Small area estimation under spatial SAR model

Transcription

1 Small area estimation under spatial SAR model Jan Kubacki Centre for Mathematical Statistics, Statistical Office in Łódź Alina Jędrzejczak Chair of Statistical Methods, Institute of Econometrics and Statistics, University of Łódź, Poland

2 Summary In the presentation the method of small area estimation under spatial Simultaneous Autoregressive (SAR) model is presented. The estimation was conducted using both spatial EBLUP and hierarchical Bayes method with SAR random effects that depend on proximity matrix and spatial autoregression coefficient (ρ). The presentation continues the idea of small area estimation by means of hierarchical Bayes method in the case of known model hyperparameters, presented in a previous work of one of the authors. The computations performed by sae package for R-project environment and special procedure prepared for WinBUGS software reveal that consistent estimates (for point estimates, for measures of variability - such as estimation error and for random effects) can be obtained. As an example the data on average per capita available income from Polish Household Budget Survey for counties (NUTS4) and auxiliary variables from Polish Tax Register (POLTAX) were used together with digital maps for Polish counties. The precision estimation for direct estimates were obtained by means of three-stage method, where Balanced Repeated Replication (BRR), bootstrap and Generalized Variance Function were used. The results of the computations reveal that for higher values of ρ some MSE reduction can be observed. This effect for HB-SAR method is even more evident than for common spatial EBLUP. The computations done with Gibbs sampler also demonstrate simultaneous nature of simulated processes especially for random effects. In our opinion, the Gibbs sampler can be a good starting point for other computation methods that seems more adequate for this kind of stochastic processes.

3 Mixed model for small areas In this work we discuss a special case of area level model (type A model), where the parameter of interest is vector θ of size m (where m is the number of small areas), which is related to the direct estimator θˆ with the following relationship Where e is the vector of independent sampling errors having mean 0 and diagonal variance matrix ψ. Parameter θ satisfies the following relationships which incorporates the spatial relationships between areas θ θˆ θ Xβ where X is the matrix of area dependent auxiliary variables of size m p, β is the regression parameters vector of size p, Z is the matrix (m m size) of known positive constants and v is the vector of independent and identically distributed variables (called random effects) with mean 0 and variance σ v e Zv () ()

4 SAR model specification There are two common variants of spatial models depending on the random effects pattern. The first one is the simultaneously autoregressive model (SAR) and the second is the conditional autoregressive model (CAR). In our presentation we assume that the random effects are described by SAR process with random effects described as follows v Wv W) Where ρ is the spatial autoregressive coefficient, W is the proximity matrix (of size m m), u is the vector of independent error term with zero mean and constant variance σ u and I is the identity matrix of size m m. The random effects have the following covariance matrix G (called SAR dispersion matrix) G and the sampling error e has the following covariance matrix u u v ( I T [( I W) ( I W )] u (3) R Ψ diag ( i )

5 Spatial EBLUP estimator When the matrix Z is neglected (or equal I) the model () with spatially-correlated random effects takes the form: θˆ Xβ ( I W ) u e Under the above model the Spatial EBLUP estimator is equal to ~ θ i x T i ~ β b T i GV (ˆ θ ~ Xβ) ~ T T Where β X V X X V θˆ is the generalized LS estimator of the i [ ] regression parameter and b T i is the m vector (0,,0,,0, 0) with in the i-th position. This estimator is dependent on the σ u and ρ which can be estimated using ML or REML method.

6 MSE for spatial EBLUP estimator The mean square error for Spatial EBLUP estimator can be expressed as the sum of four following elements ~ mse[ i ] g g g3 g 4 where g is connected with uncertainty about the small area estimate and is of order O(), g is connected with uncertainty about β and is of order O(m - ) for large m, g 3 is connected with uncertainty about σ u (or variance components) and g 4 is connected with uncertainty of spatial autocorrelation parameter ρ. The first two elements are given by gi T b i [ G GV G] b i g i b T i T T [ I m GV ] X(X V X) X [ I m V G ] b i The third element is more complicated and can be expressed as g i 3 trace{ L VL T i i I } where L i b T i b [ C T i V [ AV u u C C V V C AV V ] ] C ( I W) ( I W) m A C ( W W W W ) C T T T m u and I - is the Fisher information matrix inverse

