Proportional incremental cost probability functions and their frontiers

Abstract

The econometric analysis of cost functions is based on the analysis of the conditional distribution of the cost Y given the level of the outputs $X\in {\mathbb {R}}_+^p$ and given a set of environmental variables $Z\in {\mathbb {R}}^d$. The model basically describes the conditional distribution of Y given $X\ge x$ and $Z=z$. In many applications, the dimension of Z is naturally large and a fully nonparametric specification of the model is limited by the curse of the dimensionality. Most of the approaches so far are based on two-stage estimations when the frontier level does not depend on the value of Z. But even in the case of separability of the frontier, the estimation procedure suffers from several problems, mainly due to the inherent bias of the estimated efficiency scores and the poor rates of convergence of the frontier estimates. In this paper we suggest an alternative semi-parametric model which avoids the drawbacks of the two-stage methods. It is based on a class of model called the Proportional Incremental Cost Functions (PICF), adapted to our setup from the Cox proportional hazard models extensively used in survival analysis for durations models. We define the PICF model, then we examine its properties and propose a semi-parametric estimation. By this way of modeling, we avoid the first stage nonparametric estimation of the frontier and avoid the curse of dimensionality keeping the parametric $\sqrt{n}$ rates of convergence for the parameters of interest. We are also able to derive $\sqrt{n}$-consistent estimator of the conditional order-m robust frontiers (which, by contrast to the full frontier, may depend on Z) and we prove the Gaussian asymptotic properties of the resulting estimators. We illustrate the flexibility and the power of the procedure by some simulated examples and also with some real data sets.

A Appendix: Simulation of data

Simulating a data set $\{(X_{i}, Y_{i},Z_{i})\}_{i=1}^{n}$ should be done with care. Usually researchers specify a model for the frontier function $\varphi _{0}(x)$ then select a model to simulate values of $X_{i}$ and $Z_{i}$ and finally generate $Y_{i}$ for $X=X_{i}$ and $Z=Z_{i}$. Here we have to generate the sample according to our PICF model which specifies that the survival function $S(y | X\ge x, Z=z) = \left[ S_{0}(y|X\ge x)\right] ^{a(z,\beta (x))}$ for some basic survival function $S_{0}(y|X\ge x)$ and some given functions $a(\cdot ,\cdot )$ and $\beta (\cdot )$. So we need to recover from our model the conditional distribution of Y given $X=x$ and $Z=z$ derived from the PICF. We will denote by ${\widetilde{S}}(y |X=x, Z=z)$ this conditional survival function where we use the ${\widetilde{S}}$ notation when we condition on $X=x$, to distinguish from $S(y| X\ge x, Z=z)$ defined above, where we condition on $X\ge x$. Consider for instance the corresponding quantile function

$$\begin{aligned} y = Q(u,x,z) = {\widetilde{S}}^{-1}(y |X=x,Z=z), \end{aligned}$$

(A.1)

where Q is monotone decreasing with u. We know that $U=Q^{-1}(Y,x,z)$ is uniform on [0, 1] and independent of X and Z, so an easy way to simulate Y given $X=x$ and $Z=z$, is to generate $U_{i}$ as uniform on [0, 1] and then define $Y_{i}=Q(U_{i},X_{i},Z_{i})$.

The general form of Q(u, x, z) can be obtained as follows in order to satisfy the PICF model. Some simple algebra leads to the equation

$$\begin{aligned} \text {Prob}(Y \ge y \mid X\ge x, Z=z) =\frac{\int _{x}^{\infty } Q^{-1}(y,t,z) f_{X}(t | z) \textrm{d}t}{S_{X}(x|z)}. \end{aligned}$$

(A.2)

So, for the PICF model, the function Q must satisfy

$$\begin{aligned} \int _{x}^{\infty } Q^{-1}(y,t,z) f_{X}(t | z) \textrm{d}t = S_{X}(x|z) \left[ S_{0}(y|X\ge x)\right] ^{a(z,\beta (x))}. \end{aligned}$$

(A.3)

Taking the derivative with respect to x (with some abuse of notations below, for $x\in {\mathbb {R}}^{p}$ the derivative $\partial _{x}^{p}$ has to be understood as $\partial ^{p}/(\partial x_{1}\ldots \partial x_{p})$) and equating both sides we obtain

$$\begin{aligned} -Q^{-1}(y,x,z) f_{X}(x| z)= & {} - f_{X}(x| z) \left[ S_{0}(y|X\ge x)\right] ^{a(z,\beta (x))} \nonumber \\{} & {} + S_{X}(x|z) \partial _{x}^{p} \left\{ \big [S_{0}(y|X\ge x)\big ]^{a(z,\beta (x)} \right\} . \end{aligned}$$

(A.4)

After some tedious but simple mathematical developments, this leads to the equation

$$\begin{aligned} Q^{-1}(y,x,z)&= {\widetilde{S}}(y|X=x,Z=z) \nonumber \\&=\left[ S_{0}(y|X\ge x)\right] ^{a(z,\beta (x))} - {\frac{S_{X}(x| z)}{f_{X}(x| z)}\partial _{x}^{p} \left\{ \big [S_{0}(y|X\ge x)\big ]^{a(z,\beta (x)} \right\} }, \end{aligned}$$

(A.5)

which allows to define (at least numerically) its reciprocal Q(u, x, z) for any (u, x, z). The expression is greatly simplified if we introduce additional assumption in the model we want to simulate.

Indeed, if we assume that the joint conditional survival function satisfies the Cox model, i.e. $S_{XY}(x,y | z)= \left[ S_{0}(x,y)\right] ^{a(z,\beta (x))}$ where $S_{0}(x,y) = S_{0}(y|X\ge x) S_{0}(x)$, we have

$$\begin{aligned} S_{0}(y|X\ge x) = \frac{\int _{x}^{\infty } {\widetilde{S}}_{0}(y|t) f_{0}(t) \textrm{d}t}{S_{0}(x)}, \end{aligned}$$

(A.6)

where again ${\widetilde{S}}_{0}(y|x)$ is ${\widetilde{S}}_{0}(y|X=x)$, the correspondent of the baseline survivor $S_{0}(y|X\ge x)$ when conditioning on $X=x$. We also have $S_{X}(x|z)= (S_{0}(x))^{a(z,\beta )}$. Therefore, Eq. (A.3) simplifies into

$$\begin{aligned} \int _{x}^{\infty } Q^{-1}(y,t,z) f_{X}(t | z) \textrm{d}t = \left[ \int _{x}^{\infty } {\widetilde{S}}_{0}(y|t) f_{0}(t) \textrm{d}t \right] ^{a(z,\beta (x))}. \end{aligned}$$

(A.7)

In addition if we assume that $\beta (x)=\beta$, the derivative of both sides of (A.7) with respect to x simplifies. Note also that $f_{X}(x| z)= a(z,\beta ) f_{0}(x) (S_{0}(x))^{a(z,\beta )-1}$. After some simplifications this leads to the equation

$$\begin{aligned} Q^{-1}(y,x,z) =&{\widetilde{S}}(y|X=x,Z=z) \nonumber \\ =&\left[ S_{0}(y|X\ge x)\right] ^{a(z,\beta )-1} \widetilde{S}_{0}(y|X=x). \end{aligned}$$

(A.8)

So given the function $a(z,\beta )$, the survival $\widetilde{S}_{0}(y|X=x)$ and the baseline density of X, $f_{0}(x)$, we can compute $S_{0}(y|X\ge x)$ by (A.6), and then the conditional survival ${\widetilde{S}}(y|X=x,Z=z)$. By inverting (A.8), we have the quantile function $y=Q(u,x,z)$ for any u (at least numerically) and then we can simulate a value $Y_{i}$, for a given $(X_{i},Z_{i})$ according to the PICF model.