State-Space Modeling with Correlated Measurements with Application to Small Area Estimation Under Benchmark Constraints

Transcription

1 ae-pace Modeling wih Correlaed Measuremens wih Applicaion o mall Area Esimaion Under Benchmark Consrains Danny Pfeffermann Hebrew Universiy, Jerusalem, Israel and Universiy of ouhampon, U.K and Richard Tiller Bureau of Labor aisics, U..A.

2 . INTRODUCTION The Bureau of Labor aisics (BL) in he U..A uses sae-space models for he producion of all he monhly employmen and unemploymen esimaes for he 50 saes and he Disric of Columbia. The models are fied o he direc sample esimaes obained from he Curren Populaion urvey (CP). The use of models is necessary because he sample sizes available for he saes are oo small o warran accurae direc esimaes, which is known in he sampling lieraure as a small area esimaion problem. The coefficiens of variaion (CV) of he direc esimaes vary from abou 8% in he large saes o abou 6% in he small saes. For a recen review of small area esimaion mehods see Pfeffermann (00, ecion 6 considers he use of ime series models). The new book by Rao (00) conains a sysemaic reamen of he subjec The sae-space models are fied independenly beween saes and combine a model for he rue populaion values wih a model for he sampling errors. The published esimaes are he differences beween he direc esimaes and he esimaes of he sampling errors as obained under he combined model. A he end of each calendar year, he model dependen esimaes are modified so as o guaranee ha he annual mean esimae equals he corresponding mean sample esimae. This benchmarking procedure has, however, wo major disadvanages: - The annual mean sample esimaes are sill unsable because he monhly sample esimaes are highly correlaed due o he large sample overlaps induced by he sampling design roaion paern underlying he CP - The benchmarking is posmorem, afer ha he monhly esimaes have already been published so ha hey are of limied use, (is main use is for long erm rend esimaion) I should be menioned also in his respec ha unlike in classical benchmarking ha uses exernal (independen) daa sources for he benchmarking process, (Hillmer and Trabelsi, 987 ; Durbin and Quenneville, 997), he procedure described above Benchmarks he monhly esimaes o he mean of he same esimaes. Exernal daa o which he monhly sample esimaes can be benchmarked are no available even for single monhs.

3 In his aricle we sudy a soluion o he benchmarking problem ha addresses he wo disadvanages menioned wih respec o he curren procedure. The proposed soluion is o fi he model joinly o several homogeneous saes (saes wih similar labor force behavior, abou -5 saes in each group, see ecion 6), wih he added consrains wyˆ = s= s s, model s= s s, cps wyˆ, =,, (.) The jusificaion for he consrains in (.) is ha he direc CP esimaors, which are unreliable in single saes, can be rused when averaged over differen saes. Noe in his respec ha by he sampling design underlying he CP, he sampling errors are independen beween saes. The basic idea behind he use of he consrains is ha if all he direc sample esimaes in he same group joinly increase or decrease due o some exernal effecs no accouned for by he model, he benchmarked esimaors will reflec his change much quicker han he model dependen esimaors. This propery is illusraed very srikingly in he empirical resuls presened in his aricle using real daa. Noe also ha by incorporaing he consrains, he benchmarked esimaors for any given ime borrow srengh boh from pas daa and cross-secionally, unlike he model dependen esimaors in presen use ha only borrow srengh from pas daa. An imporan quesion underlying he use of he consrains in (.) is he definiion of he weighs{ w, s =..., =,,...}. This quesion is sill under consideraion bu possible definiions include s w = / ; w = N / N ; w = / Var ( CP) (.) s s s s= s s s where N and Var ( CP) are respecively he oal size of he labor force and he s s variance of he direc sample esimae in ae s a ime. The use of he weighs { w s } is appropriae when he direc esimaes are proporions. The use of he weighs { w or { w } guaranees ha he global benchmarked esimaes for he group of aes are he s same as he corresponding global direc esimaes in every monh. } s

