生產(chǎn)率與效率分析lecture8-隨機(jī)前沿效率1(完整版)
【正文】 cept not consistent ?ML estimator yields ?more efficient b (than OLS) ?consistent intercept ?consistent variance of (vi ui) 23 Stochastic Frontier (cont.) ?Aigner, Lovell, Schmidt (1977) ?derived loglikelihood function for model with posed error (vi ui ) ?v . N(0,?v2) ?u . truncations (at zero) N(0,?u2) ?“halfnormal” ?variance parameters ??2 = ?v2 + ?u2 ?? = [?u / ?v ] ? 0 24 Stochastic Frontier (cont.) ?Battese and Cora (1977) ?reparameterized loglikelihood function with posed error (vi ui ) ?variance parameters ??2 = ?v2 + ?u2 ? ? = (?u2 / ?2 ) ?[0,1] ?vs. ? = [?u / ?v ] ? 0 What is causing deviations from frontier when ? = 0? Deviations from frontier due entirely to noise 25 Stochastic Frontier (cont.) ?Battese and Cora (1977) ?reparameterized loglikelihood function with posed error (vi ui ) ?variance parameters ??2 = ?v2 + ?u2 ? ? = (?u2 / ?2 ) ?[0,1] ?vs. ? = [?u / ?v ] ? 0 What is causing deviations from frontier when ? = 1? Deviations from frontier due entirely to inefficiency 26 Battese and Cora (cont.) ?variance parameters ??2 = ?v2 + ?u2 ? ? = (?u2 / ?2 ) ?[0,1] ?this parameterization has advantage that can search for values of ? over [0,1] as start value for iterative maximization step 27 Battese and Cora (cont.) ln L = (N/2) ln(?/2) (N/2)ln(?2 ) + ? ln[1?(zi)] (1/2 ?2 ) ? (ln yi xi ?)2 where zi = [(ln yi xi ?) / ?] ? (? / 1 ?) and ?(.) is distribution function of N(0,1) ?maximize ln L to obtain ML estimates of ?, ?2 and ? (K+2) 28 Battese and Cora (cont.) ?Parameters: ?, ?2 and ? ?Three steps of maximization ?1. OLS to get start values for ?, ?2 ??0 ?2 estimates biased ?2. evaluate ln L for values of ? ?[0,1] ?3. use values for ?, ?2 and ? from first two steps in iterative maximization until converge (you hope!) 29 Estimating Mean TE ?Suppose u . halfnormal E[exp(ui)] = 2[1 ?(? ? ? )] exp( ??2/2). ?Substitute ML estimates of ? and ?2 into this ?This only gives mean TE across the sample
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