Nonparametric estimation of the determinants of inefficiency

Abstract

We consider the benchmark stochastic frontier model where inefficiency is directly influenced by observable determinants. In this setting, we estimate the stochastic frontier and the conditional mean of inefficiency without imposing any distributional assumptions. To do so we cast this model in the partly linear regression framework for the conditional mean. We provide a test of correct parametric specification of the scaling function. An empirical example is also provided to illustrate the practical value of the methods described here.

Appendices

Appendix 1: Theoretical properties of the proposed estimator

Here we discuss the necessary conditions to establish $\sqrt n$ consistency of the estimator for the parametric stochastic frontier as well as the nonparametric component, along with its derivatives.

Assumption 1

The triplets (Y _i , X _i , Z _i ) are independent and identically distributed with E|X−ξ| ^4+δ < ∞ for δ > 0 where ξ = E[X|Z]. Further, E[Y|X,Z] = X′β−g(Z), ∀ X,  Z.

Assumption 2

ε⊥X, Z and E[ε ²] = σ ²(X, Z) < ∞.

Assumption 3

$f(Z) \in {\cal G}_\lambda ^\infty$ where $\lambda > 0;\xi \in {\cal G}_\mu ^2$ , where μ > 0; $g(Z) \in {\cal G}_\nu ^4$ , where $\nu > 0$ . Here ${\cal G}_b^a$ is the class of functions whose elements are b-times differentiable with derivatives bounded by a function whose has ath-order finite moments.

Assumption 4

For n → ∞, h → 0, nh ^4ν → 0 and n ^1−δ h ^2q → ∞ for some δ > 0.

Assumption 5

The kernel weighting function $k( \cdot )$ is such that $k \in {{\cal K}_\nu }$ , where ${{\cal K}_\nu }$ , $\nu \ge 1$ is the class of even functions which satisfy ${\int} {s^i}k(s)ds = {\delta _{i0}}$ for $i = 0, \ldots ,\nu - 1$ where δ _i0 is Kronecker’s delta and $k(s) = O\left( {{{(1 + |s{|^{\nu + 1 + \delta }})}^{ - 1}}} \right)$ for some δ > 0.

Given that ε will have higher moments which depend on z, we allow for arbitrary heteroskedasticity in Assumption 2. We impose standard assumptions (in the context of the theory for partly linear regression) on the differentiability of the unknown function, the density of z, and the kernel function. One difference with these results is the requirement that v = X − ξ has finite fourth moment as well as ε having finite fourth moment. Robinson (1988) only requires finite second moments, but to allow for arbitrary heteroskedasticity these moment conditions need to be strengthened. Assumption 5 restricts the kernel to be a bounded, νth-order kernel and is equivalent to Definition 1 in Robinson (1988).

These conditions mimic those found in Li (1996) and this allows a generalization of Robinson (1988) main result to allow use of a second order kernel for dimension up to 5. Lastly, here we are following the approach of Fan et al. (1995) and using density weighted demeaned variables to avoid trimming. Similar assumptions are used by Tran and Tsionas (2009) and Gao et al. (2015) except that they allow for different first and second stage bandwidths. Here we are generically using the same bandwidth everywhere, but one could allow different bandwidths for each separate conditional mean that is estimated. This would require additional notation as well as assumptions on the rate of decay of both sets of bandwidths.

Theorem 1

Under Assumptions 1–4 we have that

$$\sqrt n \left( {\hat \beta - \beta } \right)\,\mathop { \to }\limits^D \,N(0,\,{\Omega _1}),$$

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where Ω₁ = Φ⁻¹ΨΦ⁻¹ with $\Phi = E[{\varphi _i}{\varphi '_i}]$, $\Psi = E[{\sigma ^2}({x_i},\,{z_i}){\varphi _i}{\varphi '_i}]$ and φ _i  = x _i −ξ _i .

