1 Enhancing Small Area Estimation Methods Applications to Istat’s Survey Data Ranalli M.G. ~ Università di Perugia D’Alo’ M., Di Consiglio L., Falorsi S., Solari F. ~ Istat Pratesi M., Salvati N. ~ Università di Pisa Q2008 ~ Rome, July 11 th

2 OUTLINE  Italian Labour Force Survey  Standard small area estimators for LFS  Small area estimators that incorporate spatial information  Model based direct estimator (MBDE)  Semi-parametric models (based on p-splines)  Experimental study  Analysis of results  Final remarks

3 Labour Force Survey description Labour Force Survey (LFS) is a quarterly two stage survey with partial overlap of sampling units according to a rotation scheme of type (2-2-2). In each province the municipalities are classified as Self-Representing Areas (SRAs) and the Non Self-Representing Areas (NSRAs). From each SRAs a sample of households is selected. In NSRAs the sample is based on a stratified two stage sampling design. The municipalities are the primary sampling units (PSUs), while the households are the Secondary Sampling Units (SSUs). For each quarterly sample about 1350 municipalities and 200,000 individuals are involved.

4 ■ Since 2000, ISTAT disseminates yearly LFS estimates of employed and unemployed counts related to the 784 Local Labour Market Areas (LLMAs). ■ LLMAs are unplanned domains obtained as clusters of municipalities cutting across provinces which are the LFS finest planned domains. ■ The direct estimates are unstable due to very small LLMA sample sizes (more than 100 LLMAs have zero sample size). SAE methods are necessary. ■ Until 2003, a design based composite type estimator was adopted. ■ Starting from 2004, after the redesign of LFS sampling strategy, a unit-level EBLUP estimator with spatially autocorrelated random area effects has been introduced. Small area estimation on LFS

5 Standard small area estimators – design based The GREG estimator is based on the standard linear model: Direct and GREG estimator and can be expressed as an adjustment of the direct estimator for differences between the sample and population area means of covariates The direct estimator is given by

6 Unit level Synthetic and EBLUP Standard small area estimators – model based The Synthetic estimator assumes a standard linear mixed model with unit- specific auxiliary variables, random area-specific effects and errors independently normally distributed and is given by The EBLUP estimator assumes the same model but is given by

7 Enhanced small area estimators 1. Unit level EBLUP with spatial correlation of area effects The matrix A depends on the distances among the areas and on an unknown parameter connected to the spatial correlation coefficient among the areas. The EBLUP-S estimator is based on the following unit level linear mixed model:

8 Enhanced small area estimators 2. Model Based Direct Estimator (Chambers & Chandra, 2006) where the weights are such that is the (E)BLUP of The MBD estimator is based on a unit level linear mixed model and is given by under the model (Royall, 1976). Calibrated with respect to the total of x. Reduces bias vs EBLUP Does not allow estimation for non-sampled areas Less efficient than EBLUP

9 In the literature there are many nonparametric regression methods (kernel, local polynomial, wavelets…) BUT difficult to incorporate in a Small area model Methods based on penalized splines (Eilers e Marx, 1996; Ruppert et al., 2003) can be estimated by means of mixed models -> promising candidate for SAE methods Enhanced small area estimators 3. Nonparametric EBLUP (Opsomer et al., 2008) Great Flexibility in definition of model Estimable with existing software using REML Hard to estimate efficiency and test for terms significance (via bootstrap?)

10 LFS empirical study The simulation study on LFS has been carried out to estimate the unemployment rate at LLMA level 500 two-stage LFS sample have been drawn from 2001 census data set. The performances of the methods have been evaluated for the estimation of the unemployment rate in the 127 LLMAs belonging to the geographical area “Center of Italy ”. GREG, Synthetic, EBLUP small area estimators have been applied considering two different sets of auxiliary variables Case A - LFS real covariates = sex by 14 age classes + employment indicator at previous census; Case B – LFS real covariates + geographic coordinates (latitude and longitude of the municipality the sampling unit belongs to).

11 ■ Spatial EBLUP: A spatial correlation in the variance matrix of the random effects has been considered (EBLUP SP) + Case A covariates ■ MBD: Model based direct estimation is performed on sampled LLMAs, while synthetic estimators based on unit level linear mixed model is considered for non sampled LLMAs (Case A covariates) ■ Nonparametric EBLUP: two semiparametric representations based on penalized splines have been applied (fitted as additional random effects): geographical coordinates of the municipality (EBLUP-SPLINE SP): this allows for a finer representation of the spatial component vs EBLUP SP (at municipality level instead of LLMA). age (EBLUP-SPLINE AGE & EBLUP SP-SPLINE AGE) Enhanced Small area estimators

12 Average Absolute RB: Average RRMSE: Maximum Absolute RB: Maximum RRMSE: Evaluation Criteria % Relative Bias: % Relative Root Mean Squared Error:

13 ESTIMATORAARB ARRMSEMARB MRRMSE DIRECT GREG A GREG B SYNTH A SYNTH B EBLUP A EBLUP B EBLUP SP MBD EBLUP-SPLINE SP EBLUP-SPLINE AGE EBLUP SP-SPLINE AGE Results – A: LFS covariates; B = A + geog. coord. mun.

14 Analysis of results  Area level estimators (not shown here) perform a little better in terms of Bias but much worse in terms of MSE.  The results of GREG, SYNTH and EBLUB in case B, when geographical information is considered in the fixed term, display better performances in terms of bias.  In terms of MSE standard estimators in case A outperform standard estimators in case B if the ARRMSE is considered as overall evaluation criteria, while better results are obtained in case B if MRRMSE is considered

15  EBLUP SP can be compared with the unit level EBLUP with geographical information included as covariates and the EBLUP-SPLINE SP. oEBLUP SP show better performances in terms of MSE, while the unit level EBLUP outperform the other estimators in terms of bias. oThe EBLUP-SPLINE SP displays performances in between the other estimators. Analysis of results  EBLUP-SPLINE AGE performs similarly to the unit level EBLUP in Case A oThe use of the age in a nonparametric way is an alternative use of auxiliary information. With respect to case A the model is more parsimonious.  As it was expected MBDE shows better results in term of bias and performs poorly in term of MSE than other SAE methods  The use of autocorrelation structure together with the spline on the variable age doesn’t improve the performances

16 Final remarks  Sensitivity to smoothing parameters’ choice in the splines approach has to be investigated.  The introduction of the sampling weighs should be considered to try to achieve benchmarking with direct estimates produced at regional level  The response in a 0-1 variable: a logistic mixed model is currently being investigated  The model group is a small portion of Italy (center); hence the area specific effects are smaller than they could be if an overall model was considered for all the country: the introduction of geographical information should be analyzed considering a larger model level group