7 Estimation of g 4 element for MSE The last term g 4 can be expressed by where the second derivatives are given by the following formulas: i b Ψ V ΨV b ) ( ) ( 4 kl l k k l T i i I V g m m u V 0 ) ( ) ( ) ( ) ( C C C V V u u T WC W C C C C C C V ) ( u u

8 Hierarchical model incorporating SAR process In the case where ρ and σ u are known (could be obtained from Spatial EBLUP estimation), a spatial hierarchical Bayes model can be specified. Its form is similar to the usual HB model (Rao, 003, eq. (0.3.)), but it accounts for spatial dependences between small areas. The model can be expressed as follows (i) (ii) (iii) (iv) θ β, ˆ θ, β, u θ f (β) u u, ~ β, θ, θˆ N( Xβ, ~ ~ u N( θ, Ψ) G - [( I ( a, b) m W ) W )] The model assumes known prior Gamma distribution parameters for σ u. In our case they were obtained from previous studies (EBLUP estimators achieved for 6 regions and two years what may be a good approximation of σ u variability). Similar approach was used in Kubacki (0), where the distribution of σ u was obtained from ordinary regression models. T ( I m )

9 Empirical distribution of model variance obtained for EBLUP estimators

10 WinBUGS computational models Conventional model for known σ u model { for(p in : N) { Y[p] ~ dnorm(mu[p], tau[p]) mu[p] <- alpha[] + alpha[] * A[p] + alpha[3] * B[p] + u[p] u[p] ~ dnorm(0, precu) } precu ~ dgamma (a0,b0) alpha[] ~ dflat() alpha[] ~ dflat() alpha[3] ~ dflat() sigmau<-/precu } Model for SAR version model { for(p in : N) { Y[p] ~ dnorm(mu[p], tau[p]) mu[p] <- alpha[] + alpha[] * A[p] + alpha[3] * B[p] + v[p] u[p] ~ dnorm(0, precu) v[p] <- inprod(rho_w[p,:n],u[:n]) } precu ~ dgamma (a0,b0) alpha[] ~ dflat() alpha[] ~ dflat() alpha[3] ~ dflat() sigmau<-/precu }

11 R-project macro determining the model parameters # fitting the Spatial EBLUP model resultsreml <- eblupsfh(direst ~ + A + B, desvar, W, method="reml", data=d, MAXITER=500) msesreml <- msesfh(direst ~ + A + B, desvar, W, method="reml", data=d, MAXITER=500) sigmas_reml <- resultsreml$fit$refvar rho_reml <- resultsreml$fit$spatialcorr # determining the model parameters I <-diag(,n) for (j in :N) { W_row <- W[j,] for (k in :lpow) { W_mat[j,k] <- W_row[k] } } rho_w <- solve(i-rho_reml*w_mat) a0 <- dochg_shape_sp b0 <- dochg_rate_sp infile <- "coda.txt" indfile <- "codaindex.txt" data <- list(n=n, Y=Y, tau=tau, A=A, B=B, a0=a0, b0=b0, rho_w=rho_w) model <- lm( ydir[,] ~ + A + B) mod_smry <- summary(model) alpha <- as.vector(mod_smry$coefficients[,]) sigma_ <- (mod_smry$sigma)*(mod_smry$sigma) precu <- /sigma_

12 R-project macro making the simulations v <- vector(mode = "numeric", length = N) u <- vector(mode = "numeric", length = N) inits <- list(list(alpha=alpha, precu=precu, u=u)) parameters <- c("mu", "alpha", "precu", "v", "u") working.directory <- getwd() # simulations - WinBUGS call and collecting the data sim_hb <- bugs(data, inits, parameters, model_hb,n.chains=, n.burnin =, n.iter=0000, n.thin =, codapkg=true, working.directory = working.directory) results <- read.coda(infile, indfile,, 0000, )

