stata软件实现随机前沿估计方法

• The model the underpredicts costs the firm money. The model the overpredicts might be catastrophic.
• Frontier estimation tries to fix this problem. However, not all data are conduce to SFA. • Uses: Production functions, cost functions, demand models, test of union effectiveness,
• We can imagine technological change over time which would be a shift in the production frontier
• Using multiple inputs, the picture changes a little:
• Allocative efficiency is measured as ON/OM ≤ 1. • Overall efficiency is measured as (OM/OP )(ON/OM ) = (ON/OP )
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History
• Farrell (1957, Journal of the Royal Statistical Society, Series A): Derives a production function approach and identifies two sources of firm inefficiency/efficiency 1. Technical Efficiency: Produce the most output with a given level of inputs 2. Allocative Efficiency: Produce a given output as cheaply as possible.
agency costs, reservation wages, school outcomes, profitability, survivorship, merger and acquisition analysis, effect of shadow inputs such as corruption. • Consider the traditional production function :
qo = f (xi; β) − ui
• To estimate this type of model, one could use a fixed effects model where ui was treated as the firm fixed effects.
• Let’s assume:
f (xi; β) = β0X1β1X2β2 · · · Xkβk lnf (xi; β) = lnβ0 + β1lnX1 + β2lnX2 + · · · βklnXk
efficient firm. • However, we never know the fully efficient production function • Farrell suggested estimating a fully efficient production function. There are two ways to
• Most of the time we focus on technical efficiency in the explanation • To determine if a firm is efficient, we have to know the production function of the fully
• Econometrics via traditional OLS would gather Price and Quantity data and estimate an average demand curve. At any Q0 the model predicts P0 but the firm could actually change PA. The average OLS approach might over or under predict price.
• Aigner, Lovell and Schmidt (1977) and van den Broeck (1977) both introduced a way to deal with SFA and production functions.
Basic Setup
• Consider a production function qi = f (xi; β) where xi is a vector of inputs, qi is output, and β is a [k × 1] vector of parameters to be estimated.
=
qi exp(xiβ)
=
exp(xiβ − ui) exp(xiβ)
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where 0 < exp(−ui) ≤ 1. • While this is a decent shot at the problem it does leave a lot to be desired. • Mainly ui is supposed to measure inefficiency but it might also be capturing other random
do this: 1. Non parametric techniques: Data envelopment analysis. This technique assumes that all deviationsБайду номын сангаасfrom the efficient frontier is a realization of inefficiency 2. Parametric techniques: Stochastic Frontier Analysis. This technique assumes that deviations from the efficient frontier can be either a realization of inefficiency or a random shock.
Stochastic Frontiers
• In this section we take the maximum likelihood approach and apply it to a fairly useful and powerful tool - stochastic frontier estimation.
j=1
However, we cannot use OLS to get at the composite error term. • Some basic assumptions vi ∼ iidN (0, σv2), ui ≥ 0 and cov(vi, ui) = 0, where v = measure-
lnqi = β1lnX1 + β2lnX2 + · · · βklnXk − ui lnqi = βX − ui
• Aigner and Chu (1968) suggested a measure of technical efficiency of
ObservedOutput F rontierOutput
shocks that are beyond the control of the firm’s management. For example, would we want to hold the firm’s management for the impacts of Katrina or an earthquake or some other weather event? • Aigner, Lovell, and Schmidt (ALS) in 1977 suggested adding a two-sided error term to the one-sided error term of AC (1968). It doesn’t seem like that big of a deal, but it did take some work to derive the likelihood function. • Now, qi = f (xiβ)ζiexp(vi) which yields
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• We can think of efficiency being measured as ζi multiplied by the theoretical norm where ζi ∈ [0, 1] such that qi = f (xi; β)ζi
If ζi = 1 then the firm is fully efficient and produces the most it can. If ζi < 1 then the firm is not fully efficient. • We can let qi = f (xi; β) be the level of output that should happen. Let q0 be the observed output where q0 < qF because of inefficiency and other factors. • As q0 < qF = f (xi; β), Aigner and Chu (1968) suggested adding a non-negative random variable to f (xiβ) which would capture the technical inefficiency of firm i: