GREG estimation with reciprocal transformation for a Polish business survey

Transcription

1 The 11 th Professor Aleksaner Zelas Internatonal Conference on Moellng an Forecastng of Soco-Economc Phenomena GREG estmaton wth recprocal transformaton for a Polsh busness survey Grażyna Dehnel 1 Abstract As a result of economc changes the scope of tasks for busness statstcs has expane. There s a growng eman for short-term statstcal ata elvere on a monthly or quarterly bass, wth mprove accuracy an coherence. To meet ths eman, t s necessary to evelop estmaton methos that can take avantage of amnstratve ata. The purpose of these efforts s to ncrease the effectveness of estmates an exten the scope of nformaton n terms of the number of varables an cross-classfcatons. The paper presents an attempt to estmate basc economc nformaton about small enterprses by applyng recprocal transformaton to one of the small area statstcs methos - GREG estmaton. Varables from amnstratve regsters were use as auxlary varables. The stuy was conucte for provnces cross-classfe by categores of economc actvty. Keywors: robust estmaton, busness statstcs, outlers, GREG JEL Classfcaton: C40, C51 1 Introucton In recent years much attenton has been evote to busness statstcs. To meet the growng nees for nformaton, t s necessary to conuct research ame at expanng the scope of busness statstcs. The man ffculty facng offcal statstcs n ths respect s the rsng nonresponse. In aton, the scope of economc nformaton that can be collecte s lmte by survey costs an responent buren resultng from statstcal reportng. Uner these crcumstances, the growng eman for nformaton can only be satsfe by explotng amnstratve sources of ata. It s expecte that the aopton of new solutons wll mprove the effcency of estmates an, above all, ncrease the number of cross-classfcatons avalable n statstcal outputs (Markowcz, 2014). In the search for new approaches to estmatng parameters of enterprses t s necessary to account for the specfc characterstcs of the target populaton. One characterstc feature of the populaton of enterprses s the presence of outlers (Schm et al., 2016; Toorov et al., 2011). For ths reason, the present artcle focuses on an estmaton metho use n small area estmaton, whch employs recprocal transformaton. The purpose of the stuy was to assess the possblty of applyng 1 Corresponng author: Poznań Unversty of Economcs an Busness, Department of Statstcs, Al. Nepoległośc 10, Poznań, Polan, e-mal: grazyna.ehnel@ue.poznan.pl. 67

2 The 11 th Professor Aleksaner Zelas Internatonal Conference on Moellng an Forecastng of Soco-Economc Phenomena a mofe generalse regresson estmator (GREG) to estmate characterstcs of small companes. In orer to mprove estmaton qualty, lagge auxlary varables from amnstratve sources were use. Estmates were calculate usng ata about small companes classfe nto actvty secton: Trae. 2 The estmaton metho One of the assumptons of a lnear regresson moel s homosceastcty. In the case of the populaton of enterprses characterse by strong asymmetry of strbuton an strong varaton, ths assumpton s often not satsfe (Zhang an Hagesaether, 2011). As a result, the varables are heterosceastc, whch results n neffcent estmates of parameters. For ths reason, t s necessary to evelop methos that mnmze the mpact of heterosceastcty on the precson of estmates. One way of achevng ths s by transformng varables. For example R. Chambers et al. (2001) propose a mofcaton of the GREG estmator whch nvolves an atonal auxlary varable z. where In ts classc form the GREG estmator of the total of varable : yˆ U ˆ yˆ w e (1) GREG x' ˆ n a populaton U s - sample sze, s moel parameter ˆ s estmate usng a mofe formula whch nclues atonal varable z (Chambers et al., 2001): where: z, x - auxlary varables, 1 ˆ w x x' / z w x y / z (2) s s - parameter selecte epenng on the egree of heterosceastcty, for 0, estmator (1) has the classc form of the GREG estmator. The estmator gven by formula (1) can be expresse n the form whch s entcal to the classc formula for the GREG estmator: Ŷ w g y. (3) GREG s The only fference s the efnton of weght auxlary varable x for sample unts an s efne as: g, whch epens on the value of 68

3 The 11 th Professor Aleksaner Zelas Internatonal Conference on Moellng an Forecastng of Soco-Economc Phenomena where: - oman, g - weght of the -th unt, ˆ GREG, 1 ' X Xˆ w ' / z z HT, x x x g 1 / (4) s - estmate of the total n oman gven by the GREG estmator, Xˆ HT - rect Horvtz-Thompson (HT) estmator of the total of auxlary varable x, X - total of auxlary varable x. In the mofe verson of the GREG estmator propose by R. Chambers et al. (2001), DFBETA s strongly correlate wth weght g. Ths means that the value of the stance measure affects the egree to whch varable y s mofe. The DFBETA measure for -th unt can be calculate usng the followng formula (Chambers et al., 2001): DFBETA ' where: h x / z x x' / z x / z s 1 x e x x' / z (5) z 1 h 1 (6) n From prevous surveys t s known that the value of parameter s nclue n the nterval <1,2> (Särnal, 1992), whch s why the followng estmators were analyse n the stuy: z an estmator base on a lnear regresson moel uner homosceastcty 0 ˆGREG 0 k (classc GREG estmator), ˆ 1 - GREG 1,5 ˆGREG - 2 ˆGREG -, 1 1 zk, 1,5 1,5 zk. 2 2 zk Numbers 2-4 enote regresson estmators base on a lnear regresson moel uner homosceastcty. Estmaton qualty was assesse wth reference to estmates obtane usng classc rect estmators: HT an the GREG estmator (Gamrot, 2014): N - Horvtz-Thompson (HT) estmator ˆ HT y (7) s n 69

4 The 11 th Professor Aleksaner Zelas Internatonal Conference on Moellng an Forecastng of Soco-Economc Phenomena where : Ŷ - estmator of the total of varable, HT N - general populaton sze, y - value of the varable of nterest for the -th unt. ' - GREG estmator ˆ ˆ X Xˆ βˆ (8) GREG HT HT where: Xˆ w x - vector of rect HT estmators of auxlary varables x, HT s X - vector of totals of auxlary varables x, 1 βˆ w xx' wx y s estmate usng the metho of weghte least s s squares usng esgn weghts, w - esgn weghts. Values of the classc GREG estmator ffer from estmates obtane usng the estmator of nterest - 0 ˆGREG. The fferences are ue to the fact that estmator 0 ˆGREG oes not take nto account all sample unts an gnores those for whch auxlary varable z s zero. If the constant s omtte, the resultng estmator s calle a rato estmator (Heln, 2004). 3 Assumptons of the stuy The emprcal stuy nclue small companes (10-49 employees) conuctng actvty classfe nto secton Trae. The response varable estmate n the moel was Revenue obtane n June Informaton about the response varable came from the DG1 survey. The survey s conucte n the form of monthly reports submtte by all large an meumsze enterprses an a 10% sample of small enterprses. (Dehnel, 2014). The followng varables were selecte as auxlary varables x an z : the number of employees from the socal nsurance regster (ZUS) an revenue form the tax regster for December The ecson to use lagge varables was motvate by techncal lmtatons nvolve n surveys conucte by the Central Statstcal Offce, namely the elay between the release of amnstratve ata for purposes of offcal statstcs. In the analyse moel t s assume that each auxlary varable can be use both as auxlary x an z (on conton that auxlary z cannot take zero values). Gven the above, n the fnal approach, auxlary varable z was taken to be the number of employees. Estmaton was conucte for omans efne by cross-classfyng provnces wth the secton of busness actvty accorng to the Polsh Busness Classfcaton (PKD). 70

5 The 11 th Professor Aleksaner Zelas Internatonal Conference on Moellng an Forecastng of Soco-Economc Phenomena 4 The metho of evaluatng precson Precson an accuracy of estmates were evaluate usng the bootstrap metho bootstrap samples were rawn, whch were then use to estmate Revenue for June 2012 for omans of nterests. Estmaton effcency was assesse usng the coeffcent of varaton of the estmator (Bracha, 2004): CV ˆ ˆ ˆ ˆ b, ˆ ˆ Var 999 b 1. (9) E E In orer to estmate bas, t s necessary to know values of the estmate parameters for the general populaton. Snce ths nformaton was not avalable n the survey t was estmate nrectly, base on ata from the tax regster for December It was assume that the followng relaton hols true: the rato of Revenue reporte n tax statements submtte by companes at the oman level to the value of Revenue reporte n the DG1 survey s constant. Revenue XII2012 Revenue_DG1 XII2012 Revenue VI2012 (10) Revenue_DG1 Revenue ( Revenue ) value of revenue reporte n the tax regster n December 2012 XII2012 VI2012 (VI2012) Revenue_DG ( Revenue_DG 1 ) value of revenue reporte n the DG1 survey n VI XII2012 December 2012 (VI2012). Ths approach mae t possble to etermne approxmate values of Revenue for June VI Contons of estmates an assessment of ther qualty The frst step of the analyss was to examne strbutons of companes epenng on the varables of nterest. The coeffcent of varaton vare from 47% to 649%. The strbutons were strongly asymmetrc, wth skewness coeffcents rangng from 0.6 to The hypothess of homosceastcty was verfe usng the Whte test an the Breusch Pagan test. For most omans of nterest test results confrme the valty of the hypothess about the varablty of the ranom component, see Table 1. Ths n turn justfe the use of the above mentone GREG estmators mofe to account for varable z. The estmates were assesse both n terms of accuracy an precson. The pont of reference for the assessment of precson were the estmates obtane usng classc, rect estmators: the HT estmator an the GREG estmator, nclung ts rato estmators. Base on CV values as a measure of effcency, t can be seen that the HT estmator exhbts the 71