13 Available income estimates with their precision and random effect estimates-part County (NUTS 4 unit) Direct estimates Parameter estimate REE % Available income EBLUP estimates REML Parameter estimate REE % Random effects HB estimates Parameter estimate REE % Random effects będziński 8,59 5,8 784,7 4,8 3,8 789,5 4,40 7,5 bielski 78,94, 778,63,0 53,9 779,40,9 54,8 cieszyński 76,78 4,99 734, 3,9 9, 738,6 3,97 33,7 częstochowski 570,65 6,54 590,44 5,00 -,00 586,37 5,08 -,73 gliwicki 693,0,4 706,85 5,9-3,38 705,97 5,6-3,95 kłobucki 539,04 8,00 574,9 5,59-8,0 568,43 5,77-30,7 lubliniecki 596,73 4,4 60,8 3,63-34, 608,03 3,63-34,85 mikołowski 796,64 5,83 793,96 5,0 0,5 796,39 5,4-0,06 myszkowski 63,46 7,74 6,49 5,35-5,9 60,0 5,4-5,43 pszczyński 69,75,4 74,5 5,4-3,85 79,7 5,79-30, raciborski 758,34 4,50 76,66 3,88 5,76 7,3 3,94 59,80 rybnicki 783,98 8,8 764,5 4,69 7, 766,0 5,08 7,97 tarnogórski 67,46 5,07 686,73 3,9-9,45 684,87 3,99 -,04 bieruńsko-lędziński 65,85,69 764,96 4,93-38,35 757, 5,3-49,9 wodzisławski 855,7 4,4 8,73 3,53 48,34 89,80 3,66 54,3 zawierciański 67,04 7,36 674,0 4,89 -,80 67,0 5,06 -,8 żywiecki 730,75,80 75,43,75 45,63 76,56,75 47,68 m.bielsko-biała 79,4 7,38 77,68 4,4 8,84 774,7 4,66 9,78

14 Available income estimates with their precision and random effect estimates-part County (NUTS 4 unit) Direct estimates Parameter estimate REE % Available income EBLUP estimates REML Parameter estimate REE % Random effects HB estimates Parameter estimate REE % Random effects m.bytom 705,56,9 705,8,7 4,93 705,3,6 5,59 m.chorzów 656,8,50 666,95,34-58,78 664,70,37-6,08 m.częstochowa 77,36 0,35 75,50 5,,94 79,7 5,57 6,93 m.dąbrowa Górnicza 777,5 4,38 78,6 3,47-6,48 783,03 3,47-8,58 m.gliwice 745,60 5,50 76,7 3,9-3,66 760,5 3,96-6,57 m.jastrzębie-zdrój 748,66 5,5 774,86 3,90 -,65 77,66 4,0-8,39 m.jaworzno 748,49 6,85 780, 4, -7,74 778,39 4,4 -, m.katowice 859,53,3 859,7, 5,66 859,38,,5 m.mysłowice 83,9,4 80,39,3 0,79 8,09,0 8,73 m.piekary Śląskie 744,4 3,4 74,89 5,3 0,35 74,68 5,88-0,4 m.ruda Śląska 67,9 5,46 77,90 5,09-3,8 768,99 5,75-9,80 m.rybnik 763,03,45 764,5,4-7,86 764,0,4-0,39 m.siemianowice Śląskie 95,69 9,5 774,56 4,93 9,68 783,96 5,63 38,39 m.sosnowiec 88,88 7,44 803,93 4,39 5,95 806,06 4,65 5,59 m.świętochłowice 686,8 9,03 708,8 4,90-8,4 706,4 5,8-0,05 m.tychy 83,03 0,07 83,03 0,07-4,80 83,0 0,07-9,04 m.zabrze 703,7 0, 703,74 0, -5,66 703,73 0, -5,83 m.żory 830,68 8,0 78,97 4,53 5,50 786,79 4,88 8,58

15 Observed vs. predicted available income for HB estimation Direct estimator (double black circles), EBLUP estimator (red circles)

16 Plot of random effects for EBLUP vs. HB method

17 Plots of distributions of model estimates for available income obtained by MCMC simulation using conventional model

18 Plots of autocorrelations of model estimates for available income per capita obtained for Silesian region by MCMC simulation using Gibbs sampler with conventional model

