrrgae croplan, rrgae croplan, or permanen pasure. The samples for he 009 an 010 Cash Ren Surveys were srafe ranom samples, where he sraa where efne accorng o he oal ollars rene repore on prevous surveys or he 007 Census of Agrculure. The srafcaon change for he 011 Cash Rens Survey such ha operaons wh greaer acres prevously rene ha hgher selecon probables. Because he samplng frame use for prevous surveys may no cover he whole populaon, an aonal sraum was ae for 011. A rec survey esmaor for a parcular lan use caegory s a rao of a weghe sum of he ollars rene o a weghe sum of acres rene. The wegh assocae wh a responen s he populaon sze of he sraum conanng he responen ve by he number of responng uns n ha sraum.. Auxlary Informaon In an effor o mprove he precson of he couny-level cash ren esmaes, auxlary varables were esre ha woul explan he varably among he rec couny esmaes. Auxlary nformaon for moelng cash renal raes s avalable from several sources exernal o he Cash Ren Survey. Afer searchng hrough many opons of covaraes, hree were chosen base upon usably an correlaon wh he em of neres. The frs covarae s he Naonal Commoy Crop Proucvy Inex (NCCPI, whch s evelope an manane by he Naural Resource Conservaon Servce. The NCCPI consss of hree fferen nexes calle NCCPI-corn, NCCPI-coon, an NCCPI-whea, whch reflec he qualy of he sol for growng non-rrgae crops n hree fferen clmae conons. The nexes are consruce a he level of a map un, a subse of a couny. (User Gue Naonal Commoy Crop Proucvy Inex (NCCPI. Verson 1.0, 008. (fp://fp-fc.sc.egov/usa.gov/nscc/nccpi_user_gue.pf. The couny-level nexes use as covaraes are weghe averages of he map-un values, where he weghs are he acres of croplan covere by a map un. The NRCS also prouces a summary nex calle max-nccpi. The max-nccpi s obane by frs ang he maxmum of he hree commoy-specfc nexes for each

3 map un an hen compung a weghe average of he maxma across map uns n a parcular couny. The NCCPI proves four poenal covaraes: he hree commoy specfc nexes an he max-nccpi. Anoher measure of he qualy of he lan n a couny s he realze crop yel. NASS publshes esmaes of crop yels for a varey of crops n many counes. Defnng a covarae base up NASS publshe yels s challengng because (1 no yels are publshe for many counes where an esmae of he cash renal rae s esre, an ( for many saes, he publshe yels for fferen counes are assocae wh fferen crops an fferen lan usages. We aemp o efne a yel covarae ha overcomes hese challenges. We sar by averagng he publshe yels across fve years n an effor o smooh he publshe yels an reuce he varance. Usng an average across years also helps reuce he number of counes wh mssng yels. Insea of efnng a sngle covarae for each crop, we efne a yel nex base on mulple crops. Ths allows us o consruc a sngle yel covarae for relavely verse sae where fferen crops are grown n fferen pars of he sae. The publshe yels for a sae fall no a mos four caegores: rrgae, non-rrgae, oal for crop, an hay. We use he publshe yels for he four caegores (non-rrgae, rrgae, oal for crop, an hay o consruc a mos four yel covaraes for each sae. One consequence of havng yel an NCCPI broen no four separae covaraes each s ha here s a enency for hgh correlaon beween covaraes of smlar ype. To accoun for hs, we efne a sngle nex for boh yel an NCCPI so ha we can avo problems resulng from mulcollneary. Ths sngle nex s consruce by usng he weghe average of he orgnal covaraes, where he wegh s base off of he correlaon beween a gven covarae an he couny cash renal rae relave o he oher covaraes. Our fnal source of auxlary nformaon s he Toal Value of Proucon (TVP. The TVP s a counylevel covarae obane from he 007 Census of Agrculure an s a measure of he value of goos prouce n a couny. 3. Cash Ren Moelng The curren NASS proceure for publshng couny-level cash ren esmaes nvolves havng expers examne he curren year survey aa an prevous year survey aa o se esmaes. They examne varous survey-base ncaons such as he rec esmaor for he curren year an he rao of he esmaors for he curren an prevous year. Our man objecve s o evelop a moel-base esmaon proceure ha uses hsorcal aa, auxlary aa, an aa from neghborng counes n a way ha s objecve an proves a measure of mean square error. We specfy a separae moel for each pracce: non-rrgae croplan, rrgae croplan, an pasure. We assume ha he sae esmaes for he curren year are alreay eermne an are avalable. Inal analyses of he aa from he 009 an 010 Cash Ren Surveys showe sgnfcan correlaons beween he cash renal raes for 009 an 010 a boh he un an couny levels. The srong correlaon suggess ha ncorporang nformaon from he prevous year may mprove he esmae for he curren year. One challenge n moelng s eermnng a meho for ncorporang aa from he prevous year ha s boh sascally efensble an compuaonally feasble.

4 Fgure 1: Drec esmaes of cash renal raes for non-rrgae croplan n Iowa counes from he 009 (x-axs an 010 (y-axs Cash Ren Surveys As an nal effor, we specfe a unvarae moel for he un-level aa. One of he covaraes n he un-level moel s he value of he cash renal rae repore n he prevous year. Ths meho s compuaonally feasble bu lacs sascal valy. One mporan problem s ha usng he cash ren per acre from he prevous survey as a covarae n a un-level moel reas he cash ren per acre from he prevous year as a fxe value. I s more approprae o rea he cash ren per acre from he prevous year as a ranom varable. Cononng on he prevous year cash renal rae can lea o mean square error esmaors for precons ha are arfcally small. Anoher unesrable propery of he un-level unvarae moel wh he prevous year cash renal rae as a covarae s ha only uns ha repor posve ollars an posve acres n boh years can be use o esmae moel parameers. Usng only a subse of he aa o esmae he moel parameers can be neffcen relave o a proceure ha uses all of he avalable aa. As a secon aemp, we specfe a un-level bvarae moel. The bvarae un-level moel overcomes he rawbacs assocae wh he unvarae un-level moel: The bvarae moel reas he prevous year cash renal rae as a ranom varable, an all of he aa from he wo years are use o esmae he parameers of he bvarae moels. One praccal problem assocae wh he bvarae un-level moel s compuaon me. We use Gbbs samplng o f he bvarae moel, an he samplng proceure can ae several hours o run for a gven sae an usage. In hs ocumen, we propose a meho base on a bvarae area-level moel. The proceure ncorporaes he cash ren per acre from he prevous year as a ranom varable (whch overcomes he sascal problems of he unvarae un-level moel, an he esmaors of moel parameers have close form expressons (whch overcomes he praccal problems of he bvarae moel. The proceure s base on wo separae unvarae area-level moels. One of he unvarae moels s base on he average of he wo me means. The secon unvarae moel s a moel for he fference of he wo me means. The precor for he average cash ren per acre for a sngle year s a sum of an esmae of he average (base on he frs moel an an esmae of he change (base on he secon moel. 3.1 Area-Level Moel

5 Le be he average cash ren per acre for couny n year. Le ŷ be he rec esmaor or ncaon of, an assume ha rec esmaors are avalable for wo consecuve me pons. Noe ha we specfy a separae moel for each pracce, so we o no nclue a subscrp for he pracce. Assume y e x (1 Where ~ (0,,( e, 1, e ~ [0, e, Re ], an (, 1, ~ [0,, R ], where R an Re are x correlaon marces wh correlaon parameers an e respecvely. We assume ha esmaes of e an e are avalable from he un-level aa. Le y 0.5( y y an ( y y. From (1,, 1, 1 y e x u ( where 0.5( 1, u 0.5( 1, an e 0.5( e 1 e. Smlarly, x v (3 Where ( 1, v 1, an ( e e, 1. The equaons n ( specfy a unvarae moel for he average of he rec esmaors for he wo me pons, an (3 s a unvarae moel for he fference. Because he varances for he wo me pons are assume o be equal n (1, ( y, s uncorrelae wh (,. One can oban precors of an along wh assocae MSE esmaors usng sanar mehoology for unvarae area-level moels. (Rao, 003, Chaper 7. If an are unbase esmaes of an, hen s an unbase esmaor precor of. If, n aon, esmae of he MSE of s 0.5( (4 an where MSE ( an MSE ( are esmaes of he MSEs of an are uncorrelae, hen an MSE ( MSE ( 0.5MS E(, (5, respecvely.

6 We assume ha he moels ( an (3 hol for rec esmaors of he average an fference, respecvely. The precor of he average for couny s (1, avg y, avg x, where 1, avg ( u e, avg u, an u s an esmaor of he varance of u of ( an esmae of he samplng varance n he rec esmaor of he average. The samplng varance s represene by he varance of e of (. The precor of he fference for couny s e, avg s an, ff (1, ff z 1 Where, ff ( v, avg v where v an, avg are esmaes of he varances of v an of (3. The proceure oulne n (7-(3 of Wang an Fuller (008 s use o oban he esmaes, (,,, u v. The mean square error esmaor has he form n (5, he wo componens of he MSE gnore he varance of he varance esmaors. 3. Esmaon of he Dfference Recall ha for an esmae of he average, we sar wh he smple average of he curren an prevous years y.5( y y. The equvalen approach o consrucng a rec esmaor of he fference 0 1 woul be o use he smple fference beween he wo rec esmaes y y 1. In he cash ren applcaon, some uns respon n boh me pons. We efne an esmaor of he fference ha reas uns ha respon n boh me pons sncly from uns ha respon n only one me pon as an effor o oban an esmaor of he fference wh a smaller varance han y y 1. To efne he esmaor of he fference, le nex he uns whn couny, an le be he number of uns n couny ha ~ repor a cash renal rae n boh me pons. Le j yj yj 1, an le j be he resul of applyng a mofcaon for oulers analogous o he meho of Appenx B o he ~ j. An esmaor of he fference base on only he observaons ha respon n boh me pons s y ( n n j1 j a j ( j1 a j 1, where aj s he average of he acres rene by un (j across years -1 an. The hree snc scenaros wh whch we can measure change from one year o he nex are as follows: (1 For a gven couny-usage combnaon, we have recors ha respone n boh years, respone n he frs year an no he secon, an respone n he secon year an no he frs. ( For a gven couny-usage combnaon, we have no encal responens n boh years, responens n he frs year an no he secon, an responens n he secon year an no he frs. (3 For a gven couny-usage combnaon, we have only recors ha respone o boh years.

7 For scenaro ( n whch we have no recors responng o boh years, we se y y 1 as our esmae of ŷ. For scenaro (3 n whch here are only recors responng o boh years, we use y as our esmae of ŷ. For scenaro (1 where here are boh recors from ( an (3, we use a weghe combnaon of an y where he weghs reflec he relave sze for recors from scenaros ( an (3 respecvely. The formula an ervaons of he esmae of ŷ for scenaro (1 can be foun n Appenx A. 3.3 Esmaon of Samplng Varances In he scusson of he small area moels n secon 3.1, we assume ha we are sarng wh an esmae of he samplng varance of he rec esmaors of he average an he fference. NASS mehoologss compue a varance esmae of he rec esmaor of average cash renal rae for each year usng a jacnfe proceure. The jacnfe esmaes of he varances are esgn unbase, bu can have large varances for areas wh small sample szes ue n par o oulers. For several saes n our suy, he jacnfe esmaes of he varances are correlae wh he rec esmaes of he cash renal raes. In hs secon, we efne a generalze varance funcon o oban esmaes of he samplng varances. Usng esmaes of samplng varances base on a generalze varance funcon GVF (Woler, 1976 nsea of he rec jacnfe esmaes of he samplng varances s common pracce n small area esmaon for several reasons. Frs, rec esmaors of varances (such as jacnfe esmaors may have large varances for areas wh small sample szes. Smoohng he esmaors of he varance hrough a generalze varances funcon can reuce he mean square error of he esmaor of he varance. Secon, a rec esmaor of a varance s equal o zero f he sample sze for he area s less han wo. Our efnon of he generalze varance funcon gves a posve esmae of he varance for areas wh a sample sze of one. Secon, correlaons beween rec esmaors of varances an rec esmaors of means can lea o bases n he esmaors of he regresson coeffcens. Use of a generalze varance funcon can reuce he correlaon beween he rec esmaor of he varance an he rec esmaor of he mean an subsequenly mprove he qualy of he precons A Generalze Varance Funcon for he Drec Esmaors of he Samplng Varances Le S be he jacnfe esmaor of he varance of ŷ, he rec esmaor of he average cash renal rae for couny n year. Le n be he number of responens n couny wh posve ollars rene an posve acres rene n year. Assume, n 0.5 S (6 0 ( x x.. 1 where x s he covarae n he moel for he average, an x m 1.. n.. xn., n. ns, 1 s1 m 1 ( n n an ~ (0, n. Le, ( 0 1 be he generalze.. 1 leas squares esmaor of (, base on (6, an le S ( [ (1 ( m n Sm n 0 n x. x.. 1] (7

8 We use S m as an esmaor of he varance of ŷ. A anger n (7 s ha he prece value for a sanar evaon may be negave. For saes presene here, all prece values for sanar evaons are posve. Anoher problem wh moelng sanar evaons nsea of varances s ha he varance esmaors ha resul from squarng he sanar evaons are no unbase even f he esmaor of he sanar evaon s unbase. Refnemens o he generalze varance funcon n (7 are an area of curren wor. To be able o ge an esmae of varance across wo years, an esmaor of he correlaon beween he samplng errors s requre. Le ~ 1 rj a ( y a (8 Where a j an j are he acres an ollars, respecvely, rene from operaor j n couny an year. Noe ha a correcon s mae for exreme values n he calculaon of r~ j an s explane furher n Appenx B. Le r j be he resul of he proceure n Appenx B. Le A be he se of (j ha repor posve ollars an posve acres n boh me pons, an -1. Le ( j A j. ( j A j 1. 1 j j ( r j A j r rj r (. ( 1. 1 (9 ( r r ( r r where 1 r. A j r ( A j an A s he number of (j n he se A. Le ag( S, S R ag( S, S (10 e m 1 m m 1 m where R s a x marx of ones on he agonal an on he off-agonals. Our esmaes of sample varance for he average an fference are obane usng hese resuls. 3.4 Two-sage Benchmarng NASS obans esmaes of cash renal raes a he sae level usng aa from a naonal survey ha s conuce n June n aon o he Cash Ren Survey. The sae esmaes are publshe before he couny-level aa from he Cash Ren Survey are fully processe. NASS also esablshes esmaes of cash renal raes for agrculural sascs srcs, groups of spaally conguous counes whn a sae. To rean nernal conssency, s mporan ha appropraely weghe sums of couny esmaes equal he src esmaes an appropraely weghe sums of src esmaes equal he prevously publshe sae esmae. The benchmarng resrcons for a me are, w (11 an D 1 pub (1