4 Applicaion of he proposed soluion o he sae-space models employed by he BL inroduces a serious compuaional problem. The dimension of he sae vecor in he separae models is of lengh 0 (see nex secion), implying ha he dimension of he sae vecor of he join model fied o a group of say aes would be 60. A possible soluion o his problem invesigaed in he presen aricle is o include he sampling errors as par of he observaion (measuremen) equaion insead of he curren pracice of modeling heir sochasic evolvemen over ime and including hem in he sae vecor. Implemenaion of his idea reduces he dimension of each of he separae sae vecors by half, because he sampling errors make up 5 elemens of he sae vecor. The use of his soluion, however, inroduces a new heoreical problem because as already menioned, he sampling errors are highly correlaed over ime, requiring he developmen of an appropriae filering algorihm for fiing he model. To he bes of our knowledge, filering of sae-space models wih correlaed measuremen errors has no been sudied previously in he lieraure. I should be emphasized ha he use of he consrains (.) invalidaes he use of he classical Kalman filer irrespecive of compuaional efficiency. This is so because he benchmark consrains conain he observaions ha depend on he sampling errors. If he sampling errors and he consrains are lef in he sae (ransiion) equaions, he model consiss of an observaion equaion and sae equaions wih disurbances ha are correlaed concurrenly and over ime. Pfeffermann and Burck (990) consider he incorporaion of consrains of he form (.) in a sae-space model and develop an appropriae filering algorihm bu in heir model here are no sampling errors so ha he measuremen errors are independen cross-secionally and over ime. The presen aricle considers herefore hree main research problems: - Develop a filering algorihm for sae-space models wih correlaed measuremen errors - Incorporae he benchmark consrains defined by (.) and compue he corresponding benchmarked sae esimaes (esimaes of he rue employmen or unemploymen figures in he presen applicaion) - Compue he variances of he benchmarked esimaors. 4

5 Noice wih respec o he hird problem ha he compuaion of he variances is under he model wihou he benchmark consrains. As menioned earlier, he benchmark consrains are imposed o proec agains sudden exernal effecs on he esimaed values bu hey are no par of he model. Indeed, he incorporaion of he consrains removes he bias of he model dependen esimaors in abnormal periods bu inflaes he variance (only mildly, see he empirical resuls). This is differen from he classical problem of fiing regression models under linear consrains where he consrains add new informaion on he esimaed coefficiens. In secion we presen he ae BL models in presen use. ecion describes he filering algorihm for sae-space models wih correlaed measuremen errors and discusses is properies. The filer is general and is no resriced o he benchmark problem considered in he remaining secions. ecion 4 shows how o incorporae he benchmark consrains and compue he variances of he benchmarked esimaors. The applicaion of he proposed procedure is illusraed in ecion 5 using real series of unemploymen esimaes. We conclude in ecion 6 by discussing some ousanding problems ha need o be addressed before he procedure can be implemened for rouine use. We assume hroughou he paper ha he model hyper-parameers are known. In pracice, he hyper-parameers will be esimaed by fiing he models separaely for each ae, see Tiller (99) for he esimaion procedures in presen use. Applicaion of he Boosrap mehod developed by Pfeffermann and Tiller (00) accouns for he use of hyperparameer esimaion in he esimaion of he predicion variances of he sae vecor predicors. 5

6 - THE BL MODEL IN PREENT UE In his secion we consider a single ae and hence we drop he subscrip s from he noaion. The model employed by he BL combines a model for he rue (esimaed) ae values and a model for he sampling errors and is discussed in deail, including hyperparameer esimaion and model diagnosics in Tiller (99). Below we provide a brief y descripion. Le denoe he direc sample esimae a ime and define by he rue Y populaion value such ha e = ( y Y) is he sampling error.. Model assumed for populaion values Y = β X + L + + I, I ~ N(0, σ ) I L L R ηl = + +, ηl ~ N (0, σ L ) ; R = R + ηr, ηr ~ N (0, σ R ) 6 j= = ; j, (.) = cosω + sin ω + ν, ν ~ N(0, σ ) * j, j j, j j, j, j, N * * * * j, = sinω j j, + cos ωj j, + νj,, νj, ~ (0, σ ) ω j = π j/ ; j =...6 The model defined by (.) bu wihou he covariae Basic rucural Model (BM). In his model L X is known in he lieraure as he is a rend level, R is he slope and is he seasonal effec operaing a ime. The disurbances * I, L, R, j, j η η ν ν are independen whie noise series. ee Harvey (989) for a deailed sudy of his kind of models. The covariae X represens he number of persons in he ae receiving unemploymen insurance benefis when modeling he oal unemploymen figures, and represens he raio beween he number of payroll jobs in business esablishmens and he populaion size in he ae when modeling employmen o populaion raios. The coefficien β is modeled as a random walk. Noe ha he rend and seasonal effecs only accoun for he remainder rend and seasonaliy no accouned for by he rend and seasonaliy of he covariae. 6