19 Available income estimates with their precision and random effect estimates-part County (NUTS 4 unit) Direct estimates Parameter estimate REE % Available income EBLUP estimates Spatial REML Parameter estimate REE % Random effects HB SAR estimates Parameter estimate REE % Random effects będziński 8,59 5,8 78,77 3,95,06 785,6 3,87,85 bielski 78,94, 779,45,0 59,60 780,,6 67,8 cieszyński 76,78 4,99 746,3 3,5 4,66 748,93 3,5 5,75 częstochowski 570,65 6,54 584,70 4,74-36,0 58, 4,68-3,99 gliwicki 693,0,4 7,7 4,55-5, 74,97 3,88 5,50 kłobucki 539,04 8,00 569,85 5,37-4,6 566,06 5,40-38,9 lubliniecki 596,73 4,4 603,87 3,58-4,7 60,7 3,54-37,00 mikołowski 796,64 5,83 788,05 4,4 4, 798,48 5,5,67 myszkowski 63,46 7,74 68,68 4,95 -,87 67,30 4,89-7,47 pszczyński 69,75,4 734,98 4,50-0,3 747,9 4,78 0,4 raciborski 758,34 4,50 70,69 4,00 49, 78,45 3,90 63,98 rybnicki 783,98 8,8 780,3 4,0 7,6 78, 4,34 6,94 tarnogórski 67,46 5,07 683,65 3,66-4,50 684,5 3,56-6,3 bieruńsko-lędziński 65,85,69 770,56 4,07-0,5 776,08 4,08-6,79 wodzisławski 855,7 4,4 83,85 3,47 4,0 89,97 3,56 56,09 zawierciański 67,04 7,36 673,66 4,56 -,07 67,67 4,64-6,88 żywiecki 730,75,80 79,35,75 5,79 79,74,69 59,34 m.bielsko-biała 79,4 7,38 799,78 3,96 4,89 797,44 4,03 47,44

20 Available income estimates with their precision and random effect estimates-part County (NUTS 4 unit) Direct estimates Parameter estimate REE % Available income EBLUP estimates Spatial REML Parameter estimate REE % Random effects HB SAR estimates Parameter estimate REE % Random effects m.bytom 705,56,9 703,46,7-6,94 703,80,6 0,8 m.chorzów 656,8,50 67,64,3-6,83 668,96,8-58,88 m.częstochowa 77,36 0,35 68,88 5,3-9,93 683,66 5,4 -,78 m.dąbrowa Górnicza 777,5 4,38 78,40 3,48-0,4 780,4 3,35 6,7 m.gliwice 745,60 5,50 753,9 3,78-7,6 753,8 3,57-9,8 m.jastrzębie-zdrój 748,66 5,5 795, 3,67 -,68 789,5 3,6 0,94 m.jaworzno 748,49 6,85 784,3 3,88-0,3 78,63 3,9-3,53 m.katowice 859,53,3 857,99, 9,63 859,,07 9,47 m.mysłowice 83,9,4 806,30,,49 806,86,5 0,7 m.piekary Śląskie 744,4 3,4 738,50 4,50 -,0 738,6 4,86-4,48 m.ruda Śląska 67,9 5,46 765,34 4,9-5,3 758,58 4,60-3,85 m.rybnik 763,03,45 768,,4-0, 767,0,39-3, m.siemianowice Śląskie 95,69 9,5 760,5 4,35 7,50 768,9 5,00 3,3 m.sosnowiec 88,88 7,44 807, 3,85 8,00 805,53 4,00 4,55 m.świętochłowice 686,8 9,03 69,04 4,5-9,30 689,37 4,76-4,46 m.tychy 83,03 0,07 83,03 0,07 0,70 83,0 0,07 9,8 m.zabrze 703,7 0, 703,73 0, -6,69 703,7 0, -9,8 m.żory 830,68 8,0 770,40 4,9 8,08 773,90 4,45 9,50

21 Observed vs. predicted available income for HB SAR estimation, Direct estimators (double black circles), Spatial EBLUP estimators (red circles)

22 Plot of random effects for Spatial EBLUP vs. HB-SAR method

23 Plots of distributions of model estimates for available income obtained by MCMC simulation using SAR model

24 Plots of autocorrelations of model estimates for available income per capita obtained for Silesian region by MCMC with SAR relationships simulation using Gibbs sampler

