Estimation of standard error of the parameter of change using simulations

Transcription

1 Esimaion of sandard error of he parameer of change using simulaions Djordje PETKOIC Saisical Offi ce of he Republic of Serbia ABSTRACT The main objecive of his paper is o presen he procedure for esimaing sandard error of parameer of change (index) of urnover in R sofware (R core eam 4) when samples are coordinaed. The problem of esimaing sandard error is deal wih in he saisical lieraure by various ypes of approximaions. In my paper I sar from he mehod presened a he Consulaion on Survey Mehodology beween Saisics Sweden and Saisical Offi ce of he Republic of Serbia (SERSTAT 3:) mae simulaions and calculae esimae of he correlaion and rue value of sandard error of change beween urnovers from wo years. I use wo consecuive sampling frames of quarerly Srucural Business Survey (SBS). These frames are updaed wih urnover from corresponding balance shees. Imporan assumpion is ha annual urnover is highly correlaed wih quarerly urnover and ha compued correlaion can be referred o when comparing mehods of esimaion of correlaion on he sample daa. Keywords: coordinaed samples wih permanen random numbers sandard error of index correlaion of oals simulaions R quarerly SBS. ITRODUCTIO The purpose of his paper is o presen a mehod based on simulaions for esimaing he sandard error of a parameer of change (index) when samples are coordinaed by permanen random numbers. The problem of esimaing sandard error of an index when samples are coordinaed is an open heoreical and pracical problem. Coordinaed samples are no independen bu overlapping so his feaure has o be aen ino accoun in he esimaion procedure. Wor on esimaion of sandard errors of indices of coordinaed samples in he Saisical Office of he Republic of Serbia (SORS) is based on simulaions conduced by A. Lindblom and S. Berg wihin Aciviy on Survey Mehodology of he Projec Parnership in Saisics funded by SIDA (Swedish Inernaional Developmen Cooperaion Agency). Differen mehods were compared and evaluaed. Some of hese mehods are based on he esimae of he correlaion on he common par of he samples which overesimaes he rue value of correlaion so ha he variance is underesimaed. Hence we used some correcion facors. This paper presens a way in which he simulaions can be carried ou in R programming language in order o calculae values of correlaion beween he 9 Romanian Saisical Review nr. / 5

2 esimaes of oals and rue variance of he index. For his purpose I use daa from quarerly SBS. In he firs par of his paper I briefly presen he frame sample design and esimaion procedure of he quarerly SBS. Then I explain he mehod for esimaion and give deails and descripions of some of he codes I use for simulaions. FRAME SAMPLE DESIG AD ESTIMATIO PROCEDURE OF SBS Quarerly SBS is a survey conduced wih he aim of examining quarerly dynamics of enerprises financial daa and changes in he economic srucure of he non-financial business secor. The populaion for his survey includes all enerprises wih he main aciviy in he manufacure and sale of goods and services for he mare. (Zamalar e al 4). Sample is sraified simple random. I is seleced from he sampling frame by a sequenial scheme using permanen random numbers. Frames unis are randomly assigned o 5 roaion groups. Roaion is achieved by an annual shif of permanen random numbers of one roaion group for.. Based on he survey daa indices are compued boh annual index (curren quarer o he corresponding quarer of he previous year) and he chain index (curren o he previous quarer). Some of he oupus of he survey are annual indices by domains defined by secions of ACE Rev.. The deails are in he Table. Table Domain umber Domains Toal Agriculure Huning Foresry and Fishing 3 Indusry and consrucion 4 Services 5 Wholesale and reail rade 6 Transpor and sorage 7 Accommodaion and food service aciviies 8 Informaion and communicaion 9 Oher services For hese simulaions I use wo daa frames one from and he oher one from 3. These frames were updaed wih auxiliary informaion on urnover from annual financial repor of he corresponding years. Afer updaing he frames urnover values were missing for some unis. I did impuaions on he level of sraum by giving value of an average of urnover in ha sraum plus a disurbance erm so ha frames used in simulaion had no missing values on urnover. In he simulaions I generae samples ha correspond o he real siuaion in SBS and compue he index of he esimae of urnover for 3 o he previous year. Based on a large number of simulaions ( housands) I calculae he correlaion beween oals and an approximaion of he rue variance of he index of urnover.. Sample coordinaion of saisical business surveys has been inroduced in SORS in 3 (Trpinac Melovsi O. e al (4) and i is based on he Swedish sysem SAMU (3).. The caegorizaion of economic aciviy in his able is based on ACE Rev. and he source is (Zamlar 4) Romanian Saisical Review nr. / 5 9