7 . Model assumed for he sampling errors The model assumed for he sampling error is approximaion o he sum of an MA(5) process and an AR() process. e ~ AR(5), which is used as an The MA(5) process accouns for he sample overlap implied by he CP sampling design. By his design, households seleced o he sample are surveyed for 4 successive monhs, hey are lef ou of he sample for he nex 8 monhs and hen hey are surveyed again for 4 more monhs. This roaion scheme induces sample overlaps of 75%, 50% and 5% for he firs hree monhly ime lags and sample overlaps of.5%, 5%, 7.5%, 50%, 7.5%, 5%,.5% a lags 9 o 5. There is no sample overlap a lags 4-8 and 6 and over. A model accouning for hese auocorrelaions is MA (5) wih zero coefficiens a he lags wih no sample overlap. The AR() process accouns for auocorrelaions no explained by he sample overlap. These auocorrelaions accoun for he fac ha households dropped from he survey are replaced by households from he same census rac. The reduced ARMA presenaion of he sum of he wo processes is ARMA(,7), which is approximaed by an AR(5) model. The separae models holding for he populaion values and he sampling errors are cas ino a single sae-space model for he observaions y (he direc sample esimaes). The resuling sae vecor consiss of he covariae coefficien, he rend level, he slope, seasonal componens accouning for he monh frequency and is five harmonics, he irregular erm and he concurren and 4 lags of he sampling errors, a oal of 0 elemens. The monhly employmen and unemploymen esimaes published by he BL are obained under he model (.) as, Yˆ = ( y ˆ ) = ˆ β X + Lˆ + ˆ + Iˆ (.) e 7

8 . FILTERING OF TATE-PACE MODEL WITH CORRELATED MEAUREMENT ERROR In his secion we assume he following sae-space model = + ; Ee ( ) = 0, Eee ( ') = Σ ; Eee ( τ ') = Σ τ (.a) y Zα e α = Tα + η ; E( η ) = 0, E( ηη ') = Q, E( ηη ') = 0 k > 0 (.b) k I is also assumed ha E( η ') = 0 for all and τ. Clearly, wha disinguishes his model e τ from he classical sae-space model is ha he measuremen errors e are correlaed over ime. Below we propose a filering algorihm o ake accoun of he covariances A ime Le 0 Σ τ. ˆ α = ( Ι KZ) Tˆ α + Ky be he filered (updaed) sae esimaor a ime where ˆ α 0 is a saring esimaor wih covariance marix P0 = E[( ˆ α ˆ 0 α0)( α0 α0)'], assumed for convenience o be independen of he observaions and K = PZF 0 is he Kalman gain wih P = TPT + Q and F = ZP% 0Z +Σ. The marix is he covariance marix of he 0 0 ' P 0 predicion errors ( Tα α ) = ( α α ) ˆ0 ˆ 0 and F is he covariance marix of he innovaions ν = ( y yˆ ˆ 0) = ( y Zα 0). ince y = Zα + e, ˆ α = ( Ι KZ) Tˆ α + KZα + Ke (.) 0 A ime ˆ Le α α = T ˆ define he predicor of α a ime wih covariance marix P = E[( ˆ α ˆ α )( α α )']. An unbiased esimaor ˆ α of α [ E( ˆ α α) = 0] based on ˆ α and y is he Generalized Leas quare (GL) esimaor of he random coefficien α in he regression model ha is, u ( ) α Z e T ˆ α Ι = y + 8 u = Tαˆ α ) (.) (

9 where ( Z ) ' ' T ˆ α = Ι Z V Ι Ι Z V y ˆ α (, ) (, ) u P C V V = ar = e C ' Σ (.4) (.5) and C = Cov[ u, e ] = TK Σ (follows sraighforwardly from (.) and he previous assumpions). Noice ha V is he covariance marix of he errors and e, and no of he predicors Tα and ˆ u y. By Pfeffermann (984), he esimaor ˆ α is he bes linear unbiased predicor (BLUP) of α based on T ˆ α and A Time ˆ α Le E[( ˆ α α )( ˆ α α )'] ' = Ι Z V ( Z ), wih covariance marix y (, ) Ι = P (.6) = T ˆ α define he predicor of α a ime wih covariance marix [( ˆ ˆ )'] TPT Q P. E α α)( α α = + = Denoe ( Ι ) = = (, Z ' V B B, B ) ˆ ˆ Tα α = PB =. ince e y P( B T ˆ α + B y ) y = Zα +, i follows from (.) ha such ha C = CovT [ ˆ α, e] = CovTPB [ TKe + TPB e, e ] = ( TPB TKΣ + TPB Σ ) (.7) An unbiased esimaor ˆ α of α is obained as he GL esimaor of he random coefficien α in he regression model Ι u ( Z ) α e T ˆ α = + y ( u = Tαˆ α ) (.8) ha is, where Ι ( Z ) ' ' T ˆ α = Ι Z V Ι Z V y ˆ α (, ) (, ) (.9) 9

10 u P, C V = Var = e C', Σ (.0) The esimaor ˆ α is he BLUP of α based on T ˆ α and A ime y wih covariance marix ' E[( ˆ α ˆ α)( α α)'] (, ) = Z V Ι Ι P Z = (.) ( ) = T ˆ define he predicor of α a ime (-) wih covariance marix Le ˆ α α ˆ α α ˆ α = + = where P = E[( ˆ α α )( ˆ α α )']. e he E[( )( α)'] TP T' Q P random coefficien regression model ( Z ) T ˆ α u Ι α y = + e ( u = Tαˆ α ) (.) and define u P, V V ar C = = e C ', Σ (.) The compuaion of C C ˆ = ov[ Tα, e] is carried ou as follows: Le, [ B j, B j ] = [ Ι, Z '] V j where conains he firs q columns and B he remaining columns wih q = dim( α j ). B j j Define, A TPB, A% = TPB, j= - ; %. Then, j = j j j j j A = TK C Cov[ T ˆ, e ] A A... A A% A A... A A%... A A% A % (.4) = α = Σ + Σ + + Σ, + Σ, The BLUP of α based on Tαˆ and y and he covariance marix of he predicion errors are obained from (.)-(.4) as, Ι ( Z ) ' ' T ˆ α Z V Z V y ˆ α = (, Ι ) (, Ι ) ' Ι = ( Z ) ; P [( ˆ )( ˆ E α α α α)'] = (, Ι Z ) V (.5) 0

11 The filering algorihm defined by (.5) has he following properies: - A every ime poin, he filer produces he BLUP of α based on he predicor α T ˆ α ˆ = from ime (-) and he new observaion y (follows from Pfeffermann, 984). - Unlike he Kalman filer ha assumes independen measuremen errors, he filer (.5) does no produce he BLUP of α based on all he observaions () y = ( y... y ). Compuaion of he laer requires join modeling of all he observaions (see commen below). - Empirical evidence so far suggess ha he loss in efficiency from using he proposed algorihm insead of he BLUP ha is based on all he observaions is mild. Commen: For arbirary covariances Σ τ beween he measuremen errors, i is impossible o consruc an opimal filering algorihm ha combines he predicor from he previous ime poin wih he new observaion. By an opimal filering algorihm we mean an algorihm ha yields he BLUP of he sae vecor a any given ime based on he observaions. To see his, consider he simples case of observaions y, y, y wih y() common mean µ and variance σ. If he hree observaions are independen, he BLUP of µ based on he firs observaions is y () = ( y + y )/ and he BLUP based on he hree observaions is y () = ( y + y + y )/ = (/) y () + (/) y. The BLUP y () is he Kalman filer predicor for ime. uppose, however, ha Cov( y, y) = Cov( y, y) = σ ρ and Cov( y, y ) = σ ρ σ ρ. The BLUP of µ based on he firs observaions is again y() = ( y + y)/, bu he BLUP of µ c ( ρ) based on he observaions is in his case y() = ay + by + ay where a = and 4ρ + ρ ( ρ + ρ) c b =. Clearly, since a b, he predicor y () canno be wrien as a linear 4ρ + ρ y ρ ρ combinaion of y () and. For example, if c = 0.5, = 0.5 y () = 0.4y + 0.y + 0.4y.

