EM Algorithm for the Stochastic Frontier Model

EM演算法在隨機邊界模型之應用

10.29963/TOJEB.200612.0005

張曉芬(Hsiao-Fen Chang)

隨機邊界模型 ； EM演算法 ； 單調遞減＼遞增函數 ； A stochastic frontier model ； EM algorithm ； confidence intervals ； monotone decreasing\increasing function

真理財經學報

15期（2006 / 12 / 01）

109 - 122

英文

一個隨機邊界模型定義如下： y(下标 it)=β0+x(下标 it)β+v(下标 it)-µ(下标 i) i=1,...N；t=1,...,T ε≡v(下标 it)-U(下标 i) y(下标 it)是第i公司，第t個時間點的生產值取對數值(logarithm)。x(下标 it)是投入變數的向量。效率這個要素(efficiency component µ(下标 i)≥0)是一個非負的常態慨。無效率是只公司的產出在邊界以下。v(下标 it)是一個不可觀測的隨機變數。模型的假設是： v(下标 it)~N(0,σ^2) µ(下标 i)~N(上标 +)(0,σ(上标 2 下标 µ))在建造各別生產效率µ(下标 i)的信賴區間有幾種估計過程： 1)JLMS和MC方法 2)MCB方法 本論文將應用EM演算法來求σ(上标 2 下标 *)和µ(上标 * 下标 it)的最大概似估計值。

A stochastic frontier model in a panel data is written as y(subscript it) =β0+x(subscript it)β+v(subscript it)-µ(subscript t) i=1,...N; t = 1,...,T ε(subscript it)≡v(subscript it)-µ(subscript t) Where y(subscript it) is the logarithm of the output of the ith firm and tth time periods, x(subscript it) is a vector of input. The efficiency component (µ(subscript i)≥0) is a one-sided, non-negative error, derived from a half-normal distribution. Technical inefficiency exists to the extent that a firm's output lies beneath the frontier. The stochastic component v(subscript it) is an unobservable random variable (a statistical noise). The model assume the following: v(subscript it)~N(0,σ^2) µ1~N(superscript +)(0,σ(superscript 2 subscript µ) There are several estimation techniques procedures about construction of confidence intervals of the individual producer's inefficiency µ(subscript i) 1) JLMS and MC method 2) MCB method In this paper, we consider the application of the EM algorithm for ML estimation of the parameters and µ(superscript * subscript it) and σ(superscript 2 subscript *).

1. Aigner, Dennis,C.A Knox Lovell,Peter Schmidt(1977).Formulation and Estimation of Stochastic Frontier production Function Models.Journal of Econometrics,6,21-37.
2. Dempster, AP.,N.M. Laird,D.M. Rubin(1977).Maximum Likelihook from Incomplete Data via the EM Algorithm.(with dicussion) Journal of the Royal Statistical Society B,39,1-38.
3. Horrace, William C.,Peter Schmidt(1996).Confidence Statements fro efficiency Estimates from Stochastic Frontier Models.Journal of Productivity Analysis,7,257-282.
4. Jerry, AH.,William, E.T.(1981).Panel Data and Unobservable Individual Effects.Econometrica,49,1377-1398.
5. Mundlak, Y.(1978).On the Pooling of Time Series and Cross Section Data.Econometrica,46,69-86.
6. Park, B.U.,L. Simar(1994).Efficient Esmiiparametric Estimation in a Stochastic Frontier Model.Journal of the American Statistical Association,89,929-936.
7. Schmidt, Peter(1978).On the Statistical Estimation of Parametric Frontier Production Functions: Rejoinder.The Review of Economics and Statistics,481-482.