Two-stage Benchmarking of Time-Series Models for. Small Area Estimation. Danny Pfeffermann, Richard Tiller

Transcription

1 Two-sage Benchmarking of Time-Series Moels for Small Area Esimaion anny Pfeffermann, Souhampon Universiy, UK & Hebrew universiy, Israel Richar Tiller Bureau of Labor Saisics, U.S.A. Small Area Conference, Trier, 2011

2 Wha is benchmarking? Y - arge characerisic in area a ime, y- irec survey esimae, Y ˆ moel - esimae obaine uner a moel. = 1,2,..., Areas, = 1,2,... Time Benchmarking: moify moel base esimaes o saisfy: ˆ moel by 1 = B = ; 12 =,,... ( B known, e.g., B = b y ). = 1 b fixe coefficiens (relaive size, scale facors, ). Coniion: B sufficienly close o rue value by = 1. Y ˆ moel no necessarily a linear esimaor. 2

3 Problem consiere in presen presenaion evelop a wo-sage benchmarking proceure for hierarchical ime series moels fie o survey esimaes. Firs sage: benchmark concurren moel-base esimaors a higher level of hierarchy o reliable aggregae of corresponing survey esimaes. Secon sage: benchmark concurren moel-base esimaes a lower level of hierarchy o firs sage benchmarke esimae of higher level o which hey belong. 3

4 Example: Labour Force esimaes in he U.S.A 4

5 Why benchmark? 1- Time series moels reflec hisorical behavior of he series. Slow in aaping o changes benchmarking provies some proecion agains abrup changes affecing he areas in a given hierarchy. 2- The publishe benchmarke esimaes a each level sum up o he publishe esimae a he higher level. Require by official saisical bureaus. 3- Anoher way of borrowing srengh across areas. 5

6 Why no benchmark secon level areas in one sep? 1- May no be feasible in a real ime proucion sysem: For U.S.A.-CPS our propose proceure requires join moeling of all he areas ha nee o be benchmarke, sae-space moel of orer elay in processing aa for one secon level area coul hol up all he area esimaes. 3- When 1 s level hierarchy compose of homogeneous 2 n level areas, benchmarking more effecively ailore o 1 s level characerisics. 6

7 Apply cross-secional benchmarking a every ime? Pro-raa (raio) benchmarking, Limiaions: Yˆ = Yˆ byˆ = moel moel, R Β / ) k 1 k k ; B = b y. = 1 1- Ajuss all he small area moel-base esimaes exacly he same way, irrespecive of heir precision, 2- Benchmarke esimaes no consisen: if sample size in area increases bu sample sizes in oher areas unchange, Yˆ, R oes no converge o rue populaion value Y. 7

8 Limiaions of inepenen pro-raa benchmarking (con.) 3- oes no len iself o simple variance esimaion. 4- If applie inepenenly a every ime poin ignores inheren ime series relaionships beween he benchmarks B b y = may a exra roughness o benchmarke = 1 esimaes an he corresponing esimae ren. Possibly similar problem wih all cross-secional benchmarking proceures when applie o a ime series. 8

9 Aiive cross-secional benchmarking Y ˆ Y ˆ + ( b y by ˆ ); moel moel, = 1 k k A a k= k= 1 k k ba = 1 = 1. Coefficiens { a } measure precision (nex slie); isribue ifference beween benchmark an aggregae of moelbase esimaes beween he areas. If a 0 n moel ˆ Y Yˆ A Y consisen., Ba news? ˆ Plim( Y, A y) = 0 Area accurae esimae n no conribuing o benchmarking in oher areas. Easy o esimae variance of Y, A. 9 ˆ

10 Examples of aiive cross-secional benchmarking Wang e al. (2008) minimize ( ˆ ) φ EY 1 Y =, A 2 uner F-H s.. by = byˆ, = 1 = 1 A. Sol: = ϕ / ϕ k = 1 k k a b b. { φ } represen precision of irec or moel-base esimaors. ˆ moel 1 φ = [ Var( Y )] Baese e al φ = b Yˆ Yˆ = Pfeffermann & Barnar moel moel 1 [cov(, )] k 1 k 1 φ = [ Var( y )] Isaki e al In pracice, moel parameers replace by esimaes. 10

11 Examples of aiive cross-secional benchmark. (con.) aa e al. (2011) minimize 2 ( ˆ E[ Y 1 Y, ) ] = A aa φ an obain soluion of Wang e al., wih ˆ moel E( Y ) Y = aa. Soluion general - no resrice o paricular moel. You an Rao (2002) propose self benchmarke esimaors for uni-level moel by moifying he esimaor of β. Approach applie by Wang e al. (2008) o area-lave moel. Ugare e al. (2009) benchmark he BLUP uner uni-level moel o synheic esimaor for all areas uner regression moel wih heerogeneous variances. 11

