This module is part of the. Memobust Handbook. on Methodology of Modern Business Statistics

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1 Ths module s part of the Memobust Handbook on Methodology of Modern Busness Statstcs 26 March 2014

2 Method: Generalsed Regresson Estmator Contents General secton Summary General descrpton of the method Propertes of GREG estmator Partcular cases and extensons Preparatory phase Examples not tool specfc The Small-Medum Enterprses Survey and the current samplng strategy Examples tool specfc Glossary References Specfc secton Interconnectons wth other modules Admnstratve secton... 16

3 General secton 1. Summary The basc estmator (see Weghtng and Estmaton Man Module ) of a target parameter expands the observed values on the sample unts usng drect weghts, whch are the nverse of the ncluson probabltes. Generalsed regresson estmator s a model asssted estmator desgned to mprove the accuracy (see Qualty Aspects Qualty of Statstcs ) of the estmates by means of auxlary nformaton. GREG estmator guarantees the coherence between samplng estmates and known totals of the auxlary varables, as well. In fact, t s a specal case of a calbraton estmator (see Weghtng and Estmaton Calbraton ) when the Eucldean dstance s used. 2. General descrpton of the method In the estmaton phase, the sample values are weghted to represent also unobserved unts. When auxlary nformaton s avalable at unt or doman level, a GREG estmator can be used n order to reduce the varance of the estmates by usng the relatonshp between the target varable and the auxlary varables. At the same tme the resultng weghts allow calbraton to the known totals. Let y, x be the target varable and the vector of auxlary varables, respectvely. The GREG estmator (Cassel, Särndal, and Wretman, 1976) can be expressed as a sum of the Horvtz Thompson estmator (HT) (see Weghtng and Estmaton Man Module ) and a weghted dfference between known totals and ther HT estmator: where tˆ GREG = d y + βˆ t X d x, (1) s s d, =1, n, s the drect weght equal to the nverse of the ncluson probablty, t X s the vector of known populaton totals; moreover βˆ s an estmate of the vector of regresson coeffcents of y on x, gven by wth 1 ˆβ = d q x x dq x y s, s q scale factors chosen properly, e.g., to account for heteroscedastcty. For example, when the varablty of the target y depends on enterprses sze, z, the q can be chosen as In general, z may also be one of the covarates n the regresson model. Alternatvely, the GREG estmator can be formulated n terms of predcted values for the target varables calculated on the bass of a lnear relatonshp between y and x. More specfcally, these predcted values are used n the estmaton together wth the resduals from the model, evaluated for sample unts,.e., the GREG estmator can be wrtten as ( y yˆ ) N N tˆ = y + d e = y + d GREG = 1 ˆ s = 1 ˆ s, (2) z. 3

4 where y ˆ = x βˆ s the predcted value accordng to the lnear model that relates y and x, e s the evaluated resdual for a unt n the sample. Fnally, the GREG estmator can be convenently formulated as a weghted sum of sample values: t ˆ = d g y = w y GREG s, s s, s (3) where the correcton factor g s of the drect weghts s gven by g, s 1 = 1+ t d x X d q x x q x j s j j j s j j j j (4) whch does not depend on the target varable y. 2.1 Propertes of GREG estmator A fundamental property of the GREG estmator s that t s nearly desgn unbased (Särndal et al., 1992). The lnear GREG estmator s motvated va the lnear assstng model (Särndal et al., 1992) E( y ) = β x, V 2 ( y ) σ =. (5) However, the knowledge of all x values s not necessary to evaluate lnear GREG, because the knowledge of totals suffces to calculate the new weghts, w, (see secton 12 below). The regresson coeffcent n (5) can be estmated at natonal level or for a dsaggregated level, e.g., NUTS2. Ths level s referred as model group { U(p) }. In case of sub-natonal model group, the known totals need to be avalable at ths level. An mportant feature of the lnear GREG s that the weghtng system does not depend on the target varable but only on x values, as (4) shows. The GREG estmator s calbrated to the known totals of the assstng model, that s w = s, s x t X. In fact GREG s a partcular case of a calbraton estmator (see Weghtng and Estmaton Calbraton ) when usng the Eucldean dstance. Moreover, all the calbraton estmators can be asymptotcally approxmated by the GREG (Devlle and Särndal, 1992). Another relevant property of GREG estmator s that the evaluaton of ts varance (see Qualty Aspects Qualty of Statstcs ) s based on the varance of the resduals ( y ˆ ) s (Särndal et al., 1992). As a consequence of ths, the hgher the fttng of the lnear workng model the lower the varance of GREG estmator and therefore the hgher ts accuracy. On the contrary, f the model underlyng the GREG s not approprate for the target varable, a too large varaton of weghts may ncrease the varance wth respect to the HT estmator. In fact, varablty of weghts unrelated wth the target varable can ncrease the varance of the estmates, an approxmaton of ths mpact s gven (Ksh, 1995) by y 4

5 2 1+ CV, w s, (6) where CV stands for the coeffcent of varaton of fnal weghts. A possble drawback of the GREG estmator s that t can produce negatve weghts (cf. secton 10 below); on the contrary, n the framework of the calbraton estmator, t s possble to obtan weghts always postve usng dfferent dstance functons (see Weghtng and Estmaton Calbraton ). 2.2 Partcular cases and extensons The rato estmator s a specal case of GREG asssted by a model wth only one covarate, obtanable f the varance of the target varable s assumed to be a lnear functon of the auxlary varable 2 V ( y ) = σ x (Devlle, Särndal, 1992). Extended GREG estmators are defned replacng the assstng model (4) wth more general (nonlnear, generalsed, or mxed) models. The non-lnear GREG estmators (e.g., Lehtonen and Vejanen, 1998) requre a separate model fttng for every target varable; hence, an mportant drawback of ths knd of model asssted estmators s that they do not produce a unque system of weghts unformly applcable. On the other sde, the nonlnear GREG may gve a consderably reducton n varance, as a result of the more refned models that can be consdered when there s complete unt level auxlary nformaton. 3. Preparatory phase 4. Examples not tool specfc 4.1 The Small-Medum Enterprses Survey and the current samplng strategy Small and Medum-szed Enterprses (SME) sample survey s carred out annually by sendng a postal questonnare wth the purpose of nvestgatng proft-and-loss account of enterprses wth less than 100 persons employed, as requested by SBS EU Councl Regulaton n. 58/97 (Eurostat, 2003) and n. 295/2008. The unts nvolved n the survey have also the possblty to fll n an electronc questonnare and transmt t to Istat va web. The survey covers enterprses belongng to the followng economc actvtes accordng to the Nace Rev.1.1 classfcaton: - Sectons C, D, E, F, G, H, I, J (dvson 67), K; - Sectons M, N and O for the enterprses operatng n the prvate sector. Man varables of nterest asked to the SME sampled enterprses are Turnover, Value added at factor cost, Employment, Total purchases of goods and servces, Personnel costs, Wages and salares, Producton value. They are also asked to specfy ther economc actvty sector and geographcal locaton n order to test the correctness of the frame wth respect to these nformaton. Totals of varables of nterest are estmated wth reference to three typologes of domans of study. 5

