stata软件实现随机前沿估计方法

• Using multiple inputs, the picture changes a little:
• Point P is inefficient relative to Point M. Point A is both technically and allocatively efficiency. • We can measure Technical Efficiency as T E = OM/OP and Technical Inefficiency as 1 − T E = 1 − OM/OP . • Allocative efficiency is measured as ON/OM ≤ 1. • Overall efficiency is measured as (OM/OP )(ON/O 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. 3
• 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 (xi β ) which would capture the technical inefficiency of firm i:
Stochastic Frontiers
• In this section we take the maximum likelihood approach and apply it to a fairly useful and powerful tool - stochastic frontier estimation. • What’s the basic idea? How to estimate economic relationships that ought to be modeled as upper or lower frontiers rather than averages. • For example: Consider a demand curve. From the theory of demand, the demand curve is a frontier which tells the firm the most it can charge for the marginal unit.
lnqi = ln(f (xi β )) + ln(ζi ) + vi
• Defining ui = −ln(ζi ) yields
lnqi = ln(f (xi β )) + vi − ui
• In a Cobb-Douglas type world
k
lnqi = β0 +
j =1
βj ln(xji ) + vi − ui
However, we cannot use OLS to get at the composite error term.
lnf (xi ; β ) = lnβ0 + β1 lnX1 + β2 lnX2 + · · · βk lnXk lnqi = β1 lnX1 + β2 lnX2 + · · · βk lnXk − ui lnqi = βX − ui
• Aigner and Chu (1968) suggested a measure of technical efficiency of qi exp(xi β − ui ) ObservedOutput = = F rontierOutput exp(xi β ) exp(xi β )
• 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. • 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, agency costs, reservation wages, school outcomes, profitability, survivorship, merger and acquisition analysis, effect of shadow inputs such as corruption. • Consider the traditional production function :
1
• This is the single input case where q = f (X ). The slope of the ray from the origin is a measure of productivity q/X . Note that q/X at Point A is less than at Point B and Point C. • We can imagine technological change over time which would be a shift in the production frontier
2
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. • 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 efficient firm. • However, we never know the fully efficient production function • Farrell suggested estimating a fully efficient production function. There are two ways to do this: 1. Non parametric techniques: Data envelopment analysis. This technique assumes that all deviations from 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. • Aigner, Lovell and Schmidt (1977) and van den Broeck (1977) both introduced a way to deal with SFA and production functions.
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: