an applied comparison of area-level linear mixed models in small area estimation
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An Applied Comparison of Area-LevelLinear Mixed Models in Small AreaEstimationLuis Nobre Pereira a b & Pedro Simões Coelho c da Escola Superior de Gestão, Hotelaria e Turismo, Universidade doAlgarve , Faro , Portugalb Centro de Investigação sobre o Espaço e as Organizações,Universidade do Algarve , Faro , Portugalc Instituto Superior de Estatística e Gestão de Informação,Universidade Nova de Lisboa , Lisboa , Portugald Faculty of Economics, Ljubljana University , Ljubljana , SloveniaPublished online: 20 Nov 2012.
To cite this article: Luis Nobre Pereira & Pedro Simões Coelho (2013) An Applied Comparison of Area-Level Linear Mixed Models in Small Area Estimation, Communications in Statistics - Simulation andComputation, 42:3, 671-685, DOI: 10.1080/03610918.2011.654029
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Communications in Statistics—Simulation and Computation R©, 42: 671–685, 2013Copyright © Taylor & Francis Group, LLCISSN: 0361-0918 print / 1532-4141 onlineDOI: 10.1080/03610918.2011.654029
An Applied Comparison of Area-Level Linear MixedModels in Small Area Estimation
LUIS NOBRE PEREIRA1,2 AND PEDRO SIMOES COELHO3,4
1Escola Superior de Gestao, Hotelaria e Turismo, Universidade do Algarve, Faro,Portugal2Centro de Investigacao sobre o Espaco e as Organizacoes, Universidade doAlgarve, Faro, Portugal3Instituto Superior de Estatıstica e Gestao de Informacao, Universidade Nova deLisboa, Lisboa, Portugal4Faculty of Economics, Ljubljana University, Ljubljana, Slovenia
This article reviews four area-level linear mixed models that borrow strength by exploit-ing the possible correlation among the neighboring areas or/and past time periods. Itsmain goal is to study if there are efficiency gains when a spatial dependence or/and atemporal autocorrelation among random-area effects are included into the models. TheFay–Herriot estimator is used as benchmark. A design-based simulation study basedon real data collected from a longitudinal survey conducted by a statistical office ispresented. Our results show that models that explore both spatial and chronologicalassociation considerably improve the efficiency of small area estimates.
Keywords Empirical best linear unbiased prediction; Linear mixed model; Small areaestimation; Spatial correlation; Temporal autocorrelation
Mathematics Subject Classification 62D05; 62G09; 62J05; 62P20
1. Introduction
Local-level planning requires reliable data for small areas, but normally due to cost con-straints, sample surveys are often planned to provide reliable estimates only for largegeographical regions and large subgroups of a population. As a result, many domains ofinterest can be unplanned at the design stage and sample sizes within these domains arerarely large enough to provide reliable direct estimates. For example, data from the Pricesof the Habitation Transaction Survey (PHTS) in Portugal only allow the production ofreliable direct estimates for the mean price of habitation transaction for the country andfor broad geographical regions called NUTSII (NUTSII is an administrative division thatsegments the state into five regions). In fact, the PHTS sample sizes are very small or evenzero in many other domains of interest (NUTSIII, municipalities). For these unplanneddomains with small sample sizes, direct estimators are either unreliable or even unfeasible.This creates a need to employ indirect estimators that “borrow information” from related
Received April 27, 2011; Accepted December 16, 2011Address correspondence to Luis Nobre Pereira, Escola Superior de Gestao, Hotelaria e Turismo,
Universidade do Algarve, Campus da Penha 8005-139, Faro, Portugal; E-mail: [email protected]
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small areas or/and past time periods through linking models, using recent census and cur-rent administrative data in conjunction with the sample survey data, in order to increasethe effective sample size and thus precision. Such indirect estimators are often based onexplicit linear mixed models (LMM) that provide a link to a related small area through theuse of supplementary data. In this context, indirect estimators could be obtained throughthe classical empirical best linear unbiased prediction (EBLUP) approach, or through otherrecognized approaches like the empirical Bayes (EB), the hierarchical Bayes (HB), or theKalman filtering. However, the EBLUP approach, using Henderson’s method (Henderson,1975), is the most popular technique for estimating small area parameters of interest. Forthat reason, this article deals with small area estimators under area-level LMM via EBLUPestimation.
The most well-known model is due to Fay and Herriot (1979). The Fay–Herriot model isused to improve the efficiency of the direct estimators using cross-sectional data. Extensiveresearch on this model can be found in the literature. For example, Datta and Lahiri (2000),Ghosh and Rao (1994), Jiang and Lahiri (2006), Lahiri (2003), and Rao (2003), amongothers, give an account of this research. Although the Fay–Herriot model borrows strengthfrom other small areas through a linear relationship between the target parameter of interestand some auxiliary variables, it assumes independent random area specific effects. However,Cressie (1991) pointed out that eventual further improvement in the EBLUP estimator couldbe obtained if spatial dependence among random area specific effects would be assumed.This avowal was mainly supported by the fact that in several applications small areaboundaries are commonly defined according to administrative criteria without consideringthe eventual spatial dependence of the target variable. So, it should be reasonable toassume correlated random effects among neighboring small areas (Pratesi and Salvati,2008; Salvati, 2004). Although the first small-area model with spatially correlated randomeffects was introduced by Cressie (1991), just a few years ago Salvati (2004) developeda model with spatially correlated random area specific effects following a simultaneousautoregressive (SAR) process. Since this work due to Salvati (2004), research on small-area estimation under spatial LMM considering correlated random area specific effects hasreceived considerable attention in recent years (Chandra et al., 2007; Pratesi and Salvati,2008; Petrucci and Salvati, 2006; Petrucci et al., 2005).
As was stated before, both the above-mentioned models only use cross-sectional data ata given point in time. However, longitudinal surveys are increasingly used presently. Thus,it also might be useful to use time-series and cross-sectional models to take advantage ofdata at other time points in order to strengthen the direct estimators. This idea was firstlydeveloped by Choudhry and Rao (1989). Their work was afterwards developed by Raoand Yu (1994), who proposed a well-known model involving chronological autocorrelatedrandom area-by-time specific effects following a first-order autoregressive [AR(1)] process.Some further research in small-area estimation on LMM with temporally autocorrelatedrandom effects may be found in Datta et al. (2002), Ghosh et al. (1996), Pereira and Coelho(2010b), and Saei and Chambers (2003), among others. Recently, it was proposed a spatio-temporal linear mixed model involving spatially correlated and temporally autocorrelatedrandom effects (SAMPLE Project, 2010). This model takes advantage of both the possiblespatial similarities among small areas and the expected time-series relationships of the datain order to improve the efficiency of the small-area estimators.
Several applied problems of small-area estimation using these kinds of area-levelLMM, in order to improve the efficiency of traditional direct and other indirect estimators,can be found in the literature. For example, Fay and Herriot (1979) used a model to estimateper capita income for small places in the United States; Datta et al. (1999) applied a model
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Comparison of the Performance of EBLUP Estimators 673
to estimate monthly unemployment rates for 49 American states (excluding the state ofNew York and the District of Columbia); You et al. (2003) used a model to estimatemonthly unemployment rates for Census Metropolitan Areas (cities with population morethan 100,000) and Census Agglomerations (other urban centers) in Canada; Datta et al.(2002) used a model to estimate the median income of four-person families for all the 50American states and the District of Columbia using time-series and cross-sectional data;Petrucci and Salvati (2006) applied a model to estimate the average sub-watershed erosionper acre on the Rathburn Lake Watershed. Nevertheless, there are very few works focused onapplied comparisons of the performance of different EBLUP estimators through a design-based Monte Carlo simulation study using real data. The latest is due to Petrucci andSalvati (2006), who have studied the relative performance of the post-stratified estimator,Fay–Herriot EBLUP estimator, and Salvati EBLUP estimator using erosion data collectedat the Rathburn Lake Watershed. In fact, most of the work on model validation in thiscontext has been based on synthetic simulation from a model-based perspective. Note thatin other small-area estimation approaches, it is possible to find some literature devotedto comparisons of the performance of different EBLUP estimators using Monte Carloexperiments. Singh et al. (2005) conducted a model-based simulation study to judge theperformance of EBLUP estimators under area-level state-space models, based on per capitaconsumption expenditure time-series data; Fabrizi et al. (2007) performed a design-basedsimulation study to evaluate the properties of EBLUP estimators under unit-level LMM,based on the European Community Household Panel survey data; and Pratesi and Salvati(2008) conducted a model-based simulation study to judge the performance of EBLUPestimators under unit-level LMM, based on Life Conditions in Tuscany survey data.
The main purposes of this article are: (1) to compare the performance of differentEBLUP estimators (cross-sectional, spatial, temporal, and spatio-temporal), based on area-level LMM, through a design-based Monte Carlo simulation study using real data; and (2)to illustrate how the inclusion of a spatial dependence or/and a temporal autocorrelationamong random-area effects into the models could improve the efficiency of small-areaestimates. In the application that we discuss here we use real data from the PHTS andthe Prices of Bank Evaluation in the Habitation Survey (PBEHS), which are longitudinalsurveys conducted by the Portuguese statistical office.
The article is organized as follows. Section 2 reviews some well-known area-levelLMM. The design of the simulation study and its empirical results on the performance ofdifferent EBLUP estimators are presented in Sec. 3. Finally, some concluding remarks aregiven in Sec. 4.
2. Area-Level Models
Let us define the following notation: θ is used to make reference to a parameter of interest,y is used to denote the direct survey estimate, ε is used to denote the sampling error withknown variance, and x is used to refer to a vector of p auxiliary variables. In addition, thesubscript i denotes the small area (i = 1, . . . , m) and the subscript t refers the time period(t = 1, . . . , T). For example, xi = (
xi1, . . . , xip
)′( p × 1) denotes a vector of auxiliary
variables associated to the ith small area (i = 1, . . . , m), θit refers a parameter of inferentialinterest for the ith small area at tth time point (i = 1, . . . , m; t = 1, . . . , T), and yit is itsdesign-unbiased direct survey estimate. Finally, β ( p × 1) denotes an unknown vector of pregression parameters and β is its generalized least-squares estimator.
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2.1. The Fay–Herriot Model
Fay and Herriot (1979) proposed the following linear mixed model:
yi = x′iβ+ vi + εi, i = 1, . . . , m (1)
where vi denotes the random small-area effect and εi the sampling error associated with yi .It is assumed that the errors {vi} and {εi} are independently distributed as vi
iid∼ N (0, A) andεi
ind∼ N (0,Di), where the model variance, A, is unknown and the sampling variances, Di ,are assumed to be known. Under model (1) and assuming first that A is known, the BLUPof θi is given by:
θi (A) = Di
A + Di
x′i β+ A
A + Di
yi. (2)
An EBLUP of θi is obtained from Eq. (2) by replacing the model variance for itsconsistent estimator, A. Several authors have been estimating that variance componentusing different approaches. Fay and Herriot (1979) proposed an iterative estimator basedon weighted residual sum of squares and the method of moments; Prasad and Rao (1990)provided an ANOVA estimator and Datta and Lahiri (2000) used maximum likelihoodand residual maximum likelihood approaches to estimate A. The important problem ofestimating the mean square prediction error (MSPE) of EBLUP has been mainly facedby Butar and Lahiri (2003), Chen and Lahiri (2008), Datta and Lahiri (2000), Datta et al.(2005), Jiang et al. (2002), Lahiri and Rao (1995), and Prasad and Rao (1990).
2.2. Model with Spatially Correlated Random Effects
Salvati (2004) proposed an extension of the Fay–Herriot model with spatially correlatedeffects in order to take into account for the relationship among neighboring small areas. Inmatrix notation, this linear mixed model can be written as:
y = Xβ+ v + ε, (3)
where y = col1≤i≤m (yi) (m × 1), X = col1≤i≤m(x′i) (m × p), v = col1≤i≤m (vi) (m × 1),
and ε = col1≤i≤m (εi) (m × 1). It is assumed that the vector of independent sampling errors,ε, has mean 0 and known diagonal covariance matrix, R = diag1≤i≤m(σ 2
i ). Salvati (2004)proposed the specification of random area specific effects following a SAR process, inorder to take into account for the spatial dependence among small areas. This process isexpressed as (Anselin, 1992):
v = φWv + u ⇒ v = (Im − φW)−1 u, (4)
where φ is a spatial autoregressive coefficient that defines the strength of the spatialrelationship among the random effects associated with neighboring areas and W = {wij}(m × m) is a known spatial proximity matrix that indicates whether the small areas areneighbors or not (i,j = 1, . . . , m). Further, u = col1≤i≤m (ui) (m × 1) is a vector ofindependent error terms with zero mean and constant unknown variance σ 2
u . Under model(3), and assuming that the variance components are known, ψ = (σ 2
u , φ)′, the BLUP of θi
is given by:
θi (ψ) = x′i β+ m′
i�(y − Xβ
), (5)
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Comparison of the Performance of EBLUP Estimators 675
where m′i = (0, 0, . . . , 0, 1, 0, . . . , 0, 0) (1 × m) is a zero vector with 1 in the ith position,
� = σ 2u B−1V−1 with B = (Im − φW)′ (Im − φW), and Im is an identity matrix of order m.
The variance components have been estimated through likelihood-based approaches in orderto obtain an EBLUP of θi (Chandra et al., 2007; Petrucci et al., 2005; Petrucci and Salvati,2006; Pratesi and Salvati, 2008, 2009; Salvati, 2004). Furthermore, the uncertainty ofEBLUP has been measured by asymptotic analytical estimators based on Prasad and Rao’s(1990) Taylor linearization approach (Pratesi and Salvati, 2008; Salvati, 2004). Recently,Molina et al. (2009) proposed parametric and nonparametric bootstrap procedures forMSPE of the spatial EBLUP.
2.3. Model with Temporally Autocorrelated Random Effects
Rao and Yu (1994) proposed an extension of the Fay–Herriot model that considers temporalautocorrelation in order to take advantage of data at other time points. The linear mixedmodel proposed by Rao and Yu (1994) is:
yit = x′itβ+ vi + uit + εit , (6)
where vi denotes the random small-area effect, uit refers the random area-by-time specificeffect, and εit represents the sampling error associated with yit . Rao and Yu (1994) proposedthe specification of autocorrelated random area-by-time effects following an AR(1) processfor each small area, in order to borrow strength across time. This process is expressed as:
uit = ρui,t−1 + ξit , |ρ| < 1, (7)
where ξit ’s are error terms such that ξit
iid∼ N (0, σ 2) and ρ is a temporal autoregressive co-efficient that measures the level of chronological autocorrelation. The errors {εit }, {vi}and {uit } are assumed to be mutually independent such that εit
ind∼ N (0, σ 2it ) and
vi
iid∼ N (0, σ 2v ), with known σ 2
it . Following Rao and Yu (1994) and assuming that ψ =(σ 2
v , σ 2, ρ)′ is known, the BLUP estimator of θit is given by:
θit (ψ) = x′it β+ (σ 2
v 1T + σ 2γt )′V−1
i (yi − Xi β), (8)
where Xi = col1≤t≤T (x′it ) (T × p), Vi = σ 2� + σ 2
v JT + Ri (T × T) is thevariance–covariance matrix of yi = col1≤t≤T (yit ) (T × 1) with Ri = col1≤t≤T (σ 2
it ) (T ×T). Furthermore, 1T (T × 1) is a column vector of 1’s, JT = 1T 1′
T , and γt is the tth row of� = {ρ|r−s|/(1 − ρ2)}, r,s = 1, . . . , T . Rao and Yu (1994) provided method of momentsestimators of σ 2
v and σ 2, through an extension of Henderson’s third method (Henderson,1953), and proposed a naıve estimator of ρ. However, their main research has focusedon derivation of an EBLUP estimator of θit and an asymptotic analytical estimator of itsMSPE along the lines of Prasad and Rao (1990), assuming known ρ. Most recently, Pereiraand Coelho (2010a) studied the use of resampling methods for MSPE estimation of thatEBLUP.
2.4. Model with Spatially and Temporally Autocorrelated Random Effects
Combining the previous spatial and temporal correlation structures of random effects, otherauthors (SAMPLE Project, 2010) proposed a linear mixed model with a spatio-temporalstructure. In matrix form, their model can be written as:
y = Xβ+ Zυ+ ε, (9)
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where y = col1≤i≤m;1≤t≤T (yit ) (mT × 1), X = col1≤i≤m;1≤t≤T (x′it ) (mT × p), Z = [ Z1 ImT ]
[mT × (m+mT)] with Z1 = Im ⊗ 1T , and ε = col1≤i≤m;1≤t≤T (εit ) (mT × 1). Furthermore,υ = [ v′ u′
2 ]′ (mT × 1) where v = col1≤i≤m(vi) (m × 1) is a vector of random area specificeffects and u2 = col1≤i≤m;1≤t≤T (u2,it ) (mT × 1) is a vector of random area-by-time specificeffects. Those authors proposed the specification of random area specific effects followinga SAR process (Anselin, 1992):
v = φWv + u1 ⇒ v = (Im − φW)−1 u1, (10)
where φ is a spatial autoregressive coefficient and u1 = col1≤i≤m(u1i) (m × 1) is a vector ofindependent error terms such that u1
iid∼ N (0; σ 2u Im). In addition, they proposed the specifica-
tion of random area-by-time specific effects following an AR(1) process for each small area:
u2,it = ρu2,i,t−1 + ξit , |ρ| < 1, (11)
where ξit ’s are the error terms satisfying ξit
iid∼ N (0; σ 2) and ρ is a temporal autoregressivecoefficient. Furthermore, the authors assumed that error terms v = (Im − φW)−1u1, u2
and ε are mutually independently distributed as v ∼ N (0; σ 2u B−1), u2 ∼ N (0; σ 2Im ⊗ �),
and ε ∼ N (0; R), where B = (Im − φW)′(Im − φW), � = {ρ|r−s|/(1 − ρ2)}, r,s = 1, . . . ,T , and R = diag1≤i≤m;1≤t≤T (σ 2
it ). Assuming first that ψ = (σ 2, σ 2u , φ, ρ)′ is fully known,
the BLUP of θit is given by:
θit (ψ) = x′it β+ h′
iV−1(y − Xβ), (12)
where h′i (1 × mT) is a vector that captures the potential spatial and temporal auto-
correlations present in the ith small area and V = R + Z1σ2u B−1Z′
1 + σ 2Im ⊗ �. Further,h′
i = σ 2u ς
′i⊗1′
T +σ 2ζ′it where ς′i = {ςii′} is the tth row of the B−1 matrix and ζ′it (1 × mT) is avector with m T-dimensional blocks, with the tth row of the � matrix, γt , in the ith block andnull vectors, 01×T , elsewhere, i,i’ = 1, . . . , m; t = 1, . . . , T . In order to obtain an EBLUPof θit , residual maximum likelihood estimators of the variance components can be used(SAMPLE Project, 2010). Note that models (1), (3), and (6) are special cases of model (9).
Following Kackar and Harville (1984) and assuming the normality of the errors, theMSE of the BLUP can be expressed as:
MSE[θit (ψ)] = g1it (ψ) + g2it (ψ) + E[θit (ψ) − θit (ψ)]2, (13)
where the first term, g1it (ψ), measures the uncertainty due to the estimation of υ and is oforder o(1), the second term, g2it (ψ), represents the variability due to the estimation of βand is of order o(m−1) for large m, and the last term measures the uncertainty due to theestimation of ψ. The first two terms can be analytically evaluated from the following exactexpressions, using Henderson’s general results (Henderson, 1975):
g1it (ψ) = σ 2u ςii + σ 2
1 − ρ2− (
σ 2u ςi ⊗ 1T + σ 2ζit
)′V−1 (
σ 2u ςi ⊗ 1T + σ 2ζit
), (14)
g2it (ψ) = [xit − X′V−1(
σ 2u ςi ⊗ 1T + σ 2ζit
)]′(X′V−1X)−1
× [xit − X′V−1(
σ 2u ςi ⊗ 1T + σ 2ζit
)]. (15)
However, the last term of Eq. (13) is intractable, due to the nonlinearity of the EBLUPin the vector y, and therefore it is necessary to approximate it. We have obtained a Taylor
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approximation of that term assuming known autoregressive coefficients and following theresults of Kackar and Harville (1984) and Prasad and Rao (1990):
E[θit (ψ) − θit (ψ)]2 ≈ tr[Lit (ψ)V(ψ)L′it (ψ)V(ψ)] = g3it (ψ), (16)
where Lit (ψ) = ∂b′it (ψ)∂ψ
with b′it (ψ) = (σ 2
u ς′i ⊗ 1′
T + σ 2ζ′it )V−1 and V(ψ) denotes theasymptotic covariance matrix of ψ. After the calculation of the derivatives and definingAit (ψ) = Lit (ψ)V(ψ)L′
it (ψ) = {akl}, we have the following elements:a11 = [ζit −(Im⊗�)V−1(σ 2
u ςi⊗1T +σ 2ζit )]′V−1[ζit −(Im⊗�)V−1(σ 2
u ςi⊗1T +σ 2ζit )],a22 = [ςi ⊗1T − (Z1B−1Z′)V−1(σ 2
u ςi ⊗1T +σ 2ζit )]′V−1[ςi ⊗1T − (Z1B−1Z′)V−1(σ 2
u ςi ⊗1T + σ 2ζit )], and a12 = a21 = [ζit − (Im ⊗ �)V−1(σ 2
u ςi ⊗ 1T + σ 2ζit )]′V−1[ςi ⊗ 1T −
(Z1B−1Z′)V−1(σ 2u ςi ⊗1T +σ 2ζit )]. So, in practical applications, one can use the following
estimator of the MSPE of the spatio-temporal EBLUP:
mspe[θit (ψ)] = g1it (ψ) + g2it (ψ) + 2g3it (ψ). (17)
3. A Monte Carlo Simulation Study
A design-based simulation study was conducted to investigate the performance of the fourEBLUP estimators presented in the previous section in the context of a real population anda realistic sampling design. These are the Fay–Herriot estimator (FH), the spatial estimator(Sp), the temporal estimator (Tp), and the spatio-temporal estimator (ST).
3.1. Data
We have used real time-series data from PHTS and PBEHS, which are quarterly surveyscarried out by the Portuguese statistical office. The PHTS is a survey that provides individualhabitation prices for the effective transactions, while the PBEHS is a survey that providesindividual bank evaluation habitation prices. The target parameter of interest was definedas the mean price of habitation transaction by square meter, for each NUTSIII at each timepoint (NUTSIII is an administrative division that segments the state into 28 regions). In allconsidered models, we have used the mean price of bank evaluation of the habitations asthe auxiliary variable. The data were available on a quarterly basis from seven time points(first quarter 2002–third quarter 2003). In this study, 28 NUTSIII were used as small areasof interest.
The sample data of PHTS were used to compute, for each NUTSIII at each quarter,the direct survey estimate of the mean price of habitation transaction (yit ) and its samplingvariance (σ 2
it ), i = 1, . . . , 28; t = 1, . . . , 7. From the PBEHS, we have obtained data tocompute the mean price of bank evaluation (xit ) for the same level. In addition, our data canbe regarded as lattice data. In this way, the centroids of the NUTSIII were taken as spatialreference points and the proximity matrix W was obtained from the neighborhood structureof the 28 NUTSIII in Portugal. Matrix W = {wij } was constructed as follows: wij = 1 ifthe NUTSIII i shares an edge with NUTSIII j and wij = 0 otherwise. Afterwards, the rowsof W were standardized so that the row elements sum up to 1. The row standardized Wmatrix was kept fixed in all replicates.
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3.2. Exploratory Analysis
We have started the exploratory analysis by studying the linear relationship between theestimates of the mean price of habitation transaction, yit , and the mean price of bankevaluation, xit , i = 1, . . . , 28; t = 1, . . . , 7. The results of a linear regression on these datashowed that yit linearly varies with xit , since the ordinary least squares (OLS) estimate ofthe regression coefficient had a p-value less than 0.001 and R2 = 0.65.
Other question of essential importance in the estimation of LMM is whether randomeffects are relevant. The significance of the random effects variance component was assessedby an exact F-test (see Demidenko, 2004), which is adequate for small samples. The nullhypothesis of this test is H0 : σ 2
υ = 0, and its statistic is given by:
(SWLS − Smin)/(k − p)
Smin/(n − k)∼ F(k−p;n−k), (18)
where SWLS is the weighted residual sum of squares under H0, Smin is the weighted residualsum of squares arising from the LMM in expression (1), k = r([XZ]), p is the number ofregression coefficients (here p = 1), and n is the sample size (in our case n = 28 for eachtime period). That test was performed for each time point and the resulting p-value was lessthan 0.001 in all cases. These results support the relevance of the random effects.
Next, we have checked the existence of autocorrelation in our data using an approachoften recommended in the literature (Verbeke and Molenberghs, 2000). The main ideais to analyze the OLS residuals of standard linear regressions to our data for each timepoint. Figure 1 shows box-and-whisker plots of these residuals by time point. The analysisof this figure suggests that there is within-area autocorrelation because we can observe aslight positive trend in the residuals average and a stable variability. This last evidence wascorroborated by the results of the Levene’s test ( p-value = 0.984). Moreover, the resultsof the Kolmogorov-Smirnov test ( p-value = 0.010) revealed deviations from normality forthe transformed residuals, which suggests that the correlation function is not adequately
Figure 1. Box-and-whisker plots of OLS residuals by time point.
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Table 1Spatial dependence in the target variable for each time point
Moran’s I Geary’s C
t Value p-value Value p-value
1 0.299 0.002 0.557 0.0012 0.318 0.001 0.537 0.0013 0.251 0.006 0.496 0.0004 0.245 0.007 0.578 0.0025 0.338 0.001 0.455 0.0006 0.309 0.001 0.493 0.0007 0.291 0.002 0.475 0.000
modeled (Verbeke and Molenberghs, 2000). We should note that the existence of autocor-relation in the data was in line with our expectations. Indeed, it is natural that the meanprice of habitation transaction and the mean price of bank evaluation of the habitations fora specific NUTSIII are strongly correlated for adjacent time periods than for time periodswith a lag greater than 1. This reasoning was corroborated by the results of the well-knownDurbin-Watson test (DW = 0.789). The first-order autocorrelation coefficient, estimatedthrough the OLS residuals, was equal to 0.37.
After that, spatial dependence in the mean price of habitation transaction was checkedby computing two standard global spatial statistics: Moran’s I and Geary’s C (Cliff andOrd, 1981; Moran, 1950). The results presented in Table 1 reveal a modest positive spatialdependence, but still statistically significant, in the target variable for each time point.
Finally, we have evaluated the errors of the four models used in this research.This model diagnostics was performed by checking the normality of the transformederrors using adequate transformation matrices obtained by the method proposed byFuller and Battese (1973). In particular, the normality of the transformed random effects,υ∗ = G−1/2υ
◦∼ N (0; I), and the transformed sampling errors, ε∗ = R−1/2ε◦∼N (0; I), was
examined by normal quantile–quantile (Q–Q) plots. Figure 2 reports the Q–Q plots of thetransformed errors of longitudinal models (6) and (9) while Fig. 3 shows Q–Q plots for thelast quarter of cross-sectional models (1) and (3). Results for all quarters of cross-sectionalmodels were also produced but in the economy of space are not reported here. We can ob-serve from Figs. 2 and 3 that there are no significant departures from the normal assumptionof the errors.
3.3. Design of the Simulation Study
For the simulation experiment, a total of L = 1,000 Monte Carlo samples were drawn froma pseudo-population of 4,655 companies of real estate mediation. We have replicated areal sample of 458 companies proportional to their sampling weight, in order to obtain thepseudo-population. Samples of 229 companies were drawn independently for each timepoint using a stratified cluster sampling without replacement. It was assumed that PHTSwas implemented according to a panel and the population of companies was stratifiedaccording to geographic regions (NUTSIII, municipalities) and gross sales classes. Thecompanies were selected through simple random sampling and all the secondary units
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Figure 2. Normal Q–Q plots to check the normality of the transformed errors of longitudinal models.