6 The 11 th Professor Aleksaner Zelas Internatonal Conference on Moellng an Forecastng of Soco-Economc Phenomena greatest egree of varaton (see Table 1). Classc GREG estmators an transforme GREG estmators are characterze by less varaton. Test: Whte Breusch-Pagan Provnce statstc p-value statstc p-value olnośląske kujaw.-pom lubelske lubuske łózke małopolske mazowecke opolske pokarpacke polaske pomorske śląske śwętokrzyske warm.-maz welkopolske zachonopom Table 1. Results of the Whte test an the Breusch Pagan test for heterosceastcty for secton Trae across provnces. Source: base on ata from the DG-1 survey. On closer analyss, t can be seen that estmaton precson of the GREG estmator epens on the sample sze. It s usually bgger n omans that have more representaton n the sample. In most omans of nterest, the CVs of the transforme estmators, regarless of the value of the coeffcent, are more or less smlar an slghtly hgher than the classc GREG estmator. Moreover, t can be notce that, as varablty an asymmetry n a gven oman ncreases, the gan n precson resultng from usng each of the GREG estmators mproves. The precson of estmatng Revenue of companes was assesse n reference to rato estmates gven by formula 10. In aton, to prove a more complete assessment, 72

7 The 11 th Professor Aleksaner Zelas Internatonal Conference on Moellng an Forecastng of Soco-Economc Phenomena transforme estmators were compare wth the HT estmators an the classc GREG estmators, see Fg. 2. The results ncate that the ncluson of varable z n the moel yels a conserable mprovement n estmaton precson, especally compare to the HT estmator, but also wth respect to the classc GREG estmator. Fg. 1. Estmaton precson (CV) for secton Trae across provnces. Source: base on ata from the DG-1 survey an amnstratve regsters. Fg. 2. Estmate revenue for June 2012 across provnces - secton Trae. Source: base on ata from the DG-1 survey an amnstratve regsters. The HT estmator overestmate Revenue for nearly all omans of nterest, whle the GREG estmator tene to unerestmate t. In contrast, estmates obtane by the transforme estmator are closest to the reference values. The largest screpances between parameter estmates prouce by the fferent estmators can be observe for omans characterze by the largest varaton an asymmetry of varables use n the moel. 73

8 The 11 th Professor Aleksaner Zelas Internatonal Conference on Moellng an Forecastng of Soco-Economc Phenomena Fgure 3 nclues hstograms for two selecte provnces showng strbutons of bootstrap estmates obtane by applyng the stue estmators. The estmators usng auxlary varables from regsters are less base. As the parameter ncreases, the strbuton of estmates tens to approxmate the reference value marke n re. Fg. 3. Dstrbuton of estmates for selecte provnces for secton Trae. Source: base on ata from the DG-1 survey an amnstratve regsters. Concluson The results obtane n the stuy lea to the followng conclusons: the ncluson of auxlary varables n the GREG estmator has conserably mprove estmaton precson compare to the HT estmator; the mofe estmaton metho yels a greater gan n precson an accuracy for omans wth greater varablty an asymmetry; mprovement n estmaton precson for the estmators usng recprocal transformaton epens on the value of the parameter. A conserable gan can be acheve f an approprate moel s selecte for a gven oman; the rawback of ths approach, however, s that the applcaton of mofe GREG estmators for a very large number of omans s tme-consumng an emanng. Acknowlegements The project s fnance by the Polsh Natonal Scence Centre, ecson DEC- 2015/17/B/HS4/

9 The 11 th Professor Aleksaner Zelas Internatonal Conference on Moellng an Forecastng of Soco-Economc Phenomena References Dehnel, G. (2014). Wnsorzaton Methos n Polsh Busness Survey. Samplng methos an estmaton, 97, 97. Gamrot, W. (2014). Estmators for the Horvtz-Thompson statstc base on some posteror strbutons. Mathematcal Populaton Stues, 21(1), Heln, D., Falvey, H., Chambers, R., & Kokc, P. (2001). Does the moel matter for GREG estmaton? A busness survey example. Journal of Offcal Statstcs, 17(4), Heln, D. (2004). Busness survey estmaton. Statstska centralbyrån. Lehtonen, R., Särnal, C. E., & Vejanen, A. (2005). Does the moel matter? Comparng moel-assste an moel-epenent estmators of class frequences for omans. Statstcs n Transton, 7(3), Lehtonen, R., Särnal, C. E., & Vejanen, A. (2008, August). Generalze regresson an moel-calbraton estmaton for omans: Accuracy comparson. In workshop on Survey Samplng Theory an Methoology (pp ). Markowcz, I. (2014). Busness Demography-Statstcal Analyss of Frm Duraton. Transformatons n Busness & Economcs, 13(2B), Rao, J. N., & Molna, I. (2015). Small area estmaton. John Wley & Sons. Schm, T., Tzavs, N., Münnch, R., & Chambers, R. (2016). Outler Robust Small Area Estmaton Uner Spatal Correlaton. Scannavan Journal of Statstcs. Toorov, V., Templ, M., & Flzmoser, P. (2011). Detecton of multvarate outlers n busness survey ata wth ncomplete nformaton. Avances n Data Analyss an Classfcaton, 5(1), Zhang, L. C., & Hagesæther, N. (2011). A oman outler robust esgn an smooth estmaton approach. Canaan Journal of Statstcs, 39(1),