Stochastic Frontier Analysis for Panel Data

Readings and References:

• G. E. Battese and T. J. Coelli, "Frontier Production Functions, Technical Efficiency and Panel Data: with Applications to Paddy Farmers in India," Journal of Productivity Analysis 3, 153-169, 1992.
• S. C. Kumbhakar and C. A. K. Lovell, Stochastic Frontier Analysis, Cambridge University Press, 2000.
• 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.

y_it = h(X_it,b)k_it, i = 1,2,...N; t = 1,2,...,T_i (time periods may be different for each producer).

where y_it is output of firm i at time t and h(X_it,b) is the production function with factor inputs X_it. The efficiency measure is defined by k_it = exp(-u_it) with 0 < k_it£ 1 for u_it ³ 0.

Therefore, in logarithm and with added random error term e_it, the regression model is

ln(y_it) = ln h(X_it,b) - u_it + e_it

Let ln(y_it) = Y_it and ln h(X_it,b) = a + ln(X_it)b. The simple case of Cobb-Douglas production function with technical efficiency is written as:

Yit	= a + Xitb + eit
	= a + Xitb + eit - uit

Time-Invariant Technical Efficiency	Time-Variant Technical Efficiency
uit = ui ³ 0	uit = exp{-h(t-Ti)}ui ³ 0 where h is the decay parameter.

Note that for cost minimizing producer, u_i is assumed to be nonpositive or u_i £ 0.

Time-Invariant Technical Efficiency

	Fixed Effects	Random Effects
Model	Yit = (a + Xitb - ui) + eit	Yit = a + Xitb + (eit - ui)
Assumption	E(Xitui) ¹ 0 E(Xiteit) ¹ 0	E(ui) = m, E(Xitui) = 0, E(Xiteit) = 0 The model is re-written as Yit = a* + Xitb + eit* a* = a-m, e* = eit-ui+m E(eit*) = 0, E(Xiteit*) = 0
Estimation	OLS with Dummy Variables Mean Deviation Regression	GLS Partial Mean Deviation Regression
Individual Effect	ai = a - ui	mi = åt=1,2,...,Tieit*/Ti
Technical Efficiency	a* = maxi=1,2,...,N(ai) ui* = a* - ai ki = exp(-ui*)	m* = maxi=1,2,...,N(mi) ui* = m* - mi ki = exp(-ui*)

At least one producer is 100 percent technical effecient. Technical efficiency is measured relative to this technical efficient producer.

Maximum Likelihood Estimation

Y_it = a + X_itb + e_it

where e_it = e_it - u_i. We assume that e_it is i.i.d. normal(0,s_e²) and distributed independently of u_i and the regressors.

Let e_i = [e_i1,e_i2,...,e_iTi]'. The density function of the T_i-element vector e_i is

f(e_i) = 1/(2ps_e²)^T_i/2 exp{-½ e_i'e_i/s_e²}

In general, time periods may be different for each producer. However, in the following for notation simplicity, we assume T = T_i and e = e_i for a given producer i.

• 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 e = [e₁,...,e_T]' is

  f(u,e) = 2/(2ps_u²)^½ (1/(2ps_e²))^T/2 exp{-½(u/s_u)² -½ e'e/s_e²}

  The joint density function of u and e=[e₁-u,...,e_T-u]' is

  f(u,e) = 2/(2ps_u²)^½ (1/(2ps_e²))^T/2 exp{-½ ((u-m_*)/s_u)² -½ e'e/s_e² + ½ (m_*/s_*)²}

  where

  m_* = -Ts_u²e^m / (s_e²+Ts_u²)
  s_*² = s_u²s_e² / (s_e²+Ts_u²)
  e^m = å_t=1,...,Te_it/T

  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)^T/2s_e^T-1(s_e²+Ts_u²)^½) exp(-½ e'e/s_e² + ½ (m_*/s_*)²) [1-F(-m_*/s_*)]

  The log-likelihood function for a sample of N producers can be written as

  ll(b,s_u,s_e) = å_i=1,2,...,Nln f(e_i)
  = N ln(2) - NT/2 ln(2p) - N(T-1)ln(s_e) - N/2 ln(s_e²+Ts_u²)
  - 1/2 å_i=1,2,...,Ne_i'e_i/s_e² + 1/2 å_i=1,2,...,N(m_*i/s_*i)²
  + å_i=1,2,...,Nln[1-F(-m_*i/s_*i)]

  For technical efficiency interpretation, we need to derive the conditional distribution

  f(u|e) = f(u,e)/f(e) ~ Normal⁺(m_*,s_*²)

  Therefore,

  E(u_i|e_i) = m_*i + s_*i [f(-m_*i/s_*i) / (1-F(-m_*i/s_*i)]
  

  M(u_i|e_i) = m_*i if e_i£ 0 and M(u_i|e_i) = 0 otherwise.

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

  k_i = exp{-E(u_i|e_i)} or exp{-M(u_i|e_i)}

  The alternative point estimator for technical efficiency of each producer due to Battese and Coelli (1988) is:

  k_i = E(exp{-u_i}|e_i)
  = [1-F(s_*i-m_*i/s_*i)] / [1-F(-m_*i/s_*i)] exp{-m_*i + ½ s_*i²}

• 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).

  Using the same approach as the above half-normal model, from f(u) ® f(u,e) ® f(u,e) ® f(e), we derive the log-likelihood function for the truncated normal model:

  ll(b,s_u,s_e) = å_i=1,2,...,Nln f(e_i)
  = - NT/2 ln(2p) - N(T-1)ln(s_e) - N/2 ln(s_e²+Ts_u²)
  - 1/2 å_i=1,2,...,Ne_i'e_i/s_e² - N/2 (m/s_u)² + 1/2 å_i=1,2,...,N(m_*i/s_*i)²
  - N ln(1-F(-m/s_u)) + å_i=1,2,...,Nln[1-F(-m_*i/s_*i)]

  Finally, the technical effeciency is interpreted based on the conditional distribution

  f(u|e) ~ Normal⁺(m_*,s_*²)

  where

  m_* = ms_e²-Ts_u²e^m / (s_e²+Ts_u²)
  s_*² = s_u²s_e² / (s_e²+Ts_u²)
  e^m = å_t=1,...,Te_it/T

Example: Airline Service Cost Function

• Technical Efficiency based on Fixed Effects and Random Effects
• Technical Efficiency based on Maximum Likelihood Estimates
  • Half Normal Model
  • Truncated Normal Model

Time-Variant Technical Efficiency

u_it = g_i(t)u_i

A simple time-varying decay function (Battese and Coelli, 1992) is

g_i(t) = exp{-h(t-T_i)}

where h is the decay parameter. For model estimation, maximum likelihood method is suggested.

Derive the log-likelihood functions for half-normal and trucated-normal models with time-variant technical efficiency. Compute and interpret the technical efficiency measures which changes over time.

(To be Continued)