9 where w n couny, D ( z z, ( z z, z s a rec esmae of he acres rene s a se of nexes for he counes n src, s he fnal esmae of he average cash renal rae for src, an pub s he publshe esmae of he sae-level cash ren per acre. The nex for he year s suppresse n (11 an (1 for smplcy. The rec esmaors of he acres rene a he couny an src levels are reae as fxe for our analyss. We use he wo-sage benchmarng proceure propose by Ghosh an Seors (011 o efne benchmare esmaes. The benchmare esmaes mnmze he quarac form, D 1 B g( c, ( (13 B D ( c 1 subjec o he consrans n (14 an (15, where c c,..., c,,...,, ( 1 m, w ( 1, w w, an an are consans selece by he analys. We choose w an, whch gves he benchmare esmaes, (14 B B ( (, w B an B 1 ( ( (1 ( ( w B (, w (15 1 (1 pub for couny an src (, respecvely, where ( s he src conanng couny. In (15, D B B w 1,, w. Each of he benchmare esmaes n (14 an (15 s a sum of he precor an an ajusmen erm. If he precor for he sae s larger (smaller han he prevously publshe sae oal, hen he ajusmen s negave (posve, an he benchmare couny an src esmaes are smaller (larger han he precors. We gnore he effec of benchmarng on he MSE of he precor. In a Bayesan seng, he mean square error of he benchmare precor s he sum of he varance of an he square fference beween an he benchmare precor. (You e al., Resuls Recall ha we are usng moels for hree usages, nonrrgae an rrgae croplan, an permanen pasure. For each sae/usage combnaon, we selec one moel for he average an one for he fference. For he purposes of smplcy, we wll focus on fve saes n our analyss: Flora, Msssspp, Mchgan, Iowa, an Kansas. These saes reflec he versy n agrculure necessary o represen a broa range of challenges wh moelng cash renal raes. In hs analyss we wll focus on he esmaes for he year 011. A common measure of wheher he moel mproves upon he nal rec survey esmae s he measure he relave roo mean square error. We efne relave roo mean square error as, ( ( (

10 RRMSE RMSE( RMSE( y where RMSE ( y s he esmae RMSE of he rec ncaon an RMSE( s he esmae RMSE of he moel esmae. Table 1 gves he means of he srbuons of RRMSEs for each sae/usage combnaon. A RRMSE less han one from our efnon occurs when he moele esmae has a lower MSE han he rec survey esmae. For each sae, he esmae MSEs of he moel-base precors are smaller han he esmae varances of he rec survey esmaors for mos of he counes n he sae. Table 1: Mean Relave Roo Mean Square Errors Sae Nonrrgae Irrgae Pasure Flora Iowa Kansas Mchgan Msssspp For nonrrgae croplan, Iowa showe he mos mprovemen, wh RRMSE less han 0.5 for half of he counes. For he rrgae croplan an pasure, we see a subsanal rop n he MSE for he moele esmaes on he whole. The smalles mprovemen s for nonrrgae croplan n Flora, where he relaonshp beween he covaraes an he average cash renal rae s relavely wea. Anoher measure n evaluaon of he moele esmaes s wheher coeffcens of varaon (CVs are reasonable. Table shows means of couny CVs across he saes for each usage. Flora has he larges amoun of varaon relave o s esmaes. No mean CV s over 30%, Table : Mean of Esmae CVs for Moel Precors (% Sae Nonrrgae Irrgae Pasure Flora Iowa Kansas Mchgan Msssspp As menone n secon 3, NASS uses a proceure where expers examne curren an prevous year survey aa o se esmaes. I s ofen esre o examne how moele esmaes compare wh NASS publshe values as publshe values are vewe as a sanar. Fgure llusraes hs comparson for nonrrgae croplan wh moel precons on he y-axs an publshe values on he x-axs. The fve saes are represene by color accorng o her sae abbrevaon. The moel precons an publshe values le close o he 45 egree lne, ncang conssency wh our comparson sanar.

