sfa (version 1.2.0)

efficiency_ci: Confidence Intervals for Individual Inefficiency and Efficiency

Description

Horrace and Schmidt (1996) intervals for the inefficiency term \(u_i\) of a fitted stochastic frontier model, and the implied intervals for technical efficiency \(\exp(-u_i)\).

Usage

efficiency_ci(object, level = 0.95, type = c("both", "u", "te"))

Value

A data frame with one row per observation, carrying u_lower, u_hat, u_upper and/or te_lower, te_hat, te_upper according to type, and a "level" attribute. u_hat is the Jondrow et al. (1982) posterior mean and te_hat the Battese and Coelli (1988) score -- exactly the values the fit already reports, taken from the fitted object rather than recomputed.

Arguments

object

an object of class "sfareg", as returned by sfm.

level

coverage of the interval, a single number strictly between 0 and 1. Defaults to 0.95.

type

which intervals to return: "both" (the default), "u" for inefficiency only, or "te" for technical efficiency only.

Details

Conditional on the fitted parameters, \(u_i \mid e_i\) is normal with mean mu_star and standard deviation sigma_star, truncated below at zero -- the same posterior the Jondrow et al. (1982) and Battese and Coelli (1988) point predictors average over. Inverting it at \((1-\alpha)/2\) and \(1-(1-\alpha)/2\) gives the bounds in closed form, so the interval costs no estimation beyond the fit itself. The efficiency bounds follow by monotonicity, with the endpoints swapped, since \(\exp(-u)\) is decreasing in \(u\).

These intervals condition on the estimated parameters. They describe where \(u_i\) sits given \(e_i\) and a known frontier, and make no allowance for sampling error in the slopes or the variance parameters. Horrace and Schmidt are explicit about this, and it is why the intervals do not narrow as the sample grows: they measure the irreducible difficulty of splitting a single residual into noise and inefficiency, not estimation uncertainty. Read a wide interval as a warning against reading much into that unit's rank.

Available for the models whose posterior really is a truncated normal: "NHN", "NHN_Z", "NE" and "NTN". Every other model_name has a posterior of a different shape, for which these formulas do not hold; the function reports that rather than returning a misleading number.

References

Horrace, W. C. and Schmidt, P. (1996). Confidence statements for efficiency estimates from stochastic frontier models. Journal of Productivity Analysis, 7, 257--282.

Jondrow, J., Lovell, C. A. K., Materov, I. S. and Schmidt, P. (1982). On the estimation of technical inefficiency in the stochastic frontier production function model. Journal of Econometrics, 19, 233--238.

Battese, G. E. and Coelli, T. J. (1988). Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. Journal of Econometrics, 38, 387--399.

See Also

sfm, sfa_diagnostics

Examples

dat <- data_gen_cs(N = 200, rand = 1, sig_u = 1, sig_v = 0.3,
                   cons = 0.5, beta1 = 0.5, beta2 = 0.5, a = 1, mu = 0.5)
fit <- sfm(y_pcs ~ x1 + x2, data = dat, model_name = "NHN")

ci <- efficiency_ci(fit, level = 0.90)
head(ci)

## How much of an efficiency ranking is real? Compare the spread of the point
## predictions against the width of a single unit's interval.
diff(range(ci$te_hat))
median(ci$te_upper - ci$te_lower)