fit.mean: Kernel smoothing with additional constraints

In tkmckenzie/snfa: Smooth Non-Parametric Frontier Analysis

Description

Fits conditional mean of data with kernel smoothing, imposing monotonicity and/or concavity constraints.

Usage

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

Arguments

X.eval	Matrix of inputs used for fitting
y.eval	Vector of outputs used for fitting
X.constrained	Matrix of inputs where constraints apply
X.fit	Matrix of inputs where curve is fit; defaults to X.constrained
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 uses kernel smoothing to fit the mean of the data while imposing specified monotonicity and concavity constraints. The procedure is derived from Racine et al. (2009), which develops kernel smoothing methods with bounding, monotonicity and concavity constraints. Specifically, the smoothing procedure involves finding optimal weights for a Nadaraya-Watson estimator of the form

\hat{y} = m(x) = ∑_{i=1}^N p_i A(x, x_i) y_i,

where x are inputs, y are outputs, p are weights, subscripts index observations, and

A(x, x_i) = \frac{K(x, x_i)}{∑_{h=1}^N K(x, x_h)}

for a kernel K. This method uses a multivariate normal kernel of the form

K(x, x_h) = \exp≤ft(-\frac12 (x - x_h)'H^{-1}(x - x_h)\right),

where H is a bandwidth matrix. Bandwidth selection is performed via Silverman's (1986) rule-of-thumb, in the function H.inv.select.

Optimal weights \hat{p} are selected by solving the quadratic programming problem

\min_p \mbox{\ \ }-\mathbf{1}'p + \frac12 p'p.

Monotonicity constraints of the following form can be imposed at specified points:

\frac{\partial m(x)}{\partial x^j} = ∑_{h=1}^N p_h \frac{\partial A(x, x_h)}{\partial x^j} y_h ≥q 0 \mbox{\ \ \ \ }\forall x, j,

where superscripts index inputs. Finally concavity constraints of the following form can also be imposed using Afriat's (1967) conditions:

m(x) - m(z) ≤q \nabla_x m(z) \cdot (x - z) \mbox{\ \ \ \ }\forall x, z.

The gradient of the estimated curve at a point x is given by

\nabla_x m(x) = ∑_{i=1}^N \hat{p}_i \nabla_x A(x, x_i) y_i,

where \hat{p}_i are estimated weights.

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
solution	Boolean; TRUE if frontier successfully estimated
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

ParmeterRacinesnfa

Examples

data(USMacro)

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

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

# Reflect data for fitting
reflected.data <- reflect.data(X, y)
X.eval <- reflected.data$X
y.eval <- reflected.data$y

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

# Plot input productivities over time
library(ggplot2)
plot.df <- data.frame(Year = rep(USMacro$Year, times = 2),
                      Elasticity = c(fit.mc$gradient.fit[,1] * X[,1] / y,
                                     fit.mc$gradient.fit[,2] * X[,2] / y),
                      Variable = rep(c("Capital", "Labor"), each = nrow(USMacro)))

ggplot(plot.df, aes(Year, Elasticity)) +
  geom_line() +
  facet_grid(Variable ~ ., scales = "free_y")
  

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