Estimating Unemployment for Small Areas in Navarra, Spain

Transcription

1 Estimating Unemployment for Small Areas in Navarra, Spain Ugarte, M.D., Militino, A.F., an Goicoa, T. Departamento e Estaística e Investigación Operativa, Universia Pública e Navarra Campus e Arrosaía, Pamplona, Spain lola@unavarra.es Abstract In the last few years, European countries have shown a eep interest in applying small area techniques to prouce reliable estimates at the county level. The EURAREA project foune by the European Union between 2000 an 2003, has investigate the performance of various stanar an innovative methos in several European countries. However, the specificity of every European country, the variety of auxiliary information as well as its accessibility, makes the use of the same methoology in the whole of Europe a very ifficult task. Navarra is a small autonomous community locate at the north of Spain. It has km 2 an only inhabitants, irregularly istribute in seven subivisions. Navarra Statistical Institute NSI) has provie ata to the Spanish Statistical Institute INE) as a member of the EURAREA project. Nowaays, NSI is intereste in proviing precise estimates of the unemployment population in every of its subivisions calle comarcas ) in the context of the Spanish Labor Force Survey. In this work we review the current estimation proceure use to provie these estimates. In aition, we iscuss the behavior of several esign-base, moel-assiste, an moel-base estimators using ifferent auxiliary information, an provie several methos for estimating the preiction error. We comment on the results an the viability of its implementation. More specifically we comment on the ifficulties of estimating in very small areas where the samples are both very scarce an unstable. 1 Introuction The Spanish Labour Force Survey SLFS) is a quarterly survey of househols living at private aresses in Spain. Its purpose is to provie information on the Spanish labour market that can then be use to evelop, manage, evaluate an report on labour market policies. It is conucte by the Spanish Statistical Institute INE). The target population inclues all persons age 16 or more living in private househols. Yet there are multiple aims achieve with this survey, the estimation of unemployment population is one of the most relevant. The survey follows a stratifie two-stage cluster esign an, for each province, a separate sample is esigne. The primary sampling units PSUs) are Census Sections areas with a maximum of 500 househols) that are grouping into h strata accoring to the size of municipality h = 1, 5, 6, 7, 8, 9). In Navarra, 91 PSUs are selecte in the first stage with probability proportional to the number of househols. For each PSU selecte, a simple ranom sampling is applie to raw 18 househols, inquirying the overall resients of the househol age 16 or more about 3000 people). This sampling esign prouces self-weighting samples at stratum level an then, every househol has the same probability of being rawn. In this work we check by simulation the benefits of using ifferent kins of auxiliary information in alternative estimators accoring to some measures of precision, later we choose the best the preiction error estimator for the chosen estimator. The scenario is the same as the one use in the SLFS survey, but with samples from the 2001 Census. The rest of the paper is organize as follows. Section 2 presents the propose estimators: esignbase methos, moel-assiste an moel-base methos for estimation purposes. Section 3 presents the ifferent inicators measures of the preiction error. Section 4 illustrates the performance of the estimators in a simulation stuy. In Section 5 we provie ifferent estimators of the mean square error an finally, we show the conclusions. 2 Alternative estimators of the unemployment The variable of interest is the number of unemploye by small areas in Navarra Spain), efine accoring to the International Labour Organization. Navarra is an autonomous community locate at the north 1