3 Romanian Saisical Review nr. / 5 9 Esimaion of he Horviz-Thompson oals is produced using survey pacage. I did esimaion on he domains level. Formulas for esimaion of he variance of indices are presened in he nex par. Because he annual daa on urnover from he final financial accouns is in he correlaion wih he quarerly survey daa compued correlaion can o be used in he esimaion of he variance of index in real life siuaions when only survey sample daa is available (see formula (7)). METHOD One of he mehods ha is used in SORS for esimaing he variance of index is based on he Taylor linearizaion. As shown in erma (993) variance of he raio r where is he level esimae for year and is he level esimae for year can be expressed as () () () () Cov r r r + () Using he relaionship beween correlaion and covariance C () we can rewrie he formula: + (3) We can see from formula (3) ha he esimaion of variance is reduced o he calculaion of variance of oals and correlaion beween oals in wo differen periods. In real life hese values are esimaed bu by large number of simulaions we can compue he rue values. Formulas ha are used for calculaion of variance and covariance from simulaions are: (4) (5) (6) These formulas are he ones ha I used in R simulaions. Even hough his procedure can be enirely applied in real life siuaions here are pars of i ha

4 can be used. Mos imporanly correlaion calculaed wih simulaions according o he formulas () (4) - (6) can be re-used in formula sample. (7) and are esimaes based on a specific real life Where COMMETS O THE R SIMULATIOS In he codes below I presen he selecion of he sequenial random samples based on permanen random numbers from he uniform disribuion. I pu he names of he variables in he braces. Firsly I do impuaions for boh of he frames f and f. I use ddply() funcion o calculae he mean (mean) and sandard deviaion (sd) in each sraum (sra) and pu he resuls in daa se. Afer i I merge frame f and frame in order o impue hose values of urnover (urnover) ha are A. I impued mean sd * unimp unimp : () generaed wih rnorm. I did impuaions 4 for he second frame in he same way. > <- ddply(f~srasummarisemeanmean(urnoverna. rmtrue)sdsd(urnoverna.rmtrue)) > f <- merge(fby sra ) > f$unimp <- rnorm(lengh(f$id) mean sd )/4 > f$urnover[is.na(f$urnover)] -> f$mean[is. na(f$urnover)]+f$sd[is.na(f$urnover)]*f$unimp[is. na(f$urnover)] The firs frame f is filled wih random numbers from uniform () disribuion. Each uni wih is unique id ges random number (prn). Afer i he frame is ordered in ascending order according o hese permanen random numbers inside each sraa using wih() funcion. In each sraa firs pnh unis is sampled (pnh is sample size seleced from each sraum h of size h). Samp is he seleced sample. The code from R is: > f$prn <- runif(lengh(f$id)) > f <- f[order(f$sraf$prn)] > f$br <- wih(f ave(f$sraf$sra FU seq_along)) > samp <- f[as.numeric(f$br)<f$pnh] Afer he firs sample is seleced he frames from boh years are merged (upu) wih merge() and hose unis from he second frame ha are in he firs as Romanian Saisical Review nr. / 5 93

