fit.sf {snfa}

Non-parametric stochastic frontier

Description

Fits stochastic frontier of data with kernel smoothing, imposing monotonicity and/or concavity constraints.

Usage

fit.sf(X, y, X.constrained = NA, H.inv = NA, H.mult = 1,
  method = "u", scale.constraints = TRUE)

Arguments

X

Matrix of inputs

y

Vector of outputs

X.constrained

Matrix of inputs where constraints apply

H.inv

Inverse of the smoothing matrix (must be positive definite); defaults to rule of thumb

H.mult

Scaling factor for rule of thumb smoothing matrix

method

Constraints to apply; "u" for unconstrained, "m" for monotonically increasing, and "mc" for monotonically increasing and concave

scale.constraints

Boolean, whether to scale constraints by their average value, can help with convergence

Details

This method fits non-parametric stochastic frontier models. The data-generating process is assumed to be of the form

\ln y_i = \ln f(x_i) + v_i - u_i,

where y_i is the ith observation of output, f is a continuous function, x_i is the ith observation of input, v_i is a normally-distributed error term (v_i\sim N(0, \sigma_v^2)), and u_i is a normally-distributed error term truncated below at zero (u_i\sim N^+(0, \sigma_u)). Aigner et al. developed methods to decompose \varepsilon_i = v_i - u_i into its basic components.

This procedure first fits the mean of the data using fit.mean, producing estimates of output \hat{y}. Log-proportional errors are calculated as

\varepsilon_i = \ln(y_i / \hat{y}_i).

Following Aigner et al. (1977), parameters of one- and two-sided error distributions are estimated via maximum likelihood. First,

\hat{\sigma}^2 = \frac1N \sum_{i=1}^N \varepsilon_i^2.

Then, \hat{\lambda} is estimated by solving

\frac1{\hat{\sigma}^2} \sum_{i=1}^N \varepsilon_i\hat{y}_i + \frac{\hat{\lambda}}{\hat{\sigma}} \sum_{i=1}^N \frac{f_i^*}{1 - F_i^*}y_i = 0,

where f_i^* and F_i^* are standard normal density and distribution function, respectively, evaluated at \varepsilon_i\hat{\lambda}\hat{\sigma}^{-1}. Parameters of the one- and two-sided distributions are found by solving the identities

\sigma^2 = \sigma_u^2 + \sigma_v^2

\lambda = \frac{\sigma_u}{\sigma_v}.

Mean efficiency over the sample is given by

\exp\left(-\frac{\sqrt{2}}{\sqrt{\pi}}\right)\sigma_u,

and modal efficiency for each observation is given by

-\varepsilon(\sigma_u^2/\sigma^2).

Value

Returns a list with the following elements

y.fit

Estimated value of the frontier at X.fit

gradient.fit

Estimated gradient of the frontier at X.fit

mean.efficiency

Average efficiency for X, y as a whole

mode.efficiency

Modal efficiencies for each observation in X, y

X.eval

Matrix of inputs used for fitting

X.constrained

Matrix of inputs where constraints apply

X.fit

Matrix of inputs where curve is fit

H.inv

Inverse smoothing matrix used in fitting

method

Method used to fit frontier

scaling.factor

Factor by which constraints are multiplied before quadratic programming

References

Aigner D, Lovell CK, Schmidt P (1977). “Formulation and estimation of stochastic frontier production function models.” Journal of econometrics, 6(1), 21–37.

Racine JS, Parmeter CF, Du P (2009). “Constrained nonparametric kernel regression: Estimation and inference.” Working paper.

Examples

data(USMacro)

USMacro <- USMacro[complete.cases(USMacro),]

#Extract data
X <- as.matrix(USMacro[,c("K", "L")])
y <- USMacro$Y

#Fit frontier
fit.sf <- fit.sf(X, y,
                 X.constrained = X,
                 method = "mc")

print(fit.sf$mean.efficiency)
# [1] 0.9772484

#Plot efficiency over time
library(ggplot2)

plot.df <- data.frame(Year = USMacro$Year,
                      Efficiency = fit.sf$mode.efficiency)

ggplot(plot.df, aes(Year, Efficiency)) +
  geom_line()