2 Figure 1: Navarra autonomous community locate in the north of Spain of Spain, see Figure 1) an it has 7 small areas, calle comarcas ), see Figure 2). The propose estimators are esign, moel-assiste an moel-base estimators, etaile in the following subsections. 2.1 Design-base estimators In the esign-base theory, the variable of interest is a fixe quantity an the probability istribution is inuce by the sampling esign. It is a istribution-free metho mainly focuse on obtaining estimates for omains with large samples. Direct estimator only use observations coming from the the omain of interest, but inirect estimators take information outsie of the omain. The use of auxiliary information borrow strength ) is a common tool to improve the precision of esign-base estimators, an frequently it comes from other omains. In this paper it consists of E) age-sex groups, with 6 categories combination of age 16 24, 25 54, > 55) an sex. S) Stratum, that represents the size of the living city an takes 9 values: 1) capital of the province, an the rest of the cities or villages epening of their population, 2) between an inhabitants, 3) between an inhabitants, 4) between 5000 an inhabitants, 5) between 2000 an 5000 inhabitants an 6) with less than 2000 inhabitants. Let us note that Navarra strata has only 6 categories 1, 5, 6, 7, 8, 9). N) eucational level has two categories 1) for illiterate, primary or seconary school an 2) for technical workers an professionals. P) previous unemployment status has three categories: 1) occupie or inactive, 2) unemploye an 3) others. D) claimant of employ that takes the value 1 if he/she is registere in the employment office of Navarra an 0 otherwise. There are four esign-base estimators consiere in this paper: a irect, a post-stratifie a synthetic an a composite estimator. In the esign-base theory unbiaseness an esign-consistency are esirable properties pursue by the majority of estimators. An estimator Ŷ of Y is esign-unbiase if E[Ŷ ] = Y an it is esign-consistent if it is unbiase an its variance tens to zero as the sample size increases Rao, 2003). The irect estimator oes not make use of any auxiliary information but only of ata in the omain. It is esign-unbiase but its variability is usually big enough to be consiere inappropriate in small-area estimation. The irect estimator of the total unemployment in the -th small area takes the form ŷ irect = ˆȳ N = H nh mhi H nh j=1 w hijy hij I h, i, j) mhi j=1 w N = hiji h, i, j) n k=1 w ky k n k=1 w N, k where h is the stratum h = 1, 5, 6, 7, 8, 9), i the cluster in the h stratum, i = 1, 2,..., n h ), j every unit of cluster i in the h stratum j = 1, 2,..., m hi ), y hij takes the value 1 for the jth unemploye person in stratum h, cluster i, an 0 otherwise, N is the total population in the -th small area = 1,..., D), 2

3 Noroeste Pamplona Pirineos Tierra Estella Navarra Meia Oriental Ribera Tuela Figure 2: 7 small areas of Navarra an n is the corresponing sample size. The inicator variable I h, i, j) takes the value 1 if person j of cluster i an stratum h is in an 0 otherwise. The esign or sampling weights w hij are the inverse of the inclusion probability but usually they are correcte by non-response effects. They also allow us to incorporate the ifferent sampling plans in the estimation process. The irect estimator oes not use any auxiliary variable. However, the rest of esign-base estimators one or more auxiliary variables are use to calibrate the final estimation. The post-stratifie estimator of -th small area incorporates the total of auxiliary variables. It is given by ŷ post = g ˆȳ g N g = g H nh mhi H nh j=1 w hijy hij I g h, i, j) mhi j=1 w N g = hiji g h, i, j) g ng k=1 w ky k ng k=1 w N g 1) k where g = 1,..., G efine the combination of auxiliary variables an varies from 1 to , the inicator variable I g h, i, j) takes the value 1 if person j of cluster i an stratum h is in g an an 0 otherwise, an n g inicates the number of sample persons in the -th region belonging to the gth group. The post-stratifie estimator may be consiere as an assiste-moel estimator because it can be erive from a linear moel where the preictor variable is the inicator variable of belonging to the gth group. The synthetic estimator is use for estimating in subareas uner the assumption that small areas have the same characteristics as the large area. Synthetic estimators are usually biase. To estimate the total unemployment in the -th small area the synthetic estimator is written as ŷ synt = g ˆȳ g N g = g H nh mhi H nh j=1 w hijy hij I g h, i, j) mhi j=1 w N g = hiji g h, i, j) g ng k=1 w ky k ng k=1 w N g, k where the inicator variable I g h, i, j) takes the value 1 if person j of cluster i an stratum h is in g an 0 otherwise. A natural way to balance the potential bias of a synthetic estimator against the instability of a irect estimator is to take a weighte average of the two estimators. The composite estimators are a linear combination of estimators. In this case composite estimators are efine by a linear combination of a irect estimator an an inirect estimator. It takes the form ŷ comp = λ ŷ post + 1 λ )ŷ synt 3

