Assessing uncertainty of the temporal EBLUP: a resampling-based - PowerPoint PPT Presentation

Assessing uncertainty of the temporal EBLUP: a resampling-based approach Luís N. Pereira, MsC 1 Pedro S. Coelho, PhD 2 1 University of the Algarve - ESGHT, Portugal 2 New University of Lisbon - ISEGI, Portugal Rome, 11 th July 2008

Outline � Background � Objectives � Methods – Rao-Yu Model – Uncertainty measures of the EBLUP � Monte Carlo Simulation Study � Application with real data � Conclusion 2

Background � Small Area Estimation (SAE): is about how to produce reliable estimates of domain characteristics when the sample sizes within the domains are very small or even zero. � We need employ indirect estimators that borrow strength from related areas and time periods through linking models based on auxiliary information, such as recent census and current administrative data. � Such estimators are often base on linear mixed models (LMM) 3

Background � The Empirical Best Linear Unbiased Prediction (EBLUP) approach is the most popular method for the estimation the parameters � When time-series and cross-sectional data is available, longitudinal LMM are useful � While EBLUP estimators are easy to obtain, measuring its quality is a challenging problem � Although there is much theory about measuring the uncertainty of EBLUP under general LMM, very little research has been done regarding the comparison of the approaches. 4

Objectives To introduce a parametric bootstrap procedure and a weighted � jackknife method for MSPE estimation of the EBLUP, under a cross-sectional and time-series stationary model � To compare the performance of the resampling-based methods with the Taylor series-based method To apply these methods to real data from the Prices of the � Habitation Transaction Survey 5

Rao-Yu Model Rao and Yu (1994) proposed the following model: = ρ + ε ˆ x ′ u u θ = θ + θ = + v + ρ < u e β 1 , , − 1 it i t it it it it it it i it , Where is the parameter of inferential interest for the i th small-area at t th time point θ it ( i =1, …, m ; t =1, …, T ) ˆ θ is its design-unbiased direct survey estimator it e θ = E e ( | ) 0 are independent sampling errors normally distributed with it it it x ( p × 1) is a column vector of area-by-time specific auxiliary variables it ( p × 1) is a column vector of regression parameters β ( ) iid v σ v N 2 are random area specific effects with ~ 0 , i i v ( ) iid u ε N σ 2 are random area-by-time specific effects with , ~ 0 , it it 6 following a common AR(1) process for each i .

Rao-Yu Model They showed that the model can be expressed in matrix form as: X β Zv u e ˆ = + + + θ ( ) v I σ N 2 ~ 0 , where v m { } ( ) ( ) − j i = ρ − ρ Γ 2 u 0 σ I ⊗ N with 1 2 Γ ~ , m ( ) R e 0 R = σ diag ~ N 2 ( ) , with ≤ i ≤ m ≤ t ≤ T it 1 ; 1 iid ( ) e ~ N 0 , 1 Assuming then: it ( ) ( ) ˆ V V V J I = = = σ + σ + Cov blockdiag θ 2 Γ 2 with ≤ ≤ i v T T i m i 1 [ ] ′ ρ = σ v ρ σ ρ ψ 2 2 ( ), ( ) Assuming that is known then: 7

Rao-Yu Model The BLUP estimator (Rao and Yu, 1994): ~ ~ ~ ( ) ′ ′ x 1 V − ˆ X θ = + σ + σ − ψ β 2 2 γ 1 θ β , ( ) ( ) it it v T t i i i ( ) ~ 1 − X ′ V − 1 X X ′ V − 1 = ˆ β θ Where { } ( ) γ − j is the t th row of i = ρ − ρ Γ 2 1 t The EBLUP estimator (Rao and Yu, 1994): ~ ˆ x ′ 1 ′ V − X θ = ˆ + σ + σ ˆ ˆ − ˆ ψ β 2 2 γ 1 θ β ˆ ˆ ˆ ( ) ( ) ( ) it it v T t i i i ( ) − 1 ˆ X ′ V − 1 X X ′ V − 1 ˆ = ˆ ˆ β θ Where ψ were estimated through a method of moments 8

Analytical approximation of the MSPE { } { } { } e v u Under the normality of the , and (Kackar and Harville, 1984): it i it [ ] ~ ( ) ( ) ( ) θ = + MSE g g ψ ψ ψ it it it 1 2 2     ~ ~ ~ ˆ ( ) ( ) ( ) ˆ ( ) ( ) θ = + + θ − θ MSE g g E ψ ψ ψ ψ ψ ˆ ˆ     it it it it it     1 2 In the context of the Rao-Yu model (Rao and Yu, 1994) σ ( ) ′ ( ) 2 ( ) = σ + − σ 1 + σ V − σ 1 + σ g ψ γ γ ( ) 2 2 2 1 2 2 O 1 it υ υ T t d υ T t , 1 − ρ 2 1 [ ] ( [ ] ( ) ) ( ) ′ ( ) ( ) − ′ ′ 1 1 ′ = x − X V − σ 1 + σ X V − X x − X V − σ 1 + σ g ψ 1 2 2 γ 1 2 2 γ , − O m 1 υ υ it it i i T t it i i T t 2 2  ~ ~  ( ) ( ) ˆ θ − θ E ψ ψ cannot be analytically evaluated. ˆ   it it   9