6 4.1.1 Frame of nterest For the reference year 2007, the populaton of nterest for SME sample surveys s about 4.5 mllons actve enterprses. The frame for SME survey s represented by the Italan Statstcal Busness Regster (SBR). It results from the logcal and physcal combnaton of data from both statstcal sources (surveys) and admnstratve sources (Tax Regster, Regster of Enterprses and Local Unts, Socal Securty Regster, Work Accdent Insurance Regster, Regster of the Electrc Power Board) treated wth statstcal methodologes. Varables n the regster are both quanttatve (Average number of employees n the year t-1, Number of employees n date 31/12/year t-1, Independent employment n date 31/12/year t-1, Number of enterprses) and qualtatve (Geographcal locaton, Economc actvty accordng to Nace Rev dgt). From the Fscal Regster s also provded the VAT Turnover, whch represents a good proxy of the varable Turnover asked to the sampled enterprses by questonnare Samplng desgn (allocaton and doman of estmates) SME s a mult-purpose and mult-doman survey and t produces statstcs on several varables (manly economc and employment varables) for three types of domans, each defnng a partton of the populaton of nterest (see Tables 1 and 2). Table 1: Types of SME Survey domans Type of doman Number of Code Descrpton Domans DOM1 Class of economc actvty (4-dgt Nace Rev.1*) 461 DOM2 Group of economc actvty (3-dgt Nace Rev.1) by sze-class of employment DOM3 Dvson of economc actvty (2-dgt Nace Rev.1) by regon 984 *Nace Rev.1 = Statstcal Classfcaton of Economc Actvtes n the European Communtes Table 2: Defnton of Sze-classes of employment for doman DOM3 of SME Survey Nace Rev dgt level Sze-classes of employment 10-45; 1-9; 10-19; 20-49; 50-99; 50-52; 1; 2-9; 10-19; 20-49; 50-99; 55;60-64;67;70-74; 1; 2-9; 10-19; 20-49; 50-99; 80; 85; 90; 92; 93; 1-9; 10-19; 20-49; 50-99; Samplng desgn of the SME survey s a one stage stratfed random samplng, wth the strata defned by the combnaton of the modalty of the characters Nace Rev.1.1 economc actvty, sze class and admnstratve regon. A fxed number of enterprses are selected n each stratum wthout replacement and wth equal probabltes. The number of unts to be selected n each stratum s defned as a soluton of a lnear nteger problem (Bethel, 1989). In partcular, the mnmum sample sze s determned n order to ensure that the varance of samplng estmates of the varable of nterest n each doman does not exceed a gven threshold, n terms of coeffcent of varaton. The varables of nterest used for sample allocaton are Number of persons employed, Turnover, Value added at factor cost, whose mean and varance are estmated n each stratum by data from the frame and data collected from the prevous survey, respectvely. 6

7 About 103,000 of small and medum-szed enterprses (unts) are ncluded n the sample. The samplng unts are drawn by applyng JALES procedure (Ohlsson, 1995) n order to take under control the total statstcal burden, by achevng a negatve co-ordnaton among samples drawn from the same selecton regster The weghtng procedure After calculatng the total non-response correctng factors as the rato of the number of sampled unts and the number of respondng unts belongng to approprate weghtng adjustment cells, the weght of every sngle enterprse s further modfed n order to match known or alternatvely estmated populaton totals called benchmarks. In partcular, known totals of selected auxlary varables on the Busness Regster (Average number of employees n the year t-1, Number of enterprses) are currently used to correct for sample-survey nonresponse or for coverage error resultng from frame undercoverage or unt duplcaton. Practcal aspects n the applcaton of the weghtng procedure n the contest of SME survey The evaluaton of fnal weghts for SME survey s usually carred out usng the selected auxlary varables, for the three types of domans descrbed n Table 1. The optmsaton problem underlyng the GREG estmaton process can be therefore formulated n the followng way: the model group { U(p) } s defned as the dvson of economc actvty (2-dgt Nace Rev.1.1) of the frame (the updated Busness Regster); the domans of nterest are represented by the three typologes of parttons (descrbed n Tables 1 and 2); the auxlary varables are dentfed by x 1 = Number of enterprses x 2 = Average number of employees n the year t-1; for each enterprse, the vector x of the auxlary varables has been defned as follows: ' ( x', x' ) x = 1 2, combnaton of two vectors 1 ' 1 { λ ( j )} x =, d { αλ (j )} x ' wth d=1,,3; j=1,.., J d, 2 = d where, accordng to the updated Busness Regster nformaton: x ' and x' 2 whose form s, respectvely: - λ j ) s a dchotomous varable whose value s equal to 1 f the unt belongs to doman j d and ( d equal to 0 otherwse; -α s the number of employed of enterprse ; for each model group { U(p) },.e., for each dvson of economc actvty (2-dgt Nace Rev.1.1), the known populaton totals calculated on the updated frame, are expressed by: X U(p) ( λ (j λ (j α λ (j α λ (j ) = U(p) x ' = U(p) ),..., U(p) 3 ), U(p) 1 ),..., U(p) 3 ). 1 7

