CALIBRATION OF SMALL AREA ESTIMATES IN BUSINESS SURVEYS

Transcription

1 CALIBRATION OF SMALL AREA ESTIMATES IN BUSINESS SURVES Rodolphe Prm, Ntle Shlomo Southmpton Sttstcl Scences Reserch Insttute Unverst of Southmpton Unted Kngdom SAE, August 20 The BLUE-ETS Project s fnnced b the grnt greement no: under Theme 8 of the 7th Frmework Progrmme (FP7) of the Europen Unon, Soco-economc Scences nd Humntes. Trer- August 20 Pge

2 BUSINESS SURVES Sttstcl unts re orgnstonl enttes n countr Interested n smll re/domn estmtes Busness regsters llow for unt level covrtes Dstrbutons re tpcll skewed wth outlers Trnsformtons, such s the log, to ensure normlt ssumptons Trer- August 20 Pge 2

3 SMALL AREA ESTIMATION Centrl problem n mn res of socl sttstcs. Recentl used n busness sttstcs. Estmton of the men n dverse domns 2 m M re ;w 2;w ; w m ; w M ; w True populton men nd desgn-bsed estmte ; Estmted smll re men (EBLUP) ; becuse of smll Trer- August 20 Pge 3

4 SMALL AREA ESTIMATION AND BENCHMARKING Smll re estmton of the totl n the dfferent domns 2 m M ; 2; ; m; M ; Problem: The totl estmted b the model ~ T = ; w should. mtch the desgn bsed estmte of the populton totlt = w ; w Soluton b benchmrkng the estmtes b pproprte method Consequence of more robust estmton to msspecfctons of the model. Trer- August 20 Pge 4

5 NESTED ERROR UNIT LEVEL MODEL The Bttese, Hrter nd Fuller (988) (BHF) model for smll res =,, M: = X β u e N The trget prmeter of nterest s the re men: = / N N The EBLUP for non-neglgble smplng frctons: f ; ( f )[ X u ] = f β c GLS Trer- August 20 Pge 5

6 BENCHMARKING AT THE LINEAR SCALE (/2) Exstng methods consdered (see for nstnce Wng & l. (2008)) The rto method b multplctve term: ~ = RT ; f f T T ; An ddtve term wth vrnce weghtng: 2 2 ( σ σ / n ) N ~ ( ) ( ) e f T T 2 2 σ σ / n VAR f u ; = ; m 2 N = u e Pfeffermnn nd Brnrd (99): PB ( )[ PB ; = f f X c βpb u ] η PB, η ( β, u,..., u ), r = T n where = η CR ( r Rη) / RCR = GLS M ( ) M N X N n, N n,, N n, N, L N = R, =, 2 2 L m m m M, Rη PB = r, Ugrte & l. (2009) ppled ths constrned model for busness surve for severl regons wth vrnce clcultons Trer- August 20 Pge 6

7 Trer- August 20 Pge 7 BENCHMARKING AT THE LINEAR SCALE (2/2) We propose the method Augmentton of the unconstrned lest-squres sstem b ddng to the orgnl GLS sstem one row nd one column: PSW s PSW s s e X X e w X w X = = β β ; ; ; ; ; where, ( ) = m w w w w ; 2; ;,,, L ; ( ) N n N w / ; = ; ( ){ } = = m c x X n N X ; ; ; ) 2 ( γ ; ( )( ) ( ) ( ) = = m n N n n N ; / 2γ ; ( ). / ) ( 2 2 ; = = m n n N w γ The benchmrkng equton s obtned b orthogonlt of the resdul to the new dded column

8 SIMULATION FOR LINEAR CASE Nested error unt level regresson model B=000 popultons generted M = 30 res (no empt res) f 4% T σ u = 0., σ e = 0.3, nd β = (2,0.25) xj ~ N (m,s) ; m ~ N(0,3) ; s = 2 ONE POPULATION GENERATED TWO AREAS IN THE POPULATION Trer- August 20 Pge 8

9 SIMULATION RESULT FOR LINEAR CASE (/2) EBLUP 2 Rto Benchmrk Vrnce Weghted Benchmrk Pfeffermnn nd Brnrd Benchmrk Proposed Method Benchmrk f RT VAR PB ; ; ; ; PSW ; BIASREL 0.06% 0.58% 0.60% 0.60% 0.60% AARB 0.04% 0.60% 0.62% 0.62% 0.62% ARMSE.3%.45%.46%.46%.47% DIFFTOT 4.0x Trer- August 20 Pge 9

10 SIMULATION RESULT FOR LINEAR CASE (2/2) EBLUP 2 Rto Benchmrk Vrnce Weghted Benchmrk Pfeffermnn nd Brnrd Benchmrk Proposed Method Benchmrk Trer- August 20 Pge 0