The proof of Theorem 1 can be found in Li (1996). This result provides $\sqrt n$-asymptotic normality for the estimator of the parametric component of the partly linear regression model as well as offering an avenue for conducting inference. Given the presence of the unknown variance term, a resampling algorithm, such as residual resampling, might be warranted to construct the variance–covariance matrix.

Before discussing the asymptotic normality of the kernel estimator of conditional inefficiency, we introduce some additional notation. First, use ${R_k} = {\int} {k^2}(u)du$ to denote the roughness of the kernel, ${c_k} = {\int} {u^2}k(u)du$ to capture the variance of the kernel, ${\nu _k} = {\int} {u^2}{k^2}(u)du$, and μ _k (x) = 0.5c _k tr{g(2)(z)}, to signify the sum of second derivatives, scaled the by the kernel’s variance of the unknown conditional inefficiency function. Lastly, let 0 _q denote a zero vector of length q and I _q an identity matrix of dimension q.

Theorem 2

Under Assumptions 1–4, if f(z) > 0,

$$D\left( n \right)\left( {\hat \delta \left( z \right)\, - \,\delta (z)\, - \,{\cal B}\left( z \right)} \right)\,\mathop { \to }\limits^D \,N\left( {0,\,{\Omega _2}\left( z \right)} \right)$$

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where

$$D(n)\, = \,\left[\begin{array}{cc}{\sqrt {n{h^q}} } & 0 \\ 0 & {\sqrt {n{h^{q + 2}}} {I_q}} \end{array}\right],\quad {\cal B}(z) = \left(\begin{array}{c} {{h^2}{\mu _k}(x)} \\ {{0_q}}\end{array}\right),$$

and

$${\Omega _2}\, = \,\left[\begin{array}{cc}{\frac{{{R_k}{\sigma ^2}(x,z)}}{{f(z)}}} & 0 \\ 0 & {\frac{{{\nu _k}{\sigma ^2}(x,z){I_q}}}{{c_k^2f(z)}}}\end{array} \right].$$

The proof of Theorem 2 appears in Li and Wooldridge (2000). The form of D(n) makes clear the slow rate of convergence of the estimator of the conditional mean of inefficiency as well as the first derivatives. As the dimensionality of Z increases the rate of convergence decreases, the well known ‘curse of dimensionality’ that many nonparametric estimators suffer from. It should be noted that despite the slow rate of convergence, the estimator is consistent and these rates are as fast as possible for the set of assumptions used here.

Appendix 2: Theoretical properties of the proposed test

Here we discuss the necessary conditions to establish the asymptotic normality of the test statistic proposed in (10).

Assumption 6

The triplets (Y _i , X _i , Z _i ) are independent and identically distributed. (X, Z) has support ${\cal L}$ that is convex in z and admits a density function f(x, z). E|Y|⁴ < ∞.

Assumption 7

$\tilde \gamma - \gamma = {O_p}({n^{ - 1/2}})$ under H ₀ while $\hat \beta - \beta = {O_p}({n^{ - 1/2}})$ under either H ₀ or H ₁.

Assumption 8

∇g(x,·) and ∇² g(x,·) are continuous in x and dominated by functions with finite second moments. f(x, z), E[ε ²|X = x, Z = z] = σ ²(x, z) and E[ε ⁴|X = x, Z = z] = μ ₄(x, z) all satisfy the Lipschitz condition: $|m(x,\,z + \upsilon ) - m(x,\,z)| \le G(x,\,z)||\upsilon ||$ with $E\left[ {{G^2}(X,\,Z)} \right] < \infty$

Assumption 9

As n → ∞, h → 0 and nh ^q → ∞.

Assumption 10

The kernel function is nonnegative, bounded and symmetric. Further, ${\int} k(u)du = 1$ .

Assumptions 6–10 are relatively weak. The support condition could be the entire Euclidean space ${{\Bbb R}^q}$ or a compact subset. Even in the case of a compact subset no knowledge of the boundary is required and the assumptions do not require that 0 < f(x,z) at the boundary. Assumption 7 requires that we use $\sqrt n$-consistent estimators for both β and γ. If a nonlinear least squares estimator is deployed to estimate the fully parametric model this assumption is trivially satisfied.