Analytical approximation of the MSPE Rao and Yu (1994) obtained the following approximation: ( ) 2  ~ ~  ( ) ( ) ( ) ˆ ( ) θ − θ ≈ A = E tr g ψ ψ Σ * ψ ˆ , − O m 1   it it t it   3 where A t (2 × 2) is a matrix with: [ ] [ ] ( ) ( ) ′ Γ V − 1 V − Γ V − 1 = − σ + σ − σ + σ a γ 1 2 2 γ 1 γ 1 2 2 γ υ υ t i T t i t i T t 11 [ ] [ ] ( ) ( ) ′ = 1 − J V − σ 1 + σ V − 1 − J V − σ 1 + σ a 1 2 2 γ 1 1 2 2 γ t T i υ T t i t T i υ T t 22 [ ] [ ] ( ) ( ) ′ Γ V − 1 V − 1 J V − 1 = = − σ + σ − σ + σ a a γ 1 2 2 γ 1 1 2 2 γ υ υ t i T t i t T i T t 12 21 [ ] Σ = σ v ρ σ ρ Cov * 2 2 and ˆ ˆ ( ) ; ( ) Rao and Yu (1994) proposed the following approximately unbiased estimator:  θ  ~ ˆ ( ) ( ) ( ) ( ) RY = + + mspe g g g ψ ψ ψ ψ ˆ ˆ ˆ ˆ 2 10   it it it it it   1 2 3

Bootstrap approximation of the MSPE Following Butar and Lahiri (2003) and González-Manteiga et al. (2005) ideas, the parametric boostrap procedure work as follows: ( ) ˆ = σ v ρ σ ρ ˆ ˆ 2 2 ˆ ˆ β β ψ θ ( ) , ( ) ˆ , 1) Estimate using the method of moments, and then estimate based on Rao-Yu model ( ) v σ N * 2 2) Generate ˆ ~ 0 , v ( ) v e N * * 3) Generate , independently of ~ 0 , 1 ( ) v e * σ * N ε * 2 4) Generate , independently of and . ~ 0 , ˆ u * , assuming that ρ ρ is known ρ ρ Then construct ˆ X ˆ Zv u e = + + + θ * β * * * 5) Construct the bootstrap data ( ) ˆ ˆ = ˆ ˆ β * β ψ θ * θ * ˆ 6) Fit the model to and estimate , B σ σ e 2 * 2 * ˆ ˆ ˆ θ * 7) Estimate from v ( ) ˆ = ˆ ˆ β * β ψ * θ * ˆ ˆ , Then fit the model to and estimate θ * 11 E

Bootstrap approximation of the MSPE ˆ θ * 8) Compute the bootstrap temporal BLUP from [ ] ~ ( ) − x ′ 1 ′ V 1 X θ = ˆ + σ + σ ˆ − ˆ * ψ β * 2 2 γ ψ θ * β * ˆ ˆ ˆ ˆ ( ) ( ) ( ) B it it B v T t i i i B , ˆ θ * 9) Compute the bootstrap temporal EBLUP from [ ] ( ) ~ ˆ − 1 x ′ ˆ 1 ′ V ˆ X ˆ θ = + σ + σ − * ψ * β * 2 * 2 * γ ψ * θ * β * ˆ ˆ ˆ ˆ ( ) ( ) ( ) E it it E v T t i i i E , ~ ~ ˆ ( ) ( ) ( ) θ b b θ b * ψ * * ψ ˆ ˆ 10) Repeat 2)-9) B times: ( ) , ( ) ( b =1, …, B ) E it B it , , g 3 11) Calculate a boostrap estimate of : it B 2  ~ ~  ∑ ˆ ( ) ( ) ( ) − = θ b b − θ b g B * 1 * ψ * * ψ ˆ ˆ ( ) ( )   it E it B it   3 , , = b 1 Following the lines of Butar and Lahiri (2003), a bias corrected bootstrap estimator is: [ ] B   ~ [ ] ˆ ∑ ( ) ( ) − B θ = + − b + b + mspe g g B g g g ψ ψ ψ 1 ψ * ψ * * ˆ ˆ ˆ ˆ ˆ ( ) 2 ( ) ( ) ( ) ( )   12 it it it it it it it   1 2 1 2 3 = b 1

Jackknife approximation of the MSPE Following Jiang et al. (2002) and Chen and Lahiri (2008) ideas, the Taylor series approximation of the jackknife MSPE estimator of the EBLUP is:   ~ ( ) ( ) ( ) ( ) [ ] ˆ ′ θ = + − c ∇ + A + J mspe g g g tr ψ ψ ψ ψ ψ υ ˆ ˆ ˆ ˆ ˆ ˆ ( )   it it it it WJ t t WJ   1 2 1 [ ] [ ] ′   ( ) ( ) ( ) ( ) ′ + L y − X ˆ y − X ˆ L tr ψ β ψ β ψ ψ υ  ˆ ˆ ˆ ˆ  t i i i i t WJ   ( ) m ∑ c = − − w ψ ψ ˆ ˆ ˆ where: WJ it j it j , , = j 1 ( )( ) m ′ ∑ = − − w υ ψ ψ ψ ψ ˆ ˆ ˆ ˆ WJ it j it − j − j , , = j 1 ψ ψ after deleting the j th small-area data ˆ is the estimator of − j 13 ( ) w , = + − w O m 1 1 are weights that satisfy j it j it ,