Small domain estimation of business parameters using skew- normal models*

Small area estimation has been till now applied esp ecially to poverty and employment parameters, however the demand for reliable information for res tricted domains is growing also with reference to business parameters. The aim of this work is to provide estimates of business parameters for small domains defined by cross-classifying geograph ical areas, economic activities and firms' size. We use data on the Small and Medium Enterprises sample survey (1-99 employees) conducted by the Italian National Statistical Institute (Istat) in 2008. We focus on the manufacturing sector and o n two basic outcomes, value added and labor cost, whi ch are used to calculate important economic competitiveness indicators. As for many of the cons idered domains the number of sampled firms is too low to obtain reliable estimates using the "dir ect" estimation strategy currently employed by Istat, a small area method estimation is advisable. We use model-based small area estimators relying on area level models, that may be represent ed as a sampling model, which accounts for the sampling variability of the direct survey estimates , and a model linking the small domain parameters of interest to area-specific auxiliary d ata. We adopt a Hierarchical Bayesian approach to estimation. Some peculiar issues arise in business surveys (Cox et al., 1995; Riviere, 2002). One of the most relevant is the asymmetry of outcomes' distribution s due to the presence, above all in Italy, of a majority of small firms. To take into account of th is peculiarity, we relax the normality assumption of the classic Fay-Herriot model (Fay and Herriot, 1979) and consider the skew-normal distribution. The specification of a skew-normal distribution off ers some advantages with respect to other non- symmetric distribution, such as allowing for modeli ng zero and negative values. The skew-normal distribution has been already considered by Ferraz and Moura (2011) for the sampling error. We propose: i) to assign a skew-normal distribution also to the random area effects; ii ) a bivariate extension of such skew-normal models, which enables us to take into account of the high correlation between the target variables. Moreover, we control for firms' heterogeneity by including the information on the size (available from the ASI A administrative archive) as auxiliary variable. To estimate area level models direct estimates as w ell as their associated variance are necessary. Direct estimates are obtained by using a Horvitz-Th ompson estimator because the domains of interest are collections of strata. Instead, we use a bootstrap strategy to estimate variances and covariance between direct estimates, because the un availability of some design information hinders the replication of the calibration procedure used b y Istat. Results highlight the importance to take into accou nt of the asymmetry of data, as the skew-normal specification allows for a considerable gain in eff iciency, about 30% on average with respect to the direct estimator. Moreover, we find that it is not useful to specify a non symmetric distribution for the random effects when it is specified for the sam pling errors, while it is more effective to specify a bivariate distribution for the random effects the n for the sampling errors.