Theorem 3

Under Assumptions 6–10 we have

• (1) Under H ₀, as n → ∞

  $$n{h^{q/2}}{I_n}/\sqrt {\hat \Omega \,} \mathop { \to }\limits^D \,N(0,\,1),$$

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  where $\hat \Omega \, = \,(1/{n^2}{h^q})\mathop{\sum}\nolimits_{i = 1}^n \mathop{\sum}\nolimits_{j = 1,j \ne i}^n \varepsilon _i^2\varepsilon _j^2{K_h}({Z_i},\,{Z_j})$ is a consistent estimator of Ω and ε = Y−X′β − g(Z, γ).

• (2) Under H ₁, $Pr(n{h^{q/2}}{I_n}/\sqrt {\hat \Omega } > {B_n})\, \to \,1$ for any nonstochastic sequence ${B_n}\, = \,o(n{h^{q/2}})$.

See Li and Wang (1998) for a proof of Theorem 3. Theorem 3 demonstrates that the use of a normalized version of I _n produces a test statistic that is asymptotical normal under H ₀ and has power in all directions, making it a consistent test.

Appendix 3: Extension to an additive nonparametric setup

The additive model would look like

$${Y_i}\, = \,m({X_i}) + {v_i} - {u_i} + E[{u_i}|{Z_i}] - E[{u_i}|{Z_i}] = m({X_i}) \\ - g({Z_i}) + {\varepsilon _i},$$

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where ${\varepsilon _i} = {v_i} - \left( {{u_i} - g({Z_i})} \right)$ and $g({Z_i}) = E[{u_i}|{Z_i}]$. Several alternative approaches to estimate additive models exist, including global approximation (splines, series, sieves, etc.) and local approximation methods (kernel smoothing). The most common kernel-based methods, consistent with the discussion in our paper, include marginal integration (Linton and Nielsen 1995) and backfitting (Buja et al. 1989). We could also use the oracle estimator of Kim et al. (1999).

If we estimate the conditional mean of y given X in a nonparametric fashion, we have that, for a constant c,

$$E\left( {Y|X} \right) = c + m\left( X \right) - E\left[ {g\left( Z \right)|X} \right].$$

The final term E[g(Z)|X] introduces a bias. In order to eliminate this bias, Kim et al. (1999) propose an instrument function, w(X, Z), such that

$$\begin{array}{ccccc}\\ E\left[ {w\left( {X,Z} \right)|X} \right] = & 1\\ \\ E\left[ {w\left( {X,Z} \right)g\left( {{X_{ - 1}}} \right)|X} \right] = & 0.\\ \end{array}$$

If we have such an instrument, then it can be shown that

$$E\left[ {w(X,Z)y|X} \right]\, = \,c + m(X).$$

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The exact instrument proposed is

$$w(X,Z) = \frac{{{f_X}\left( X \right){f_Z}\left( Z \right)}}{{f\left( {X,Z} \right)}},$$

where ${f_X}\left( X \right)$, ${f_Z}\left( Z \right)$ and $f\left( {X,Z} \right)$ are the marginal and joint probability densities of X and Z, respectively. We can think of these instruments as a measure of the degree of independence between the two sets of regressors, X and Z, influencing the two objects of interest, the frontier and the mean of inefficiency. Note that in the simple case where X is independent from Z, $w( \cdot , \cdot )\, = \,1$. This has intimate connections to the construction of the copula function.w(X, Z) is a valid instrument given that

$${\int} w\left( {X,Z} \right)\frac{{f\left( {X,Z} \right)}}{{{f_X}\left( X \right)}}{\mathrm{d}}Z = {\int} {f_Z}\left( Z \right){\mathrm{d}}Z = 1$$

and

$${\int} w\left( {X,Z} \right)g\left( Z \right)\frac{{f\left( {X,Z} \right)}}{{{f_X}\left( X \right)}}dZ = {\int} g\left( Z \right){f_Z}\left( Z \right)dZ = 0.$$