4 Table 1: Mean of the absolute value of the relative bias SRAM) an mean of the square root of the relative mean square error RECMRM) for the post-stratifie an synthetic esign-base estimators evaluate in the 8 groups of auxiliary variables. Design-Base Men Women Total SRAM RECMRM SRAM RECMRM SRAM RECMRM Poststratifie E ED EN EP ES ESD ESN ESP Synthetic E ED EN EP ES ESD ESN ESP Direct where 1 if ˆN αn λ = ˆN αn otherwise verifying 0 λ 1. ˆN = w j is the estimate total size in region, an α = 2/3, 1, 1.5, 2). These values provie the names of composite 1, 2, 3 an 4 respectively. 2.2 Moel-Assiste estimators These estimators use regression moels a mean to obtain consistent estimators from the esign-base point of view Särnal, Swensson an Wretman, 1989). The most well known moel-assiste estimators are the generalize regression estimators GREG). Here they have been obtaine assiste in a linear an a logit moel. Here, the linear moel is y j = x t jβ + ɛ j, j = 1,..., n, 2) where for every small area, y j takes the value 1 if the jth person is unemploye, x j = x i,1, x i,2,..., x i,p ) t is the vector of the p auxiliary variables an ɛ i N0, σ 2 ). The GREG estimator of the total number of unemploye is given by where Ŷ GREG = N X = 1 n w j x t j = N j=1 X ˆβ + 1 ˆN jɛn w j y j x t j ˆβ) N1, N 2,, N ) p = X1, N N N X 2,..., X ) t p 4

5 is the vector of the p auxiliary variable population means, N 1,..., N p are the population of these p auxiliary variables. The β coefficients are estimate with observations coming from the overall areas an then n ˆβ = w j x j x t j j=1 1 n j=1 w j x j y j. 3) Assuming that y j Bn, p j ), it is more appropriate to be assiste in a logit moel given by ) pj logitp j ) = log = x t 1 p jβ. 4) j The GREG estimator of the total number of unemploye in the th area is given by ) N Ŷ GREG e xt ˆβ j = j=1 1 + e ˆβ + N ˆβ w j y j ext j xt j ˆN jɛn 1 + e xt ˆβ. j Usually β is estimate by iteratively weighte least squares metho. 2.3 Moel-base estimators These estimators use regression moels for estimation, preiction an inferential purposes. The moelbase theory is calle preiction theory an consiers y 1,..., y N as realizations of the ranom variables Y 1,..., Y N. Splitting the population in sampling observations s) an non-sampling observations r), the total of Y, calle T, can be expresse as the sum of sample an non-sample observations, T = j s y j + j r y j. The preiction theory preicts the non-observe variable, an therefore it provies the estimator ˆT = j s y j + j r Ŷ j. The common preictors of the non-sampling total are linear combinations of y j an are base on ifferent moels. In this paper, linear moels an logit moels have been consiere. Assuming a linear moel given by y j = x T jβ + ɛ j j = 1,..., n = 1,..., 7 where x j = x j,1, x j,2,..., x j,p ) T is a vector of p covariates. The estimator of the total number of unemploye base on a linear moel is given by Ŷ = X ˆβ 5) where X = X,1, X,2,..., X,p ) T is the total population vector of the p covariates. a) The synthetic estimator, calle Linear Synthetic, estimates β as ˆβ = x j x T j jɛs 1 b) The synthetic estimator base on a weighte linear moel, calle Linear Synthetic W, assumes that ɛ j N0, σ 2 /w i ) an jɛs x j y j. 5

6 Table 2: Mean of the absolute value of the relative bias SRAM) an mean of the square root of the relative mean square error RECMRM) for the composite esign-base estimators evaluate in the 8 groups of auxiliary variables. Design-Base Men Women Total SRAM RECMRM SRAM RECMRM SRAM RECMRM Composite 1 E ED EN EP ES ESD ESN ESP Composite 2 E ED EN EP ES ESD ESN ESP Composite 3 E ED EN EP ES ESD ESN ESP Composite 4 E ED EN EP ES ESD ESN ESP Direct

7 ) 1 ˆβ = w j x j x T j w j x j y j. iɛs c) The estimator base on a linear moel with a fixe effect of the area, calle Linear F, assumes that one of the explanatory variable is a fixe-effect of the area an β is estimate as ˆβ = x j x T j jɛs jɛs 1 ) The estimator base on a weighte linear moel an fixe-effect of the area, calle Linear WF, assumes that ˆβ is estimate with weights w j an ˆβ = jɛs w jx j x T j jɛs x j y j. ) 1 jɛs w jx j y j Assuming a logit moel ) pj log = x T 1 p jβ j where p j is the probability of being unemploye, x j = x j,1, x j,2,..., x j,p ) T is a vector of p covariates. The estimator of the total number of unemploye base on a logit moel is given by Ŷ = N j=1 e xt j ˆβ 1 + e ˆβ xt j e) The synthetic estimator base on a logit moel, calle Logit Synthetic, estimates β by iteratively weighte least squares. f) The synthetic estimator base on a weighte logit moel, calle Logit Synthetic W, assumes that ˆβ is estimate with weights w j. g) The estimator base on a logit moel with fixe-effect of the area, calle Logit F, assumes that one of the explanatory variables is a fixe-effect of the area. h) The estimator base on a weighte logit moel with fixe-effect of the area, calle Logit WF, assumes that ˆβ is estimate with weights w i. 3 Inicators measures of the preiction error We consier the following inicators measures of preiction error: the absolute value of the relative bias SRA ) an its mean SRA) over the D small areas, given by SRA ŷ) = 1 K ŷ k) Y K Y 100, k=1 SRAMŷ) = 1 SRA ŷ), D an the square root of the relative mean square error RECMR ) an its mean RECMR) over the D small areas, given by 1 K ) ) ŷ k) Y RECMR ŷ) = 100, RECMRMŷ) = 1 EMCR ŷ). K D k=1 Y 6) 7

8 Table 3: Mean of the absolute value of the relative bias SRAM) an mean of the square root of the relative mean square error RECMRM) for moel-assiste estimators evaluate in the 8 groups of auxiliary variables. Moel-Assiste Men Women Total SRAM RECMRM SRAM RECMRM SRAM RECMRM Linear GREG E ED EN EP ES ESD ESN ESP Logit GREG E ED EN EP ES ESD ESN ESP Mixe Logit GREG E Direct Simulation Results We have conucte 500 simulations from the 2001 Census base on the same scenario as the one use in the Spanish Labour force Survey) SLFS. The aim of these simulations is to choose the best estimator of the unemploye people by small areas between those with less SRAM an RECMRM. There are a total of 7 small regions in Navarra for which we have evaluate 4 composite estimators, 3 GREG estimators an 3 moel-base estimators. In all of them 8 combinations of auxiliary variables have been use: E) age-sex, ED) age-sex-claimant, EN) age-sex-eucational level, EP) age-sex-previous employment status, ES) age-sex-stratum, ESD), age-sex-stratum-claimant, ESN) age-sex-stratum-eucational level an ESP) age-sex-stratum-previous employment status. There is no optimal estimator with the smallest SRAM an RECMRM simultaneously, but a goo trae off between both criteria has bean reache by composite 4 with α = 2) estimator. Table 1 an 2 show the mean of the absolute value of the relative bias SRAM) an the mean of the square root of the relative mean square error RECMRM) of the esign-base estimators for the 8 combinations of auxiliary variables. Composite 4 ED α = 2) presents a goo balance of bias an mean square error for men an total. Synthetic ES an ESN are also competitive, but Composite 4 EP α = 2) presents the best balance between bias an mean square error for men, women an total, because a SRAM less than 4 is consiere negligible. Table 3 shows the same inicators for assiste-moel estimators. The bias SRAM) of all the estimators, is roughly the same as the irect one, but with the Linear GREG ESP a less RECMRM with regar to the irect one is attaine. Table?? shows again the inicators measures of the precision error for the moel-base estimators in the 8 combinations of auxiliary variables. The bias of all the estimators are higher than the irect one but the Logit Synthetic ES estimator presents the smallest REMRM. Summing up, the composite 4 EP estimator is the simplest an more precise small area estimator for the total number of unemploye in Navarra. Table 6 provies the estimate of the number of unemploye with Composite 4 EP Age-Sex-unemployment population register in the SNE) by small areas. It is evient the goo approximation provie by the estimator in all the small areas, 8

9 Table 4: Mean of the absolute value of the relative bias SRAM) an mean of the square root of the relative mean square error RECMRM) for the linear moel-base estimators evaluate in the 8 groups of auxiliary variables. Moel-Base Men Women Total SRAM RECMRM SRAM RECMRM SRAM RECMRM Linear Synthetic E ED EN EP ES ESD ESN ESP Linear Synthetic W E ED EN EP ES ESD ESN ESP Linear F E ED EN EP ES ESD ESN ESP Linear WF E ED EN EP ES ESD ESN ESP Direct

10 Table 5: Mean of the absolute value of the relative bias SRAM) an mean of the square root of the relative mean square error RECMRM) for the logit moel-base estimators evaluate in the 8 groups of auxiliary variables. Moel-Base Men Women Total SRAM RECMRM SRAM RECMRM SRAM RECMRM Logit Synthetic E ED EN EP ES ESD ESN ESP Logit Synthetic W E ED EN EP ES ESD ESN ESP Logit F E ED EN EP ES ESD ESN ESP Logit WF E ED EN EP ES ESD ESN ESP EB mixe logit E Direct

11 Number of unemploye Men Women Total Composite Composite Composite Census 4 EP Direct Census 4 EP Direct Census 4 EP Direct Pirineo Navarra Meia Oriental Tierra Estella Ribera Noroeste Tuela Pamplona Navarra Table 6: Estimate of the number of unemploye with Composite 4 EP Age-Sex-unemployment population register in the SNE) even in those with a scarce populations such those of Pirineos an Noroeste. 5 Estimators of the mean square error Three methos are presente to calculate the MSE of the composite 4 estimator: two re-sampling methos jakknife an bootstrap) an the variance linearization metho. Jackknife an bootstrap use sub-samples from the original sample. In jackknife metho we take as many sub-samples as clusters we have in the sample, because they are obtaine leaving out the clusters from the original sample. For every sub-sample new weights are efine an with them the composite 4 estimator is calculate. To obtain the variance an bias of these estimators an therefore its MSE we procee as is inicate in subsection 5.2), the jackknife is applie to the overall expression of the MSE. In the bootstrap, the sub-samples are obtaine by ranom sampling, but we nee to etermine how many we nee. Analogously for every sub-sample new weights are efine an then the estimator is calculate. The MSE is estimate as etaile in subsection 5.3). The variance linearize metho consists of applying the Taylor series as etaile in subsection 5.1). 5.1 Variance Linearization Metho The linearization metho or elta metho consist of applying a Taylor series to the function of the total estimators Wooruff, 1971). Let us efine the following inicator variables I k h, i, j) = 1 if person j of cluster i an stratum h is in group k, z hij = y hij I k h, i, j) an v hij = w hij I k h, i, j). Post-stratifie an synthetic estimators of the mean θ k can be written as θk = H n h m hi v hij z hij /v..., where v... = j=1 H n h m hi v hij 7) j=1 If θ k is the post-stratifie estimator, k is the group g in the omain but, when θ is the synthetic estimator k = g. The linearize estimator of the variance is the following 11

12 Var L θ k H ) = Var h θ k ) where Varh θ k ) = n h nh n h 1 Uhi. Ūh..) 2, a U hi. = 1 ) mhi v... j=1 v hij z hij θ k an Ū h.. = 1 nh n h U hi. 8) The variance of the total θ k is calculate as Var L ˆθ k ) = k Var L ˆ θ k )N 2 k 9) The synthetic estimator is biase, then we procee to calculate the bias Ghosh an Särnal, 2001) given by Biasŷ sint ) = N j=1 ɛ j an its estimator Therefore Biasŷ sint ) = N 1 n n j=1 ɛ j where ˆɛ j = y j ȳ g 10) an finally MSE L ŷ sint ) = Var L ŷ sint ) + Bias 2 ŷ sint MSE L ŷ comp ) = λ 2 ŷpost MSE L ) ) + 1 λ ) 2 MSEL ŷsint ). 11) 5.2 Jackknife Estimator Jackknife metho was introuce by Quenouille 1949, 1956) as a metho to reuce the bias, an later Tukey 1958) propose to use it for estimating the variance an confience intervals. To apply jackknife we nee to rop a cluster post-coe section) each time. Let ˆθ hi) k be the estimator ˆθ k obtaine from ropping a cluster i from the h stratum. To calculate ˆθ hi) k we efine the new weights w jhi) = w j if j is not in the h stratum 0 if j is in cluster i of the h stratum if unit j is in the h stratum but not in cluster i n h n h 1 w j 12) The jackknife estimator of the MSE of ˆθ estimator can be obtaine as MSE JK ˆθ k ) = H n h 1 n h n h [ˆθ k hi) ˆθ k h.) ]2 13) where ˆθ h.) k = 1 nh ˆθ n h hi) k. The jackknife estimator of the post-stratifie estimator is given by ] MSE JK ŷ post ) = H [ n h 1 n h n h [ŷ post hi) ŷpost h.) ]2 where ŷ post h.) = 1 nh n h ŷpost hi), an ŷpost hi) is similar to ŷpost 12)). The jackknife estimator of the MSE of a synthetic estimator is given by MSE JK ŷ sint ) = H [ n h 1 n h n h [ŷ sint hi) ŷsint 12 14) but substituting v hij by w jhi) see expression h.) ]2 + ] ) 2 n h 1)ŷh.) sint ŷsint ), 15)