(price of habitation transactions) within each selected company were observed. Althoughthe average real sample size of companies was 458 per wave, we have decided to usesamples of 229 companies of real estate mediation per wave in order to compare theperformance of the four EBLUP estimators in critical situations, that is, when they are usedin areas with very small sample sizes. The pseudo-population and sample sizes were keptfixed in all replicates. The whole simulation experiment was carried out with the use ofnew procedures in SAS.
3.4. Evaluation Measures
For the evaluation of the estimators’ performance, we have adopted a commonly usedapproach based on the following global design-based measures: average absolute relativebias (AARB), average relative root mean square error (ARRMSE), and average relative
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Figure 3. Normal Q–Q plots to check the normality of the transformed errors of cross-sectionalmodels.
efficiency (ARE). They are given by:
AARB = 1
mT
m∑i=1
T∑t=1
∣∣∣∣∣ 1
L
L∑l=1
μe(l)it − μit
μit
∣∣∣∣∣, (19)
ARRMSE = 1
mT
m∑i=1
T∑t=1
√MSE(μe
it )
μit
, (20)
ARE = 1
mT
m∑i=1
T∑t=1
[MSE
(μFH
it
)MSE
(μe
it
)]1/2
, (21)
where μe(l)it is the mean estimate for small area i at time period t produced by estimator e, e ∈
{FH ; Sp; Tp; ST }, using sample replicate l (l = 1, . . . , L), μit is the empirical value of thetrue mean for small area i at time period t, and MSE(μe
it ) = L−1 ∑Ll=1 (μe(l)
it − μit )2. Notethat AARB measures the bias of an estimator, whereas ARRMSE measures its accuracy,both expressed as percentages. To summarize results, we have produced global measuresfor groups of small areas. The 28 NUTSIII were divided into six mutually exclusive groupsbased on their expected sample size (secondary sampling units). Table 2 presents the sixgroups of small areas.
3.5. Empirical Results
The main results of the design-based simulation study are summarized in Table 3, whichcontains the values of AARB, ARRMSE, and ARE obtained for the four EBLUP estimators.
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Table 2Groups of small areas
Group Mean sample dimension Number of small areas
1 1 transaction 22 2–5 transactions 43 6–9 transactions 34 10–19 transactions 85 20–29 transactions 56 At least 30 transactions 6
Comparing the AARB the ARRMSE measures, among the four EBLUP estimators,we observe lower levels of bias and better levels of accuracy when an estimator based oncorrelated random-area effects is used than when the Fay–Herriot estimator is used. This istrue for both the spatial (Sp) and the temporal (Tp) estimators, although bias and accuracygains tend to be more significant for the latter. This fact suggests that the introduction oftemporal autocorrelation in small-area estimation models has a better effect on the bias andefficiency of the estimators than the introduction of spatial correlation. This is probably dueto a better ability of the temporal model to borrow strength from related small areas thanthe spatial model when both types of correlation among the neighboring areas are modest.Taking an insight analysis into the evaluation measures reported in Table 3, we see thatthe overall AARB of the spatial and the temporal estimators are considerably lower thanthe overall AARB of the Fay–Herriot estimator (4.47% and 3.36%, respectively, comparedto 6.18%). However, we notice that the overall ARRMSE of those two estimators arejust slightly lower than the corresponding overall measure for the Fay–Herriot estimator(7.28% and 6.43%, respectively, compared to 7.81%). The spatio-temporal estimator (ST)exhibits results that outperform all other included estimators, showing that the simultaneousconsideration of spatial and temporal correlation has the ability of still improving both biasand accuracy, when compared to estimators that include only one of these dependences. Forthis estimator, the overall AARB and ARRMSE are equal to 2.25% and 5.84%, respectively,while these measures are equal to 6.18% and 7.81% for the Fay–Herriot estimator. FromTable 3, we can also see that all estimators have the same behavior in terms of AARB and
Table 3Performance measures of bias, accuracy, and relative efficiency for all estimators
AARB(%) ARRMSE(%) ARE
Group μFHit μ
Spit μ
Tpit μST
it μFHit μ
Spit μ
Tpit μST
it μFHit μ
Spit μ
Tpit μST
it
1 10.10 5.52 3.39 2.31 11.59 6.47 4.70 3.31 1.00 3.79 3.49 6.422 12.90 8.76 8.09 6.82 14.90 14.44 12.30 11.66 1.00 1.12 1.25 1.383 6.10 4.29 3.57 2.50 7.37 6.55 6.02 5.31 1.00 1.24 2.08 4.834 5.42 4.76 3.04 1.55 7.50 7.63 6.47 5.61 1.00 1.04 1.33 1.605 3.97 2.75 1.77 1.20 5.26 5.10 4.99 5.00 1.00 1.04 1.07 1.066 3.29 2.38 1.86 0.89 4.58 4.47 4.44 4.08 1.00 1.07 1.06 1.15Overall 6.18 4.47 3.36 2.25 7.81 7.28 6.34 5.84 1.00 1.28 1.45 2.07
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ARRMSE when sample size increase: the bias tends to decrease and the accuracy has apropensity to improve with the increase of the sample size.