11 Prece NONIRRIGATED 0 19 Publshe Fgure : Plo of moel prece esmaes vs. publshe values for 011 nonrrgae croplan 5. Conclusons an Fuure Wor We use emprcal Bayes mehos o esmae couny-level cash renal raes usng prevous year nformaon an auxlary aa. We specfy wo separae unvarae area-level moels: one for he average of he wo years an one for he fference beween he wo years. In hs approach, he precor for eher me pon s jus a lnear combnaons of he esmaors of he average an fference. We can oban a MSE esmae by assumng no correlaon beween he esmaors of average an fference. An avanage o hs approach s ha he processng me s que small, whch s mporan for use n a proucon envronmen. In he saes examne, he moel-base proceure leas o an mprovemen n roo mean square error relave o he rec survey esmaes. We also emonsrae relavely sable coeffcens of varaon for he hree usages an a srong relaonshp wh he publshe values of he same year. Alhough hese moels emonsrae sgnfcan mprovemens, here are sll several challenges. One s he poenal for he rare case of a negave esmae o be prouce. Ths s possble manly for counes wh relavely small survey ncaons an large varances ue o he lnear relaonshp wh he covaraes. A way o guaranee posve esmaes ha we are loong no s o mae a sngle, posve covarae nex wh a posve correlaon wh he cash renal rae. Anoher poenal for nnovaon s o combne conguous saes wh small realze samples o pool from a larger area. We may also nvesgae he use of nonlnear moels o hs applcaon. Acnowlegemens FL MI MS IA KS The auhors han Wayne A. Fuller for hs conrbuons o hs research. Aonal hans are n orer o Ange Consne, Sharyn Lavener, Cur Soc, Sco Shmmn, an Dan Becler from he Naonal an Rch Iovanna from he Farm Servce Agency.

12 References Baese, G.E., Harger, R.M., an Fuller, W.A. (1988, An error-componens moel for precon of couny crop areas usng survey an saelle aa, Journal of he Amercan Sascal Assocaon, 83, Dhuyveer, D., an Kasens, T. (010. Kansas Lan Values an Cash Rens a he Couny Level. hp://www/agmanager.nfo/farmm/lan/couny/counvalues &Rens(Sep010.pf. Ghosh, M. an Seors, R. C., Two Sage Bayesan Benchmarng as Apple o Small Area Esmaon, (011. To be subme. Woler, K.(1976, Inroucon o Varance Esmaon, Sprnger seres n sascs Wooar, S., Paulson, N., Bayls, K., an Woar, J. (010. Spaal Analyss of Illnos Agrculural Cash Rens, The Selece Wors of Kahy Bayls. hp://wors.bepress.com/ahy_bayls/9. Rao J.N.K.(003: Small Area Esmaon, New Wor: John Wley an Sons. Wang, J., an Fuller, W.A. an Qu, Y. (008, Small area esmaon uner a resrcon. Survey Mehoology, 34, You, Y., Rao, J.N.K., Dc, P. (00. Benchmarng Herarchcal Bayes Small Area Esmaors wh Applcaons n Census Unercoverage Esmaon. Proceengs of he Survey Mehos Secon, Sascal Socey of Canaa. Appenx A: Dervaon for he Esmae of he Dfference Recall from secon 3. ha we are ryng o ge an esmae of Where y y j 1 j 1 ~, R ŷ for scenaro 1. Suppose 1 R s a x correlaon marx wh parameer. Then, V{ y } (1 n an 1 1 V{ yu} n 1, 1 n,, where y u y, y 1, 1, an y s, s s he smple average of he responens wh posve ollars an posve acres n years s bu no n year. A generalze leas squares esmaor of 1 s, op u 1, y y (1 y (16

13 where (1 n 1, [(1 n 1, n 1, 1 n, ]. If n1 n, hen n n 1, y (17 n n bb y, u, bb 1, where y y y 1, an y s he smple average of y j. If we solve for y u n (0, we oban, y u n n 1, y (18 n, bb n, bb where n n, bb n1,. By subsuon of he rgh han se of (1 no (19, n n y (19 y, op (1 n, bb n, bb Fnally, by replacng n ( wh we ge our esmae of Appenx B: Mofcaon o Exreme Taylor Devaes ŷ use n scenaro 1. Le S r, be he sample sanar evaon of r~ j efne n (11. Le me be he mean of r~ j. If ~ rj me 3. 3S r, hen se ~ rj me 3. 3S r. If ~ rj me 3. 3S r, hen se ~ rj me 3. 3S r. Oherwse, se r j ~ rj. We o no erae he proceure for smplcy.