12 Firs-sage ime series benchmarking Pfeffermann & Tiller (2006) consier he following moel for unemploymen census ivision series obaine from CPS. Le Y = ( Y1,..., Y ) = rue ivision oals, 1 irec esimaes, 1 y = ( y,, y ) = e = ( e,, e ) = sampling errors. 2 2 ( ) σ σ y = Y + e; E( e ) = 0, E e e = Σ = iag [,, ] τ τ 1, τ,, τ,. ivision sampling errors inepenen beween ivisions bu highly auo-correlae wihin a ivision an heerosceasic. (4 in, 8 ou, 4 in roaion paern) 12

13 Time series moel for ivision Toals Y assume o evolve inepenenly beween ivisions accoring o basic srucural moel (BSM, Harvey 1989). Moel accouns for sochasic ren, sochasically varying seasonal effecs an ranom irregular erms. Moel wrien: Y = zα; α = Tα, 1+ η. (sae-space) Errors η muually inepenen whie noise, E( η η ) = Q. ARIMA, regression wih ranom coefficiens an uni & area level moels can all be expresse in sae-space form. 13

14 Combining he separae ivision moels y = Y + e = Zα + e (measuremen eq.) ; y = ( y,..., y ) 1, α = T α + η (sae eq.) ; α = ( α,..., α ) 1, 1 Z = Ι z, T =Ι T ; - block iagonal ( ) ( ) ( ) E η = 0, E ηη = Q =Ι Q, E ηη = 0, τ. τ Benchmark consrains: MOEL b 1 y = b 1 z α = b 1 Y = = =, = 1,2,... Bu in ruh, b 1 y = b 1 z α = = + =1 be. 14

15 Aing benchmark equaions o moel A b y = bz α + be o measuremen eq. = 1 = 1 = 1 ( ) y = Zα + e ; y = y, b 1 y =, Z Z =,, e = e b e bz = 1 1 bz, ( ), 1 Sae equaions α = T α 1 + ηunchange.. 15

16 Se up ranom coefficiens regression moel T α I 1 u 1 = α +, u 1= T α 1 α; y Z e P C Var u 1 1 = = e C Σ V. Σ Σ h ( ) = E ee = h v ; h = Cov( e, b e ) ; = 1 v = Var( b e ). = 1 1 ( 1 ) τ = 1 C E u e τ = = Σ covariance marices of sampling errors. τ linear combinaion of 16

17 Imposing benchmark consrains Impose, = 1 = 1 b y = b z α be = 0 when =1 esimaing he sae vecor uner RCR moel. efine, e, = ( e,0) 0, Σ, = E( e 0, 0e, 0), C, = E( u 1 e 0, 0), V, 0 P 1 C, 0 = C 0 Σ,, Ι 1 T α 1 α = (, Ι Z ) V, (, Ι Z ) V 0, Z 0 sanar GLS. y Benchmarke preicor for ivision : Yˆ = z α. 17

18 Seing Variance of benchmarke esimaor = 1 = 1 b y = b z α be = 0 only for compuing benchmarke preicor bu no when compuing Var( α - α ). =1 ( ˆ Var Y Y ) accouns for variances an auocovariances of ivision sampling errors, variances an auocovariances of benchmark errors, = 1 be = b y = 1 = 1 b z α, an heir covariances wih ivision sampling errors, an variances of moel componens. 18

19 Alernaive expression for benchmarke preicor enoe by α ˆ,u he sae preicor wihou imposing he consrain be = 0 a ime (bu imposing consrains =1 in previous ime poins). efine, Λ [ ( ˆ f = Var bz α, u α)] ; = 1 δ = Cov[( ˆ α α ), b z ( ˆ α α )]. f, u = 1, u The benchmarke preicor of oal in ivision is, Y ˆ z ˆ z ( b y b z ˆ ) 1 = α, u + δfλ f 1 1 α = =, u = 1 b 1 z δ = fλ f 1. 19

20 Properies of ivision benchmarke preicor Y ˆ z ˆ z ( b y b z ˆ ) 1 = α, u + δfλf 1 1 α = =, u. Y ˆ member of cross-secional benchmarke preicors, Y ˆ Y ˆ a ( b y by ˆ ) (Wang e al. 2008) moel moel, = + k= 1 k k A k= 1 k k / k 1 k k a = φ b φ b. In presen case, = ˆ moel Y = z αˆ un-benchmarke preicor a ime ;,u ˆ moel moel -1 φ = = [ cov( Y, b / zδf by k k )] Pfeffermann & k=1 ˆ Barnar (1991). 20

21 Properies of ivision benchmarke preicor (con.) ETα 1 (a)- unbiaseness: if E( α 1 α 1) = 0 E( α α ) = 0. ( α) = 0 To warran unbiaseness uner moel, suffices o iniialize a ime = 1 wih unbiase preicor. (b)- Consisency: Plim( y Y ) = 0 & Plim( ˆ Y y ) = 0 (by GLS) n Plim( ˆ Y Y ) = 0 (even if moel misspecifie). n n area no helping benchmarking oher areas. n 21