25 Relative Estimation Error Reduction and Spatial Gain for Silesian region obtained using EBLUP and HB method with Gibbs sampler County (NUTS-4 unit) Relative estimation error reduction EBLUP HB Spatial EBLUP HB-SAR Spatial estimation error reduction Spatial EBLUP HB-SAR będziński,36,3,476,504,084,37 bielski,03,06,05,049,00,03 cieszyński,73,57,4,40,7,30 częstochowski,307,86,379,396,055,086 gliwicki,45,987,448,869,4,444 kłobucki,430,386,488,48,04,069 lubliniecki,40,38,56,67,03,05 mikołowski 3,59,94 3,737 3,074,83,05 myszkowski,446,49,563,583,08,08 pszczyński,4,097,698,54,03, raciborski,6,44,5,55 0,969,00 rybnicki,746,60,950,886,6,7 tarnogórski,94,7,385,43,07,8 bieruńsko-lędziński,374,99,87,867,09,303 wodzisławski,99,56,,90,09,09 zawierciański,506,454,64,588,07,09 żywiecki,04,04,08,065,004,040 m.bielsko-biała,669,58,864,89,7,56

26 Relative Estimation Error reduction and Spatial Gain for estimation error for Silesian region obtained using EBLUP and HB method with Gibbs sampler County (NUTS-4 unit) Relative estimation error reduction EBLUP HB Spatial EBLUP HB-SAR Spatial estimation error reduction Spatial EBLUP HB-SAR m.bytom,04,0,0,09 0,997 0,998 m.chorzów,066,05,08,093,04,040 m.częstochowa,0,858,949,908 0,964,07 m.dąbrowa Górnicza,63,6,6,306 0,999,035 m.gliwice,407,388,456,543,035, m.jastrzębie-zdrój,47,375,504,53,06,3 m.jaworzno,64,550,767,749,088,8 m.katowice,04,04,00,053 0,996,039 m.mysłowice,077,095,085,7,007,00 m.piekary Śląskie,473,34,9,706,8, m.ruda Śląska 3,035,689 3,60 3,359,87,50 m.rybnik,07,09,03,046,005,07 m.siemianowice Śląskie,857,66,03,83,3,6 m.sosnowiec,693,598,93,858,4,6 m.świętochłowice,844,70,000,897,085,09 m.tychy,000,0,000 0,994,000 0,983 m.zabrze,000 0,99,000 0,99,000,00 m.żory,8,68,959,845,08,097

27 Absolute values of random effects for available per capita income (Silesian region )

28 Distribution of relative estimation error for: Direct estimator, EBLUP (spatial and ordinary - REML variant) and HB (both ordinary and SAR)

29 Distribution of REE reduction for EBLUP (spatial and ordinary - REML variant) and HB (both ordinary and SAR) estimators

30 Distribution of REE reduction due to spatial relationships for Spatial EBLUP and HB-SAR estimators

31 REE reduction for Spatial EBLUP and HB-SAR estimators obtained for different a priori spatial autoregressive coefficient values (simulation study)

32 REE reduction due to spatial relationships for Spatial EBLUP and HB-SAR estimators for different a priori spatial autoregressive coefficient values

33 Trace of results for Gibbs sampler simulation obtained for first 3 values of random effects (u description) obtained for ordinary HB method and first 3 values of random effects (v description) obtained for HB-SAR method for 004 year and Silesian region, assuming (for v values) that the spatial autoregressive coefficient is equal 0.95

34 Conclusions The results presented in the presentation show that, using the auxiliary information on spatial neighborhood, one can obtain relatively precise estimates based on small area hierarchical models. In particular, it is possible by means of spatial Simultaneous Autoregressive Process (SAR). The advantage of Spatial HB estimation observed in terms of a realworld example prepared for Silesian region and the data from Polish HBS and Polish tax register may not be so clear for other regions. The simulation-based calculations also confirmed noticeable gains in efficiency for Spatial HB estimators, when some assumptions on spatial autoregressive coefficient have been made. This effect, however, is more evident for Spatial HB method than for Spatial EBLUP technique.

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