Looking at the efficiency measure, we can observe from Table 3 that gains are alwaysachieved when an estimator based on correlated random-area effects is used over theFay–Herriot estimator. Although the gains in efficiency of the spatial and the temporalestimators are generous (the overall average gain in efficiency is equal to 28% and 45%,respectively, for spatial and temporal estimators), the gains attained by the spatio-temporalestimator are more substantial mainly for Groups 1, 3, and 4. The average gain in efficiencyassociated with this latter estimator compared with the Fay–Herriot estimator is about107%. Let us also observe from Table 3 that the relative efficiency of the spatial and/ortemporal estimators (when compared to the Fay–Herriot benchmark) tends to decrease withthe increase of sample size. Particularly, these gains are small for larger areas in terms ofexpected sample size (Groups 5 and 6). Nevertheless, the bias and accuracy gains of thespatio-temporal (ST) estimator when compared with the spatial (Sp) and the temporal (Tp)estimators tend to be stable and consistent for all groups of areas despite the expectedsample size. All these results confirm the superiority of the spatio-temporal estimator, incomparison to other well-known EBLUP estimators, for estimating the mean price of thehabitation transaction in Portugal.
4. Conclusion
There are some excellent papers recently published in which were proposed theoreticalsmall-area methodologies to explore all kinds of available auxiliary information throughthe use of LMM. The underlying idea of proposed model-based methodologies is to “bor-row information” from related small areas, past samples, or both, apart from that givenby auxiliary variables. Most of the proposed LMM with spatially or/and temporally au-tocorrelated random effects have been validated by the results of synthetic model-basedsimulation studies. Yet, few studies have shown the applicability of these models with realdata and from a design-based perspective. Therefore, more work is needed to understandif there are efficiency gains when a spatial dependence or/and a temporal autocorrelationamong random-area effects are included into the LMM, based on design-based simulationstudies using real data and realistic sampling designs. This was the main motivation of thisresearch.
In this article, we have reviewed four area-level LMM that borrow strength by exploitingthe possible correlation among the neighboring areas or/and past time periods. Furthermore,we have noted that the main reason for using this kind of small area LMM is to make thebest use of the available spatial or/and temporal auxiliary information in order to use themost efficient estimator in real-life applications.
In a design-based simulation study, we have compared the performance of three EBLUPestimators based on correlated random-area effects (spatial, temporal and spatio-temporal)with the well-known Fay–Herriot EBLUP estimator. Our empirical results have showedthat EBLUP estimators with spatially or/and temporally autocorrelated random effects out-perform the Fay–Herriot EBLUP estimator in terms of bias, accuracy, and efficiency. Inparticular, our results have confirmed the superiority of the spatio-temporal estimator, incomparison to other well-known EBLUP estimators. As expected, these gains are partic-ularly conspicuous for smaller areas in terms of expected sample sizes. Globally, resultsshow that the use of “borrowed information” from related small areas and past samples atthe same time, apart from that given by auxiliary variables, strengths small-area estimates.
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In addition, it is important to note that our findings are consistent with other achievedin design-based simulation studies based on real data. Fabrizi et al. (2007) have concludedthat, in general, estimators connected to unit-level LMM that take temporal autocorrelationinto account lead to slight gains in efficiency over an estimator based on a model with inde-pendent time and area effects (cross-sectional model). Likewise, Chandra et al. (2007) haveobserved only marginal gains when spatial dependence among small areas is introducedinto unit-level small area models. Finally, Singh et al. (2005) have also confirmed, but undera model-based simulation study based on real data, the superiority of an estimator basedon an area-level spatio-temporal state-space model in comparison to estimators deductedfrom other state-space models (cross-sectional, spatial, and temporal).
Finally, it should be noticed that this study is not free of limitations. In particular,caution is advised when trying to extrapolate the conclusions of the study, as they may bepartially contingent on the: (1) framework of the application, (2) the intensity of spatialand temporal association exhibit by data, and (3) the type of covariance structures that arechosen to model this association. Therefore, further applications and simulations of thesearea-level models with other types of real data are to be encouraged.
Acknowledgments
The authors acknowledge the Portuguese statistical office for the availability of the dataused in the research. The views expressed here are solely those of the authors. The authorsalso thank the Fundacao para a Ciencia e a Tecnologia (Portugal) for financial support.
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