22 Secon-sage benchmarking Suppose S saes in ivision an similar moel; irec Toal y = Y + e ; E( e ) = 0, Cov( e, e ) =δ σ Y 2 s, s, s, s, s, τ s*, s, s* s, τ = z α s, s, s, ; α = T α + η, E( η ) = 0, E( η η ) = δ Q s, s s, 1 s, s, s, s,*,* s, S S Benchmark: b ˆ ˆ s, ys, = bs, z s, αs, = Y s= 1 s= 1 Benchmark error: ˆ S r ( Y Y ) ( z ˆ α b z = = s, s, αs, ) No longer simple linear combinaion of sampling errors. s= 1 22

23 Benchmarking of Sae esimaes (con.) (,...,, ˆ y = y y Y ) = 1, S, (, ˆ y Y ) e = ( e,..., e, r ) = ( e, r ). 1, S, α = ( α,..., α ), 1, S, = ΙS T, s Z S z s, T =Ι, Z Z =. b1, z 1,,..., bs, z S, Combine moel: y = Z α + e ; α = T α + η 1 E( ηη ) = Ι Q = Q ; E( e e ) =Σ S s, τ τ. 23

24 Benchmarking of Sae esimaes (con.) T α I u RCR Moel: u P C Var = =,, 1 1,, = u 1 T 1 y α α α Z + ; = e,, 1 1 V e C Σ, Σ h C = E( u 1 e ); Σ = Eee ( ) =. e = ( e, r ) h v ( ˆ r = Y Y ) correlae wih moel errors,, u 1, an Sae sampling errors, e, s, in complicae way (see paper). 24

25 Compuaion of Sae benchmarke preicors S Impose b, y, = b, z, α, s= 1 s s s= 1 s s s S r =0. efine,, e = ( e, o), C, = E( u 1 e 0, 0), Σ, 0 = Ee (, 0 e, 0 ), V, 0, P 1 C, 0 =. C, Σ 0, 0 1, 1 Ι 1 T α 1 α, = (, Ι Z )( V, ) (, Ι Z )( V 0, ) Z 0 GLS. y ˆ Benchmarke preicor for Sae (,s): Ys, = z, s, α s,. 25

26 Variance of benchmarke preicors Marices, C, = E( u 1 e 0, 0) & Σ, 0 = E( e, 0 e, 0 ) only use for compuing benchmarke preicors. True Var (ˆ s, Y -Y s, )accouns for variances an auocovariances of Sae sampling errors, e, 26 s, variances an auo-covariances of ivision benchmark preicion errors, r, an heir covariances wih Sae sampling errors, an variances of moel componens, η s,.

27 Empirical resuls Toal unemploymen, CPS-USA, Jan ec2009. Firs level- Census ivisions, Secon level- Saes 1. Compare smoohness of ime series benchmarking an inepenen proraing; R = 1 R 1 R / 1 R + 1 R /-1 /-1, pr /-1, moel /-1, pr / -1, moel R /-1,... monh o monh raio of benchmarke preicor 2. Illusrae consisency of benchmarke preicors; 3. Illusrae robusness; 4. Illusrae variance reucion. 27

28 Raios R /-1 when esimaing oals, New Hampshire 28

29 Raios R /-1 when esimaing rens, New Hampshire 29

30 Raios R /-1 when esimaing oals, New Mexico 30

31 Raios R /-1 when esimaing rens, New Mexico 31

32 isribuion of Ι [ R/- >0] over Saes, Toal Esimae oal Esimae ren

33 [S.E.( y s, ) /y s,] (lef) & [ Y ˆ s, / y s, ] (righ) Massachuses Jan-90 Jan-94 Jan-98 Jan-02 Jan Jan-90 Jan-94 Jan-98 Jan-02 Jan-06 33

34 [S.E.( y s, ) /y s,] (lef) & [ Y ˆ s, / y s, ] (righ) New Hampshire Jan-90 Jan-94 Jan-98 Jan-02 Jan Jan-90 Jan-94 Jan-98 Jan-02 Jan-06 34

35 irec, Benchmarke an Unbenchmarke esimaes of Toal Unemploymen, Georgia (numbers in 000 s) CPS BMK UNBMK Jan-00 Jan-02 Jan-04 Jan-06 Jan-08 35

36 irec, Benchmarke an Unbenchmarke esimaes of Toal Unemploymen, Alabama (numbers in 000 s). 170 CPS BMK UNBMK Jan-00 Jan-02 Jan-04 Jan-06 Jan-08 36

37 Relaive S errors of irec, Benchmarke an Unbenchmarke es. of Toal Unemploymen, Georgia 0.25 CPS BMK UNBMK Jan-00 Jan-02 Jan-04 Jan-06 Jan-08 37

38 Relaive S errors of irec, Benchmarke an Unbenchmarke es. of Toal Unemploymen, Alabama CPS BMK UNBMK Jan-00 Jan-02 Jan-04 Jan-06 Jan-08 38