12 4. INCORPORATION OF THE BENCHMARK CONTRAINT 4. Join modeling of concurren sample esimaes and heir weighed mean In his secion we model joinly he direc esimaes in aes and heir weighed mean. We follow for convenience he BL modeling pracice and assume ha he rue populaion values and heir direc sample esimaes are independen beween aes. In ecion 6 we consider exensions of he join model o allow for cross-secional correlaions beween componens of he separae sae vecors operaing in he various aes. uppose ha he separae ae models are wrien as in (.) wih he sampling errors placed in he observaion equaion. Below we add he subscrip s o all he model componens o disinguish beween he various aes. Noe ha he observaions y s (he direc sample esimaes) and he measuremen errors e s (he sampling errors) are scalars and Z is a row vecor (denoed hereafer as z ' ). Le y% = ( y... y, w )' s sy = s define he concurren esimaes in he aes (belonging o he same homogeneous group ) and heir weighed mean (he righ hand side of he benchmark equaions (,)). The corresponding vecor of sampling errors is... * e% = ( e e, wses)'. Le Z =Ι zs '(block diagonal marix wih s= z ' in he s h * block), T = Ι T, s % * Z = '... wz ' wz, α ( ' = α '... α ') and η = ( η '... η ')'. By (.) and he independence of he sae vecors and sampling Z errors beween he aes, he join model holding for y% is, h y% Σ ; ( ) 0, ( ') = Z% % α + e% E e% = E e% τe% =Σ % τ = h τ ' τ ντ (4.a) % α = T% % α + % η ; E( % η) = 0, E( %% ηη ') =Ι Qs, E( %% ηη k ') = 0, k > 0 (4.b) Σ = Diag[ σ... σ ) ; σ = Cov[ e, e ), ν τ τ τ sτ sτ s τ = w [, s sτwsσsτ = Cov w s sτesτ wse = = s= s τ τ τ s τ sσ s τ sτ s= s s h = ( h... h )'; h = w = Cov[ e, w e ] ] Commen: The model (4.) is he same as he separae models defined by (.). There is no new informaion in he observaion equaion by adding he model holding for wsys. s=

13 4. Incorporaing he benchmark consrains Under he model (.) wih he sampling errors in he observaion equaion, he model dependen esimaor for ae s a ime akes he form Yˆ s,model and.). Thus, he benchmark consrains (.) can be wrien as, = z ' ˆ α (see equaions. s s wz ' ˆ α = wy, =,, s s s s s s= s= (4.) where y = Yˆ defines as before he direc sample esimae. By (4.a) s s= s s= s,cps s + w = s s y = zs ' α s e. Hence, a simple way of incorporaing he benchmark consrains is by imposing w y = w z ' α, or equivalenly, by seing s= s s s= s s s s= s s s s= s s, =,, (4.) Var [ w e ] = Cov [ e, w e ] = 0 This is implemened by replacing he covariance marix Σ% in he observaion equaion (4.a) by he marix Σ % * Σ, 0( ) =. Thus, he benchmarked esimaor akes he form, 0 ( )', 0 ' * Ι ' * ˆ α (, )[ ] (, )[ ] T = Ι Z% V Z V α Z Ι % % % % y% (4.4) * % α, where T% % α P * C V Var = = ' * ; Q e C, Σ % P = TP % T % ' + % Noe ha is he rue covariance marix of ( % α T % % ) under he model. imilarly, P α and C = C ov[ T% %, e% ]. α C [ % %, % ] = Cov Tα e is he covariance under he model. ee below for he compuaion of P and C.

14 4. Compuaion of P = Var( % α α ) and C = C ov[ T% %, e% ] α Le * ' * Ι P = (, Ι Z% )[ V ] Z% % % * * * = PB % T% α + PB Z % α + PB e. such ha α T% % % α % * ' * * * (, )[ ] % = P Ι Z% V = P B Tα + P B y y % % By definiion of * P, B and B, * * * * PB + PB Z% = P [ P ] =Ι. Hence, % α = PB % α + PB Z % % α and * * * * ( % α % α) = PB ( T% α % α) + PB e% % (4.5) I follows ha, P = E[( % α % α )( % α % α )'] = P B P B ' P + P B Σ% B ' P * * * * + PB C B ' P + PB C ' B ' P * * * * + (4.6) The compuaion of C = Cov[ T% %, e% ] α is carried ou by use of formula (.4), wih T ~ *, P, B, B ) replaced by T, P ( B, B ) in he definiions of A and j ( j j and defining A ~ ~ = T. * P B, j, j j j A ~ j, j= -, 5. EMPIRICAL ILLUTRATION For he empirical illusraions we fied he BL model defined in ecion bu wihou he covariae X, o he direc (CP) unemploymen esimaors in he 9 Census divisions of he U..A. The observaion period is January, 976 December, 00. The las year is of special ineres since i is affeced by a sar of a recession in March and he bombing of he New York World Trade Cener in epember. These wo evens provide an excellen es for he performance of he proposed benchmarking procedure. The individual Division models, along wih heir esimaed hyper-parameers, are combined ino he join model (4.). The benchmark consrains are as defined in (.) wih w s =, so ha he model dependen esimaors of he Census Divisions 4