8 An example In Table 3A the NACE code of every sngle doman of nterest s lsted n each cell; n the nput data set of the weghtng procedure each of them s replaced by the respectve populaton total, n terms of the auxlary varable Average number of employees n the year t-1 (a smlar specfcaton s done n terms of the auxlary varable Number of enterprses): Table 3A: Example of benchmark specfcaton (known totals) DOMAIN DOM1: Nace-4 dgt (codes) DOM2: Nace-3 dgt * Sze class (codes) DOM3: Nace-2dgt *Nuts Nace 2 dgt Tx1 Tx2 Tx3 Tx4 Txj d Tx15 Tx16 Tx17 Tx18 Tx180 Tx181 Tx * * * *1-9.. north central south * * * * north central south * * * *1-9.. north central south * * * *50+.. north central south * * * *1-9.. north central south * north central south * * * *1-9.. north central south * * * *50+ north central south For each respondng unt (enterprse), the vector of the auxlary varable Average number of employees n the year t-1 can be expressed as n Table 3B, whether or not the unt belongs to the doman represented on the cell: Table 3B: Example of sample data specfcaton (α k = number of persons employed of enterprse k) DOM1:Nace-4 dgt (x 2 -values) DOM2: Nace-3 dgt * Sze class (x 2 --values) DOM3: Nace-2dgt *Nuts Unt dentfer Doman Nace2 q k Drect weght x1 x2 x3 x4 xj d x15 x16 x17 x18 X180 X181 X α 1 22 α α α α 2 1,4 0 α α α α 3 10, α α α α 4 3 α α α α 5 6,4 0 α α α α k.... n p 14 α np α np 0.. α np α np 0 0 8

9 The overall number of benchmarks (constraned estmates) n the optmsaton process s equal to 182. In spte of the consderable number of constrants to be satsfed, the weghtng process ends wth a good convergence between fnal estmates and know populaton totals (see Table 4). Table 4A: Example of output of the weghtng procedure for a doman of nterest (2 dgt NACE): Check on the auxlary varables Doman code =14 Constrant code Known Totals (1) Fnal Estmates of X varables (2) Drect Estmates of the X varables (3) (2)-(1) (3)-(1) Samplng unts Table 4B: Example of output of the weghtng procedure for a doman of nterest (2 dgt NACE): Fnal weghts Unt dentfer Doman Nace2 q k Drect weght Fnal weght 1 10 α , α 2 1, α 3 10, α α 5 6,4 4,2 α k n p 14 α np The estmator effect for the fnal weghts has been calculated on the sample of respondng enterprses wth less than 100 persons employed at dvson of actvty level (NACE Rev dgt), for the followng subset of target varables: 1. Turnover (code ) 9