In practice, we estimate the instruments with standard kernel density estimators. We estimate (22) by choosing a method which eliminates the random denominator term. Formally, if we have $n$ observations for Y, X and $Z$, $\left\{ {{Y_i},{X_i},{Z_i}} \right\}_{i = 1}^n$, our estimator of (22), which we define as ${\gamma _m}\left( \cdot \right)$, is

$${\hat \gamma _m}\left( {{X_i}} \right)\, = \,\frac{1}{n}\mathop{\sum}\limits_{j = 1}^n K_{ij}^X\frac{{{{\hat f}_Z}\left( {{Z_j}} \right)}}{{\hat f\left( {{X_j},\,{Z_j}} \right)}}{Y_j},$$

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where $K_{ij}^X$ is the product kernel with bandwidth h _X for X and the kernel estimators of the marginal and joint densities are

$${\hat f_Z}\left( {{Z_j}} \right)\, = \,\frac{1}{n}\mathop{\sum}\limits_{i = 1}^n K_{ji}^Z$$

and

$$\hat f\left( {{X_j},{Z_j}} \right) = \frac{1}{n}\mathop{\sum}\limits_{i = 1}^n {K_{ij}}.$$

Each element of ${\hat \gamma _g}\left( Z \right)$ is similarly estimated. The first stage estimator of γ _g (Z) (at the point Z _i ) is

$${\hat \gamma _g}\left( {{Z_i}} \right)\, = \,\frac{1}{{n{h_Z}}}\mathop{\sum}\limits_{j = 1}^n K_{ji}^Z\frac{{{{\hat f}_X}\left( {{X_j}} \right)}}{{\hat f\left( {{X_j},\,{Z_j}} \right)}}{Y_j}.$$

Appendix 4: Enforcing positivity on the estimated mean of inefficiency

Du et al. (2013) have proposed a general methodology for imposing various smoothness constraints on an estimated nonparametric regression surface. Here, our smoothness constraints are on the boundedness of the function itself as opposed to ensuring monotonicity or concavity, say. To begin, note that the estimator in (9) can be written as

$$\hat g({Z_i})\, = \,{n^{ - 1}}\mathop{\sum}\limits_{j = 1}^n A_j^{LL}({Z_i})n{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \varepsilon } _i},$$

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which highlights that each observation in the sum contributes uniformly to the estimator of g(Z _i ). The idea of Du et al. (2013) is to replace the uniform 1/n weights with weights selected to ensure that the constraint of interest is satisfied, here nonnegativity. Thus, we have the constraint weighted estimator

$$\hat g({Z_i}) = \mathop{\sum}\limits_{j = 1}^n {p_j}A_j^{LL}({Z_i})n{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \varepsilon } _i},$$

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where the p _j s are selected such that $\hat g({Z_i}) \ge 0$ for every Zj. The weights can be selected by minimizing some distance metric between the uniform weights and the constraint weights. Du et al. (2013) recommend that L ₂ metric, $D(p)\, = \,({p_u} - p)\prime ({p_u} - p)$, which then suggests that the weights, p, should be selected that minimize D(p) subject to $\hat g(z) \ge 0$; as it turns out this is a quadratic programming problem.

The procedure to estimate the original model is to use the kernel methods to estimate $E[{Y_i}|{Z_i}]$ and $E[{X_i}|{Z_i}]$ and use these estimated conditional means to obtain consistent estimates of β. Then, construct ${\tilde \varepsilon _i}$ and use the local linear estimator in (9) to estimate g(Z _i ). If the unconstrained estimates satisfy nonnegativity then proceed to direct analysis of the marginal effects of each of the determinants. If the constraints are not satisfied globally then use the constrained kernel regression procedure described above to impose the constraints prior to investigating the marginal impact of each of the determinants of inefficiency. This procedure works because of the $\sqrt n$-consistency of the OLS estimator established in Theorem 1. Thus, the main results of Du et al. (2013) hold even though a first stage estimation was conducted.