Topic 4e

Stochastic Frontier Production Function

Readings and References:

W. H. Greene, Econometric Analysis, 5th Ed., Chapter 17: Maximum Likelihood Estimation, Prentice-Hall, 2002.
S. C. Kumbhakar and C. A. K. Lovell, Stochastic Frontier Analysis, Cambridge University Press, 2000.
D. Aigner, C. A. K. Lovell, and P. Schmidt, "Formulation and Estimation of Stochastic Frontier Production Function Models," Journal of Econometrics, 1977, 21-37.
W. Meeusen and J. van den Broeck, "Efficiency Estimation from Cobb-Douglas Production Function with Composed Errors," International Economic Review, 1977, 435-444.
J. Jondrow, I. Materov, K. Lovell, and P. Schmidt, "On the Estimation of Technical Inefficiency in the Stochastic Frontier Production Function Model," Journal of Econometrics, 1982, 233-238.

Consider a production function with technical efficiency:

y = h(X,b)k

where y is output and h(X,b) is the production function with X consisting of factor inputs. The efficiency measure is defined by k = exp(-u) so that 0 < k £ 1 for u ³ 0. Therefore, in logarithm and with added random error term v, the regression model is written as:

ln(y_i) = ln h(X_i,b) - u_i + v_i (i = 1,2,...,N)

Let e_i = v_i - u_i. The simple case of Cobb-Douglas production function with technical efficiency is

ln(y_i) = ln(X_i)b + e_i and e_i = v_i - u_i

We assume that v_i is i.i.d. normal(0,s_v²) and distributed independently of u_i and the regressors. That is,

f(v_i) = 1/(2p)^½ 1/s_v exp{-½ (v_i/s_v)²}

Because of the requirement u_i ³ 0, several model specifications for technical efficiency are presented below. Depending on the density function of u_i and the fact e_i = v_i - u_i, we define the marginal density of e_i as

f(e_i) = ò₀^¥ f(u_i,e_i+u_i)du_i

where f(u_i,v_i) = f(u_i,e_i+u_i) is the joint density of u_i and v_i = e_i+u_i. It is clear that e_i is not normally distributed. The model parameters of the production function and the distribution function are estimated by maximizing the non-normal log-likelihood function for a sample of N producers defined by

ll = å_i=1,2,...,Nln f(e_i)

Model Estimation: Maximum Likelihood

Assume v ~ i.i.d. Normal(0,s_v²).

