Wrong skewness and finite sample correction in the normal-half normal stochastic frontier model

Abstract

In parametric stochastic frontier models, the composed error is specified as the sum of a two-sided noise component and a one-sided inefficiency component, which is usually assumed to be half-normal, implying that the error distribution is skewed in one direction. In practice, however, estimation residuals may display skewness in the wrong direction. Model respecification or pulling a new sample is often prescribed. Since wrong skewness may manifest as a finite sample problem, this paper proposes a finite sample adjustment to existing estimators to obtain the desired direction of residual skewness. This provides an alternative empirical approach to deal with the wrong skewness problem that does not require respecification of the model.

Appendix: Constrained COLS

Proof of Proposition 1

As defined in Sect. 3, the constrained COLS is the restricted least squares with the linear constraint

$$\begin{aligned} R\beta =q(k_{0}) \end{aligned}$$

where $R=\frac{1}{N}{\tilde{e}}^{\prime }M_{0}X$ and $q(k_{0})=R{\hat{\beta }} _\mathrm{OLS}+\frac{{\hat{\mu }}_{3}^{\prime }}{3}+\frac{\Pi }{3}k_{0}^{3/2}({\hat{\mu }} _{2}^{\prime })^{3/2}$, with the slope estimator

$$\begin{aligned} {\hat{\beta }}_{r}(k_{0})={\hat{\beta }}_\mathrm{OLS}-(X^{\prime }X)^{-1}R^{\prime }[R(X^{\prime }X)^{-1}R^{\prime }]^{-1}[R{\hat{\beta }}_\mathrm{OLS}-q(k_{0})] \end{aligned}$$

and the corresponding sum of squared residuals

$$\begin{aligned} SSR_{r}(k_{0})=SSR+[R{\hat{\beta }}_\mathrm{OLS}-q(k_{0})]^{\prime }[R(X^{\prime }X)^{-1}R^{\prime }]^{-1}[R{\hat{\beta }}_\mathrm{OLS}-q(k_{0})], \end{aligned}$$

where ${\hat{\beta }}_\mathrm{OLS}=(X^{\prime }X)^{-1}X^{\prime }y$. Thus,

$$\begin{aligned} \frac{dSSR_{r}(k_{0})}{dk_{0}}=-2[R{\hat{\beta }}_\mathrm{OLS}-q(k_{0})]^{\prime }[R(X^{\prime }X)^{-1}R^{\prime }]^{-1}\frac{dq(k_{0})}{dk_{0}}. \end{aligned}$$

Given the facts that ${\hat{\mu }}_{3}^{\prime }>0$ in the presence of wrong skewness, the scalar

$$\begin{aligned} R{\hat{\beta }}_\mathrm{OLS}-q(k_{0})= & {} R{\hat{\beta }}_\mathrm{OLS}-[R{\hat{\beta }}_\mathrm{OLS}+\frac{ {\hat{\mu }}_{3}^{\prime }}{3}+\frac{\Pi }{3}k_{0}^{3/2}({\hat{\mu }}_{2}^{\prime })^{3/2}] \\= & {} -[\frac{{\hat{\mu }}_{3}^{\prime }}{3}+\frac{\Pi }{3}k_{0}^{3/2}({\hat{\mu }} _{2}^{\prime })^{3/2}]<0. \end{aligned}$$

In addition,

$$\begin{aligned} \frac{dq(k_{0})}{dk_{0}}=\frac{1}{2}({\hat{\mu }}_{2}^{\prime })^{3/2}\Pi k_{0}^{1/2}>0, \end{aligned}$$

and the scalar $R(X^{\prime }X)^{-1}R^{\prime }>0$ since the matrix $X^{\prime }X$ is positive definite. Therefore,

$$\begin{aligned} \frac{dSSR_{r}(k_{0})}{dk_{0}}>0. \end{aligned}$$

$\square$

Since $C(k_{0})=\frac{1}{N}SSR_{r}(k_{0})-k_{0}{\hat{\sigma }}_{\varepsilon }^{2}\frac{\ln N}{N}$, the FOC is