10 2. Value added at factor cost (code ) 3. Personnel costs (code ) 4. Gross nvestment n tangble goods (code ) 5. Number of employees(code ) 6. Wages and salares (code ). The estmator effect values confrm the hgher effcency ganed by usng the GREG estmator nstead of the drect estmaton for most of the consdered dvsons of actvtes and target varables; the man excepton concerns the varable Gross nvestment n tangble goods, whch s hardly predctable by a model. Moreover, the varables Turnover and Value added at factor cost have an estmator effect hgher than 1 for some dvsons,.e., 73- Research and development and 74- Other busness actvtes, that are charactersed by specalsed actvtes where the hgh amounts nvoced by the enterprses can be attaned by a relatvely small number of sklled employees. In concluson, apart from a small group of economc actvty classes, the varable average number of employees has shown a good correlaton wth the followng target varables of nterest: turnover, producton value, whereas t s not enough correlated wth Gross nvestment n tangble goods. 5. Examples tool specfc 6. Glossary For defntons of terms used n ths module, please refer to the separate Glossary provded as part of the handbook. 7. References Bethel, J. (1989), Sample Allocaton n Multvarate Surveys. Survey Methodology 15, Bredt, F. J. and Opsomer, J. D. (2000), Local Polynomal Regresson Estmators n Survey Samplng. The Annals of Statstcs 28, Cassel, C. M., Särndal, C.-E., and Wretman, J. H. (1976), Some Results on Generalzed Dfference Estmaton and Generalzed Regresson Estmaton for Fnte Populatons. Bometrka 63, Devlle, J. C. and Särndal, C.-E. (1992), Calbraton Estmators n Survey Samplng. Journal of the Amercan Statstcal Assocaton 87, Devlle, J. C., Särndal, C.-E., and Sautory, O. (1993), Generalzed Rakng Procedures n Survey Samplng. Journal of the Amercan Statstcal Assocaton 88, Hedln, D., Falvey, H., Chambers, R., and Kokc, P. (2001), Does the Model Matter for GREG Estmaton? A Busness Survey Example. Journal of Offcal Statstcs 17, Ksh, L. (1995), Methods for Desgn Effects. Journal of Offcal Statstcs 11, Lehtonen, R. and Vejanen, A. (1998), Logstc Generalzed Regresson Estmators. Survey Methodology 24,

11 Montanar, G. E. and Ranall, M. G. (2005), Nonparametrc Model Calbraton Estmaton n Survey Samplng. Journal of the Amercan Statstcal Assocaton 100, Ohlsson, E. (1995), Coordnaton of Samples usng Permanent Random Numbers. In: Busness Survey Methods (eds. Cox, B. G., Bnder, D. A., Chnnapa, B. N., Chrstanson, A., Colledge, M. J., and Kott, P. S.), Wley, New York, Särndal, C.-E., Swensson, B., and Wretman, J. (1992), Model Asssted Survey Samplng. Sprnger Verlag, New York. 11

12 Specfc secton 8. Purpose of the method The method s used for estmaton, when auxlary nformaton s avalable at unt or doman level. It can be used to reduce the varance of the estmates f a strong correlaton between the target varable and the auxlary varables exsts. At the same tme, GREG allows to calbrate to the known populaton totals of the auxlary varables x. Ths means that GREG s a partcular case of a calbraton estmator when the dstance functon s lnear,.e., the fnal weghts that satsfy the calbraton equatons w are chosen to mnmse the followng dstance wth the ntal weghts d: 9. Recommended use of the method ( w d ) 2d s 1. GREG s recommended when a lnear relatonshp between target y and covarate varables x s present, y = β x + ε. 10. Possble dsadvantages of the method 1. GREG can ntroduce a large varaton n weghts that can cause an ncrease n varance, see formula (6) to quantfy the mpact. 2. Possbly correcton weghts g too far from unty or negatve fnal weghts as the correcton factors (see formula (4)) can be n some cases a negatve quantty. 3. Even beng asymptotcally unbased, bas can be ntroduced f sample sze s too small (see also secton 14). 4. GREG can be very senstve to presence of outlers (see Weghtng and Estmaton Outler Treatment ); an llustratve example wth dscusson can be found n Hedln et al. (2001). Ths ssue s very relevant to busness survey where target varables are typcally non-normal and very skewed. 11. Varants of the method 1. Specfc case: Rato regresson. 2. Non-lnear GREG estmators. Expresson (2) can be appled on general models. In fact, the predcton 2. ŷ, that for GREG s based on lnear model can be based on more complex models f the target varable for example s not normal. An example of non-lnear GREG s logstc GREG whch s based on logstc model when the target varable s a bnary varable. The use of more complex models, however, requres more detaled nformaton on the x varable w.r.t. the knowledge of populaton total that s needed by (lnear) GREG. 12. Input data 1. Ds-nput1 = elementary sample data contanng covarates, drect weghts and scale coeffcents q, model group (.e., level for whch the model s specfed). 12