Half Normal Model: u_i ~ i.i.d. Normal⁺(0,s_u²)
The half-normal density of u is given by
f(u) = 2/(2p)^½ 1/s_u exp{-½ (u/s_u)²}
Given the independence assumption, the joint density function of u and v is
f(u,v) = 2/(2ps_vs_u) exp{-½ (u/s_u)² -½ (v/s_v)²}
Since e = v-u, the joint density function of u and e is
f(u,e) = 2/(2ps_vs_u) exp{-½ (u/s_u)² -½ ((e+u)/s_v)²}
The marginal density of e is obtained by integrating u out of f(u,e), which is
f(e) = ò₀^¥f(u,e)du
= 2/(2p)^½ 1/s exp(-½ (e/s)²) [1-F(le/s)]
= (2/s) f(e/s) F(-le/s)
where s = (s_u²+s_v²)^1/2, l = s_u/s_v, and F(.) and f(.) are the standard normal cumulative distribution and density functions. The marginal density function f(e) is asymmetrically distributed with mean and variance:
E(e) = -E(u) = -s_u(2/p)^½
Var(e) = (p-2)/p s_u²+s_v²
The log-likelihood function for a sample of N producers can be written as
ll(b,s_u,s_v) = å_i=1,2,...,Nln f(e_i)
= N ln(2) - N/2 ln(2p) - Nln(s) - 1/2 å_i=1,2,...,N(e_i/s)² + å_i=1,2,...,Nln[F(-le_i/s)]
Exponential Model: u_i ~ i.i.d. Exponential(1/s_u) with the density function
f(u) = 1/s_u exp{-u/s_u}
Given the independence assumption, the joint density function of u and v is
f(u,v) = 1/[(2p)^½s_vs_u] exp{-u/s_u -½ (v/s_v)²}
Since e = v - u, the joint density function of u and e is
f(u,e) = 1/[(2p)^½s_vs_u] exp{-u/s_u -½ ((e+u)/s_v)²}
The marginal density of e is
f(e) = ò₀^¥f(u,e)du
= (1/s_u) F(-e/s_v-s_v/s_u) exp{e/s_u + ½ (s_v/s_u)²}
where F(.) is the standard normal cumulative distribution function. The marginal density function f(e) is asymmetrically distributed with mean and variance:
E(e) = -E(u) = -s_u
Var(e) = s_u²+s_v²
The log-likelihood function for a sample of N producers can be written as
ll(b,s_u,s_v) = å_i=1,2,...,Nln f(e_i)
= - Nln(s_u) + N/2 (s_v/s_u)² + å_i=1,2,...,N(e_i/s_u) + å_i=1,2,...,Nln[F(-e_i/s_v-s_v/s_u)]
Truncated Normal Model: u_i ~ i.i.d. Normal⁺(m,s_u²)
The truncated-normal density for u ³ 0 is given by
f(u) = a/(2p)^½ 1/s_u exp{-½ ((u-m)/s_u)²}
where a = 1/F(m/s_u). Given the independence assumption, the joint density function of u and v is
f(u,v) = a/(2ps_vs_u) exp{-½ ((u-m)/s_u)² -½ (v/s_v)²}
Since e = v - u, the joint density function of u and e is
f(u,e) = a/(2ps_vs_u) exp{-½ ((u-m)/s_u)² -½ ((e+u)/s_v)²}
The marginal density of e is
f(e) = ò₀^¥f(u,e)du
= a/(2p)^½ 1/s exp{-½ ((e+m)/s)²} F(m/(sl)-le/s)
= (a/s) f((e+m)/s) F(m/(sl)-le/s)
where s = (s_u²+s_v²)^½, l = s_u/s_v, and F(.) and f(.) are the standard normal cumulative distribution and density functions. f(e) is asymmetrically distributed with mean and variance:
E(e) = -E(u) = -m(a/2)-s_ua/(2p)^½ exp{-½ (m/s_u)²}
Var(e) = m²(a/2)(1-a/2) + (a/2)((p-a)/p)s_u²+s_v²
The log-likelihood function for a sample of N producers can be written as
ll(b,m,s_u,s_v) = å_i=1,2,...,Nln f(e_i)
= - N/2 ln(2p) - Nln(s) - Nln[F(m/s_u)] - 1/2 å_i=1,2,...,N((e_i+m)/s)² + å_i=1,2,...,Nln[F(m/(sl)-le/s)]
If m = 0, then a = 2 and the log-likelihood function for the truncated normal model equals to that of the half normal model.

Model Interpretation: Technical Efficiency

The individual producer's technical efficiency is the point estimator of u derived from the conditional distribution f(u|e). Based on the distribution assumption of u discussed above, f(u|e) is distributed as Normal⁺(m_*,s_*²) (see Jondrow et al. 1982):

f(u|e) = f(u,e)/f(e) = 1/(2p)^½ 1/s_* exp{-½ ((u-m_*)/s_*)²} / F(m_*/s_*)

Half Normal Model Exponential Model Truncated Normal Model

m_* -es_u²/s² -e-s_v²/s_u (-es_u²+ms_v²)/s²

s_*² s_u²s_v²/s² s_v² s_u²s_v²/s²

	Half Normal Model	Exponential Model	Truncated Normal Model
m_*	-es_u²/s²	-e-s_v²/s_u	(-es_u²+ms_v²)/s²
s_*²	s_u²s_v²/s²	s_v²	s_u²s_v²/s²

Either the mean or the mode of this distribution can serve as the estimator for u:

E(u|e) = m_* + s_* [f(-m_*/s_*) / F(m_*/s_*)]

M(u|e) = m_* if m_*³ 0 and M(u|e) = 0 otherwise.

Once the point estimate of u is obtained, estimate of the technical efficiency of each producer can be obtained from

k = exp{-E(u|e)} or exp{-M(u|e)}

The alternative point estimator for technical efficiency of each producer was proposed by Battese and Coelli (1988):

k = E(exp{-u}|e)
= [F(m_*/s_* - s_*) / F(m_*/s_*)] exp{-m_* + ½ s_*²}

In either case the estimates of technical efficiency are unbiased but inconsistent.

Example

In this example, we estimate and compare the above three model specifications of stochastic production function using 25 statewise production data (value added, capital, and labor) in the transportation equipment manufacturing industry studied by Zellner and Revankar (1970). To keep the model simple, we assume the Cobb-Douglas functional form for the production technology. For model interpretation, compute and compare the point estimates of technical efficiency for all model specifications.