fit.sf: Non-parametric stochastic frontier

In tkmckenzie/snfa: Smooth Non-Parametric Frontier Analysis

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, σ_v^2)), and u_i is a normally-distributed error term truncated below at zero (u_i\sim N^+(0, σ_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{σ}^2 = \frac1N ∑_{i=1}^N \varepsilon_i^2.

Then, \hat{λ} is estimated by solving

\frac1{\hat{σ}^2} ∑_{i=1}^N \varepsilon_i\hat{y}_i + \frac{\hat{λ}}{\hat{σ}} ∑_{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{λ}\hat{σ}^{-1}. Parameters of the one- and two-sided distributions are found by solving the identities

σ^2 = σ_u^2 + σ_v^2

λ = \frac{σ_u}{σ_v}.

Mean efficiency over the sample is given by

\exp≤ft(-\frac{√{2}}{√{π}}\right)σ_u,

and modal efficiency for each observation is given by

-\varepsilon(σ_u^2/σ^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

\insertRef

AignerLovellSchmidtsnfa

\insertRefParmeterRacinesnfa

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()
  

tkmckenzie/snfa documentation built on June 11, 2020, 4:34 a.m.