15 unemploymen are benchmarked o he oal naional unemploymen. The CV of he CP esimaor of he oal naional unemploymen is %, which is considered o be sufficienly precise. Figure compares he sum of he model dependen predicions over he 9 Divisions wihou he benchmark consrain wih he CP naional unemploymen esimaor. In he firs par of he observaion period he sum of he model predicors are close o he CP esimaor. In 00 here is evidence of sysemaic model underesimaion. This is beer illusraed in Figure, which plos he difference beween he oal of he model predicors and he CP esimaor. As can be seen, saring in March, 00, all he differences are negaive and in some monhs he absolue difference is larger han wice he sandard deviaion of he CP esimaor. Figures - display he model dependen predicors, he benchmarked predicors and he direc CP esimaors from January 000 for each of he 9 Census divisions. Excep for New England, he Benchmarked esimaors are seen o correc he underesimaion of he model dependen esimaors in he year 00. The reason ha his bias correcion does no occur in New England is ha in his division, he model dependen predicors are acually higher han he CP esimaors, which serves as an excellen illusraion for he need o apply he benchmarking in homogeneous groups (see ecion 6). Table shows he means of he monhly raios beween he benchmarked predicors and he model dependen predicors for each of he 9 Census divisions in he year 00. The means are compued separaely for he esimaion of he oal unemploymen figures and L for esimaion of he rend levels ( in equaion.). As can be seen, he means of he raios are all greaer han one bu he larges means are abou 4% indicaing ha he effec of he benchmarking is generally mild. 5

16 6. CONCLUDING REMARK, OUTLINE OF FUTURE REEARCH Benchmarking of small area model dependen esimaors o agree wih he direc sample esimaes in large areas is a common requiremen by saisical agencies producing official saisics. This aricle shows how his requiremen can be implemen wih saespace models. When he direc esimaes are obained from a survey wih correlaed sampling errors like in labor Force surveys, he benchmark consrains canno be incorporaed wihin he framework of he Kalman filer, requiring insead he developmen of a filer wih correlaed measuremen errors. This filer is needed o allow he compuaion of he variances of he benchmarked esimaors under he model. Unlike he Kalman filer, filering wih correlaed measuremen errors does no produce he BLUP predicors based on all he observaions bu empirical evidence obained so far indicaes ha he loss of efficiency by use of he proposed filering algorihm is mild. Furher empirical invesigaion is needed o ascerain his propery. An imporan condiion for he success of he benchmarking procedure is ha he small areas (aes in he presen applicaion) are homogeneous wih respec of he behavior of he rue (esimaed) quaniies of ineres (he rue employmen or unemploymen figures in he presen applicaion). The need for he fulfillmen of his condiion is illuminaed in he empirical illusraions where he benchmarking of he Census Division esimaes o he direc (CP) naional esimae increased he model dependen predicors in New England insead of decreasing hem. This happened because unlike in all he oher divisions, he model dependen predicors in New England were already higher han he corresponding CP esimaors. ince he benchmarking of he employmen and unemploymen esimaes In he U..A. is currenly planned for he ae esimaes, our nex major ask is o classify he 50 aes and he Disric of Columbia ino homogeneous groups. everal facors need o be aken ino accoun when defining he groups. Geographic proximiy o accoun, for example, for weaher condiions, breakdown of he Labor Force ino he major caegories of employmen (percenages employed in manufacuring, services, farming ec.) and he size of he aes (o avoid he possibiliy ha large aes will dominae he benchmarking in small aes) are obvious candidae facors ha should 6

