Spatial stochastic frontier model

Transcription

1 Spatial stochastic frontier model Svetlana Mladenovi and Vincenzo Atella December 16, 2013

2 Idea Propose spatial stochastic frontier model for panel data (SFM + spatial interactions) Deriving ML estimators of the model parameters Application to Hospital data

3 Stochastic frontier model Stochastic (production) frontier model is: y it = α + x itβ + ε it ε it = υ it u i where υ N(0, σ 2 υ) u i 0 truncated normal, half normal, exponential, gamma Estimated usually by ML

4 Stochastic frontier model Ineciency: time invariant u i or time-varying u it, modeled in dierent ways exogenous determinants of ineciency, usually not inputs nor outputs, for example: u i N + (µ i, σ 2 u) µ i = z i δ

5 Spatial models Measure spatial dependence between geographical units. Spatial eects through dierent channels General model y = ρwy + X β + γwx + υ υ = λw υ + ɛ Spatial weight matrix W with rows: w i = (n x 1) vector of spatial connections of unit i, elements w ij nonnegative, i-th element 0 w i y linear combination of all j neighbours of i, if w i normalized ( w j ij = 1) it is weighted average of neighbours' outcomes W taken as known specication of weights is research specic

6 Spatial models Identication problem Manski (1993): model parameters not identied, need to exclude at least one spatial eect in the model Additional assumptions on W and model parameters needed Identication is possible relies on strong assumption that W is correctly specied and that model specied is true data generating process

7 Spatial models Spatial autoregressive model (SAR) y it = ρ N w ij y jt + x itβ + υ it j=1 i = 1...N t = 1...T with spatial lag linear and constant across observations ρ shows causal eects of neighbours' outcomes on y i β shows causal eect of exogenous characteristics on y i

8 Spatial models Spatial lag of X model (SLX) N y it = x itβ + w ij x jtγ + υ it Spatial error model (SE) j=1 y it = x itβ + υ it N υ it = λ w ij υ jt + ɛ it j=1

9 Spatial models Spatial Durbin model (SAR + SLX) N N y it = ρ w ij y jt + x itβ + w ij x jtγ + υ it j=1 j=1 Spatial Durbin error model (SDE) (SLX+SE) N y it = x itβ + w ij x jtγ + υ it j=1 N υ it = λ w ij υ jt + ɛ it j=1

10 Spatial stochastic frontier models Stochastic frontier model augmented with spatial eects General form: y = ρw 1 y + X β + γw 2 X + ε ε = υ u υ = ξw 3 υ + υ u = ξw 4 u + λw 5 y + δz + ζw 6 z + ũ estimation by ML

11 Spatial stochastic frontier models Stochastic frontier model augmented with spatial eects Related literature: 1. Druska and Horace (2004): generalized moment estimation for panel data model with spatially correlated errors, applied to SF framework in agriculture 2. Jeleskovic,Schwanebeck (2012): 2 step estimation of FE spatial models 3. Scmidt et al. (2009): Bayesian approach to estimating stochastic frontier models with latent spatial structure 4. Lavado, Barrios (2008,2010): Stochastic frontier models with spatial autoregressive component in eciency for cross section and spatial-temporal component for panel data

12 Spatial stochastic frontier models 5. Auso (2010): Stochastic production frontier + spatial autoregressive term in production equation applied to cross section data in agriculture spatial eects: techological diusion, similar soil characteristics limitation: no estimation of spatial spillovers in ineciency spatial model perform better than conventional SFM 6. Brehm (2013): Link between scal decentralization and eciency using SF aproach with spatial eects in ineciency ineciency depends on public expenditures (education, infrastructure and administration) and their spatially lagged values as potential sources for regional clustering of ineciency u = zδ + λwz + ω

13 Spatial stochastic frontier models 7. Pavlyuk (2009,2010,2012): Spatial stochastic frontier models applied to regional touristm and airport competition Paper (2010): competition in regional tourism: spatial autoregressive term in production frontier and ineciency y = ρwy + X β + υ u u = λwy + δz + ũ Paper (2012): ML estimation for: y = ρwy + X β + υ u υ N or υ = ρw υ + υ cross-section data additional research needed for eects of spatial dependence in ineciency

14 Spatial stochastic frontier models Objectives: Applying existing models to panel data on hospitals measuring hospital eciency with spatial stochastic frontier model Change in theoretical model including spatial eects both in production and eciency equation using ML estimation instead of 2SLS

15 Spatial stochastic frontier models Model to start with: y = ρwy + X β + υ u u = δz + ζwz + ũ estimation by ML

16 Further Extending to Multiple output distance functions instead of single output stochastic frontier function

17 The end