5 well ge he same permanen random number hey had in he firs frame. Afer ha he roaion is performed on hose unis from inersecion ha are in he firs roaion group. Those unis ha don have PR afer merging ge a new one and he selecion of he second sample is done in he same way as he selecion for he firs sample. Funcion sapply() is used o idenify hose unis from he second frame ha are no in he inersecion. This is code for roaions: > upu$cif <- as.numeric(subsring(as.characer(*upu$prn) 67)) %% > upu$rgr <- upu$cif %% 5 > upu$prn[upu$prn+. < && upu$rgr] <- upu$prn+. > upu$prn[upu$prn+. < && upu$rgr] <- upu$prn+.- This is he selecion of he second sample based on coordinaed samples: > upu <- merge(ffby id alltrue) > upu.prnna <- upu$id[is.na(upu$prn)] > index <- sapply(upu.prnna funcion(x) which(upu$id x)) > prn <- runif(lengh(upu$prn)) > upu$prn[is.na(upu$prn)] <- prn[index] > f_pom <- upu[!is.na(upu$ind) ] > f <- f_pom[order(f_pom$sra.yf_pom$prn)] > f$br <- wih(f ave(f$sra.y f$sra.y FU seq_along)) > samp <- as.daa.frame(f[as.numeric(f$br)<f$pnh ]) Afer I selec samples I mae nine domains based on Table. I use he funcion subse() for defining 9 subses from he firs sample. ariables saopred and saopred define domains. This is he code for he firs sample: > ds <- samp > ds <- subse(sampsaopred) > ds3 <- subse(sampsaopred3) > ds4 <- subse(sampsaopred5 saopred6 saopred7 saopred8 saopred9) > ds5 <- subse(sampsaopred5) > ds6 <- subse(sampsaopred6) > ds7 <- subse(sampsaopred7) > ds8 <- subse(sampsaopred8) > ds9 <- subse(sampsaopred9) The sub seing is done in he same way for he second sample samp. I ge esimaes of he oals using survey pacage (Lumley 4) (svyoal() funcion) afer I define he design wih svydesign().here I give an example for he definiion of he survey design for he firs domain in he firs sample and esimaion of he oals of urnover. > ddom <- svydesign(id~ sraa~sradaadsfpc~nh) > o[i] <- svyoal(~ds$urnoverddom)[] 94 Romanian Saisical Review nr. / 5

6 I sore all he oals from all simulaions in one daa frame (s). I repea he procedure of selecing he samples and esimaing he oals for imes. In his way I mimic he real life siuaion imes in order o ge esimaes. In he end I calculae he variances and correlaions based on formulas () (4) (6) and ge he resuls using he final formula (3) for he variance of indices. > s$var <- (s$o/s$o)^*(s$v/s$o^+s$v/ s$o^-*s$corr*sqr(s$v*s$v)/(s$o*s$o)) o and o are he esimaes of he oals from he firs and second sample respecively in he firs domain while v and v are esimaes of variance. Corr is he esimae of correlaion beween oals in wo ime periods. COCLUSIO In his paper I presened a way of calculaing approximaion of he rue correlaion and variance beween oals in wo consecuive ime periods when samples are coordinaed by permanen random numbers. I used simulaions for his purpose. I implemened he mehod for esimaion in R using survey pacage. Based on simulaions I compued he correlaion beween oals and he rue variance of he index of urnover. Aside from he benefi of having programs for esimaing he variance of index his projec produced values for correlaion ha can be used for furher esimaion when only survey daa is available for quarerly SBS. Compuaion of he rue values is jus par of he aciviy concerning he esimaion of he variance of index in SORS. arious oher mehods are invesigaed by simulaions and compared beween hemselves. The values produced in he simulaions ha are presened in his paper are benchmars for oher mehods. One of he fuure projecs would be o implemen some of he mehods for esimaion of variance of change parameer in R as well. REFERECES. Melovsi Trpinac O. M.inić M.Panović (4) Sampling Coordinaion of Saisical Business Surveys Woring paper of he Saisical Offi ce of he Republic of Serbia o. 89 Year L (4).. Mission repors of he Saisics Sweden Inernaional Consuling Offi ce: SERSTAT :9 SERSTAT 3: 3. Ohlsson E. SAMU The Sysem for Co-ordinaion of Samples from he Business Regiser a Saisics Sweden. R&D Repor Saisics Sweden 8 (99). 4. R: A language and environmen for saisical compuing. R foundaion for Saisical Compuing ienna Ausira. URL hp:// 5. SAMU The sysem for co-ordinaion of frame populaions and samples from he Business Regiser in Sweden Deparmen of Economic Saisics Saisics Sweden (3). 6. T. Lumley (4) survey: analysis of complex survey samples. R pacage version erma. Sampling errors in household surveys U (993). 8. Zamalar G. O. Melovsi Trpinac M. Quarerly business aciviies of enerprises in he Republic of Serbia Woring paper of Saisical Offi ce of Republic of Serbia o. 88 Year L (4). Romanian Saisical Review nr. / 5 95