17 be considered. Obviously, he behavior of pas esimaes and heir componens like he rend and seasonal effecs should be invesigaed for a successful classificaion of he aes. Accouning for all he facors menioned above for he grouping process migh resul in very small groups bu i should be emphasized ha he groups mus be sufficienly large o jusify he benchmarking o he corresponding global CP esimae in he group. Thus, he sensiiviy of he benchmarking process o he definiion of he groups needs o be invesigaed. Anoher area for fuure research is he developmen of a smoohing algorihm ha accouns for correlaed measuremen errors. Clearly, as new daa accumulae i is desirable o modify pas predicors, which is paricularly imporan for rend esimaion. Las, he presen BL models assume independence beween he sae vecors operaing in separae aes. I can be surmised ha changes in he rend or seasonal effecs are correlaed beween homogeneous aes and accouning for hese correlaions migh improve furher he efficiency of he predicors. In fac, he exisence of such correlaions underlies implicily he use of he proposed benchmarking procedure. Accouning explicily for he exising correlaions is simple wihin he join model defined by (4.) and may reduce quie subsanially (bu no eliminae) he effec of he benchmarking on he model dependen predicors. REFERENCE Durbin, J. and Quenneville, B. (997). Benchmarking by ae pace Models. Inernaional aisical Review, 65, -48. Harvey, A.C. (989). Forecasing rucural Time eries wih he Kalman Filer. Cambridge: Cambridge Universiy Press. Hillmer,.C., and Trabelsi, A. (987). Benchmarking of Economic Time eries, Journal of he American aisical Associaion, 8,

18 Pfeffermann, D. (984). On Exensions of he Gauss-Markov Theorem o he case of sochasic regression coefficiens. Journal of he Royal aisical ociey, eries B, 46, Pfeffermann, D. (00). mall area esimaion- new developmens and direcions. Inernaional aisical Review, 70, 5-4. Pfeffermann, D., and Burck, L. (990). Robus small area esimaion combining ime series and cross-secional daa. urvey Mehodology, 6, 7-7. Pfeffermann, D., and Tiller, R. B. (00). Boosrap approximaion o predicion ME for sae-space models wih esimaed parameers. Working Paper, Deparmen of aisics, Hebrew Universiy, Jerusalem, Israel. Rao, J. N. K. (00), mall Area Esimaion. New York: Wiley. Tiller, R. B. (99). Time series modeling of sample survey daa from he U.. Curren Populaion urvey. Journal of Official aisics, 8, Means of Raios Beween Benchmarked and Model Dependen Predicors of Toal Unemploymen and Trend in Census Divisions, 00 Division Predicion of Predicion Unemploymen of Trend New England Middle Alanic.0.0 Eas Norh Cenral Wes Norh Cenral ouh Alanic Eas ouh Cenral Wes ouh Cenral Mounain Pacific

19 Figure. Monhly Toal Unemploymen Naional CP and um of Division Model Esimaes (00,000) Figure. Monhly oal Unemploymen Difference beween um of Division Model Esimaes and CP D(CP).5 (00,000) CP DivModels DivModels Jan- 98 ep- 98 May- 99 Jan- 00 ep- 00 May Jan- 98 ep- 98 May- 99 Jan- 00 ep- 00 May- 0 Figure. CP, Model and Benchmark Esimaes of Monhly Toal Unemploymen New England (0,000) CP BMK Model Jan-00 ep-00 May-0 Figure 4. CP, Model and Benchmark Esimaes of Monhly Toal Unemploymen Middle Alanic (00,000) CP BMK Model Jan-00 ep-00 May-0 Figure 5. CP, Model and Benchmark Esimaes of Monhly Toal Unemploymen Eas Norh Cenral (00,000) CP BMK Model Jan-00 ep-00 May-0 Figure 6. CP, Model and Benchmark Esimaes of Monhly Toal Unemploymen Wes Norh Cenral (0,000) CP BMK Model Jan-00 ep-00 May-0 9

20 Figure 7. CP, Model and Benchmark Esimaes of Monhly Toal Unemploymen ouh Alanic (00,000) CP BMK Model Jan-00 ep-00 May-0 Figure 8. CP, Model and Benchmark Esimaes of Monhly Toal Unemploymen Eas ouh Cenral (0,000) CP BMK Model Jan-00 ep-00 May-0 Figure 9. CP, Model and Benchmark Esimaes of Monhly Toal Unemploymen Wes ouh Cenral (00,000) CP BMK Model Jan-00 ep-00 May-0 Figure 0. CP, Model and Benchmark Esimaes of Monhly Toal Unemploymen Mounain (0,000) CP BMK Model Jan-00 ep-00 May-0 Figure. CP, Model and Benchmark Esimaes of Monhly Toal Unemploymen Pacific (00,000) CP BMK Model Jan-00 ep-00 May-0 0