11 LOG TRANSFORMATION FOR SKEWED VARIABLE In BHF model, = β j xj u e In busness surves, dstrbutons re skewed o Log norml trnsformton ( x ) = β j exp j u e o New formulton of the predctors Trer- August 20 Pge

12 BACK-TRANSFORMATION WITH BIAS CORRECTION Formulton of nerl unbsed estmtor s: ( f ) exp( f, sum = f ; j α ) j U \ s () The bs correcton s α nd cn be defned t the unt level or re level (see Chmbers, Dorfmn (2003) nd Moln (2009)) Other formulton from Kurn, Notodputro, Chmbers (2009): *,exp exp( * = ~ α ) ; ; (2) o The bs correcton s the modfed term t the re level ~ o We propose the correctve term α 2 nd compre to ~ α α ~ where Σ s the covrnce mtrx of the covrtes. Trer- August 20 Pge 2

13 BACK-TRANSFORMATION WITH BIAS CORRECTION Approches under model () Chmbers, Dorfmn (2003) ntroduce severl estmtors: the rst predctor nd smerng predctor Fbr, Ferrnte, Pce (2007) compre estmtors to nïve predctor wthout bs correcton. The twced smered estmtor performed best n smulton Chndr, Chmbers (20) dscuss clbrton fter logtrnsformton Trer- August 20 Pge 3

14 BENCHMARKING AFTER BACK-TRANSFORMATION Compre benchmrkng t dfferent stges wth bck trnsformton 2 2 α = σ σ / ~ or (b) = / 2 nd bs correcton b: () ( ) u e 2 Rto method under dfferent scenros α Σ 2 α β β No benchmrk t log scle, bck-trnsformed method (2), bs correcton () Benchmrk t log scle, bck-trnsformed method (2), bs correcton () PB, RT PSW, RT ; ; f, RT ; VAR, RT ; f sum RT No benchmrk t log scle, bck-trnsformed method (), bs correcton (),, ; No benchmrk t log scle, bck- trnsformed method (2), bs correcton (b) f 2, RT ; A mxmton of the log-lkelhood of the BHF model under constrnts, bck trnsformed method (2) nd bs correcton (b) MLC ; Trer- August 20 Pge 4

15 SIMULATION RESULT FOR NON-LINEAR CASE (/2) No benchmrk t log scle, bck-trnsformed method (2),,bs correcton (), rto djusted Benchmrk t log scle, bck- trnsformed method (2), bs correcton (), rto djusted No benchmrk t log scle, bck- trnsformed method (), bs correcton (), rto djusted No benchmrk t log scle, bck- trnsformed method (2), bs correcton (b), rto djusted MLC djustment, bck- trnsformed method (2), bs correcton (b) NOT BENCHMARKED BENCHMARKED b 2b 3b 4b 5b 6b 7b f, sum f f f, sumrt, 2 VAR PB PSW ; f 2, RT f, RT VAR, RT ; PSW, RT ; PB, RT ; ; ; ; ; ; BIASREL 0.39%.6% 0.47% 8.77% 8.77% 8.75% 2.99% 2.84% 3.03% 2.83% 2.87% 2.90% 2.58% AARB 0.66% 0.89% 0.28% 8.50% 8.49% 8.49% 3.30% 3.5% 3.34% 3.5% 3.8% 3.20% 2.89% ARMSE 5.8% 2.05% 5.75% 0.0% 0.0% 0.02% 6.87% 6.84% 6.90% 6.84% 6.86% 6.90% 6.69% DIFFTOT 5.6x x0 5 7.x x x x ; ; ; MLC ; Trer- August 20 Pge 5

16 SIMULATION RESULT FOR NON-LINEAR CASE (2/2) A C B D b 2b 3b 4b 5b 6b 7b -0.4 Group A: All benchmrk estmtes to orgnl scle usng the Rto Method or the MLC method ( b 7b ) Group B: No benchmrk, bck- trnsformed method () nd bs correcton () ( ) nd bck- trnsformed method (2) nd bs correcton (b) ( 3 ) Group C: Benchmrk t log-scle nd no benchmrk to orgnl scle, bck- trnsformed method (2) nd bs correcton () ( 4, 5, 6 ) Group D: No benchmrk, bck-trnsformed method (2) nd bs correcton () ( 2 ) Trer- August 20 Pge 6

17 CONCLUSION We hve used the nested error unt level regresson model Benchmrkng methods for the lner cse perform smlrl Benchmrkng methods for non-lner cse dffer dependng on bck-trnsformton nd stge of benchmrkng Rto djustment to benchmrked log-scle nd bck trnsformton provde comprble results to the cse when logscle s not benchmrked Future reserch: Performnce under more relstc popultons, empt res Comprson wth lterntve methods, for exmple robust methods of smll re models Incluson of surve weghts, vrnce estmtes Thnks for our ttenton Trer- August 20 Pge 7