13 where ŷh.) sint = 1 nh n h ŷsint hi), an ŷsint hi) is similar to ŷsint but substituting v hij by w jhi).finally MSE F ŷ comp ) = λ Bootstrap Estimator ŷpost MSE F ) + 1 λ ) 2 MSEF ŷsint ). 16) The re-scale bootstrap estimator in a stratifie ranom sampling has been provie by Rao an Wu 1988). It assumes the following steps 1) Given the h stratum we have a sample of n h clusters. From the sample of the h stratum, we raw a sub-sample of n h 1 clusters by ranom sampling with replacement 2) For every sub-sample r r = 1, 2,..., R) we reefine the new weight n h w hij r) = w j n h 1 m ir) 17) where m i r) is the number of times that cluster i is chosen in the sub-sample an we calculate ˆθ r using the new weight w hij r). 3) Repeat steps 1 an 2 R times. 4) To erive the bootstrap estimatorwe calculate MSE B ˆθ) = 1 R 1 The bootstrap estimator of the post-stratifie estimator is given by MSE B ŷ post ) = 1 R 1 R r=1 R r=1 ˆθ r ˆθ ) 2 18) ŷ post ) r) ) 2 ŷ post 19) where ŷ post ) r) is similar to ŷ post but substituting v hij by w hij r) etaile in expression 17). The bootstrap estimator of the synthetic estimator is given by where ŷ sint ) r) is similar to ŷ sint MSE B ŷ sint ) = 1 R 1 MSE B ŷ comp ) = λ 2 R r=1 ŷ sint ) r) ) 2 ŷ sint, 20) but substituting v hij by w hij r) etaile 17). Finally ŷpost MSE B ) + 1 λ ) 2 MSEB ŷsint ). 21) We obtain 500 simulations with post-stratifie, synthetic an composite 4 estimator using the auxiliary variables: age-sex E) an unemploye accoring to the SNE P). We consier R = 200, 500, 1000 an From small values of R we fin ifferent performance of the estimator, but from R = 1000 an higher, the performance of the estimators are similar. In figures 3 an 4 we see the all the coefficients of variations obtaine for men an women respectively, where RMSEˆθ) ĈV ˆθ) = ˆθ We also provie the real coefficient of variation obtaine from the Census ata. All the methos propose here ten to overestimate the MSE, particularly when the sample size is small, but in this case the best performance is attaine by the jackknife estimator. Acknowlegments: We thank to the Navarra Statistical Institute for proviing the ata. Research supporte by the Ministry of Science an Eucation project MTM ) an OTRI project ). 13

14 CV Men Sim Jakknife Bootstrap Linearization Small Areas Figure 3: Coefficient of Variation of Composite 4 EP in Men CV Women Sim Jakknife Bootstrap Linearization Small Areas Figure 4: Coefficient of Variation of Composite 4 EP in Women 14

15 References Ghosh, M. an Rao, J.N.K., 1994), Statistical Science, 9, González, M.E., 1973). Use an Evaluation of Synthetic Estimates. Proceeings of the Social Statistics Section American Statistical Association. Washington, D.C. Militino A. F.,Ugarte, M. D., an Goicoa,T., 2007), A BLUP Synthetic Versus an EBLUP Estimator: An Empirical Stuy of a Small Area Estimation Problem. Journal of Applie Statistics, 34, Rao, J.N.K. 2003). Small Area Estimation. Wiley Series in Survey Methoology. Särnal, C. E., Swensson B. an Wretman, J. H., 1992). Springer. Moel assiste survey sampling. 15