13 2. Ds-nput2 = known totals on the covarates for each model group. 13. Logcal precondtons 1. Mssng values 1. GREG s calculated on sample values on DS-nput1 after mputaton anyway, varance estmaton s affected by the mputaton. 2. Ds-nput2 cannot contan mssng values. 2. Erroneous values Other qualty related precondtons Other types of precondtons 1. If the auxlary varables are categorcal, the known totals for dfferent parttons should not be n conflct. 14. Tunng parameters 1. Choce of the auxlary covarates n the model, a rule of thumb for the choce of categorcal varable s to defne categores so that the sample totals are greater than Choce of the model group level. 3. Choce of q. 15. Recommended use of the ndvdual varants of the method 1. Non-lnear GREG can be used when auxlary varables are avalable for each unt n the populaton and the relatonshp wth the target varable s markedly non-lnear. 16. Output data 1. Ds-output1 = elementary sample data set contanng the new fnal weghts. 17. Propertes of the output data 1. The fnal weghts allows to satsfy the mplct constrants gven by the known totals of the auxlary varables. 18. Unt of nput data sutable for the method Sample unts, also separately by model group. 19. User nteracton - not tool specfc 1. Choce of auxlary covarates. 2. Choce of the group level. 13

14 3. Choce of q. 20. Loggng ndcators 1. The run tme of the applcaton. 2. Iteratons to attan convergence n the estmaton process. 3. Characterstcs of the nput data, for nstance problem sze. 21. Qualty ndcators of the output data 1. The coeffcent of varaton of the fnal weghts n comparson wth the basc weghts. 2. Presence of negatve weghts, n ths case t may be approprate to consder a dfferent underlyng model or to use a calbraton estmator wth a functon that allows to restrct the range of fnal weghts (see the module Weghtng and Estmaton Calbraton ). 3. Varance, coeffcent of varaton of produced estmates. 4. Check of equalty of sample estmates of x and known populaton totals. 22. Actual use of the method 1. Interconnectons wth other modules 23. Themes that refer explctly to ths module 1. Weghtng and Estmaton Man Module 2. Qualty Aspects Qualty of Statstcs 24. Related methods descrbed n other modules 1. Weghtng and Estmaton Calbraton 2. Weghtng and Estmaton Outler Treatment 25. Mathematcal technques used by the method descrbed n ths module 1. Matrx algebra 26. GSBPM phases where the method descrbed n ths module s used Calculate weghts Calculate aggregates 27. Tools that mplement the method descrbed n ths module 1. CALMAR (Devlle, Särndal and Sautory 1993) 2. CLAN (Statstcs Sweden) 3. BASCULA (The Netherlands) 14

15 4. GES (StatCan) 5. GENESEES (ISTAT) 6. Survey, an R package downloadable from the CRAN 7. Samplng, an R package downloadable from the CRAN 8. REgenesees (ISTAT), an R package downloadable from the JonUP: Process step performed by the method Estmaton 15

16 Admnstratve secton 29. Module code Weghtng and Estmaton-M-Generalsed Regresson Estmator 30. Verson hstory Verson Date Descrpton of changes Author Insttute frst verson Loredana D Consglo, Clauda De Vts, Crstna Cascano second verson Loredana D Consglo, Clauda De Vts, Crstna Cascano second verson wth correctons second verson after EB revew prelmnary release fnal verson wthn the Memobust project Loredana D Consglo, Clauda De Vts, Crstna Cascano Loredana D Consglo, Clauda De Vts, Crstna Cascano ISTAT ISTAT ISTAT ISTAT 31. Template verson and prnt date Template verson used 1.0 p 4 d.d Prnt date :32 16