$$\begin{aligned} \frac{1}{N}\frac{dSSR_{r}(k_{0})}{dk_{0}}-{\hat{\sigma }}_{\varepsilon }^{2} \frac{\ln N}{N}=0, \end{aligned}$$

or

$$\begin{aligned} -2[R{\hat{\beta }}_\mathrm{OLS}-q(k_{0})]^{\prime }[R(X^{\prime }X)^{-1}R^{\prime }]^{-1}\frac{dq(k_{0})}{dk_{0}}-{\hat{\sigma }}_{\varepsilon }^{2}\ln N=0. \end{aligned}$$

Substituting $R{\hat{\beta }}_\mathrm{OLS}-q(k_{0})=-[\frac{{\hat{\mu }}_{3}^{\prime }}{3} +\frac{\Pi }{3}k_{0}^{3/2}({\hat{\mu }}_{2}^{\prime })^{3/2}]$ and $\frac{ dq(k_{0})}{dk_{0}}=\frac{1}{2}({\hat{\mu }}_{2}^{\prime })^{3/2}\Pi k_{0}^{1/2}$ into the equation above, we obtain

$$\begin{aligned} \left[ \frac{1}{\Pi }\frac{{\hat{\mu }}_{3}^{\prime }}{({\hat{\mu }}_{2}^{\prime })^{3/2}}+k_{0}^{3/2}\right] k_{0}^{1/2}=\frac{3}{\Pi ^{2}}\frac{{\hat{\sigma }} _{\varepsilon }^{2}}{({\hat{\mu }}_{2}^{\prime })^{3}}\ln N\cdot [ R(X^{\prime }X)^{-1}R^{\prime }]. \end{aligned}$$

The LHS $k_{0}^{2}+\frac{1}{\Pi }\frac{{\hat{\mu }}_{3}^{\prime }}{({\hat{\mu }} _{2}^{\prime })^{3/2}}k_{0}^{1/2}$ is a monotonic increasing function of $k_{0}$, with a minimum 0 at $k_{0}=0$ and a maximum of $1+\frac{1}{\Pi } \frac{{\hat{\mu }}_{3}^{\prime }}{({\hat{\mu }}_{2}^{\prime })^{3/2}}$ at $k_{0}=1$ . Since the OLS residual skewness $\frac{{\hat{\mu }}_{3}^{\prime }}{({\hat{\mu }} _{2}^{\prime })^{3/2}}$ is usually a very small positive number in the presence of wrong skewness, $1+\frac{1}{\Pi }\frac{{\hat{\mu }}_{3}^{\prime }}{( {\hat{\mu }}_{2}^{\prime })^{3/2}}$ is slightly bigger than 1.

Consider the RHS $\frac{3}{\Pi ^{2}}\frac{{\hat{\sigma }}_{\varepsilon }^{2}}{( {\hat{\mu }}_{2}^{\prime })^{3}}\frac{\ln N}{N}\cdot N[R(X^{\prime }X)^{-1}R^{\prime }]$. It is positive. In addition, the positive scalar

$$\begin{aligned} N\cdot R(X^{\prime }X)^{-1}R^{\prime }= & {} \frac{1}{N}{\tilde{e}}^{\prime }M_{0}X(X^{\prime }X)^{-1}X^{\prime }M_{0}{\tilde{e}}=\frac{1}{N}({\tilde{e}}- {\hat{\mu }}_{2}^{\prime }\iota )^{\prime }X(X^{\prime }X)^{-1}X^{\prime }( {\tilde{e}}-{\hat{\mu }}_{2}^{\prime }\iota ) \\= & {} \frac{1}{N}[{\tilde{e}}^{\prime }P_{X}{\tilde{e}}-2{\hat{\mu }}_{2}^{\prime } {\tilde{e}}^{\prime }P_{X}\iota +({\hat{\mu }}_{2}^{\prime })^{2}\iota ^{\prime }P_{X}\iota ] \\= & {} \frac{1}{N}[{\tilde{e}}^{\prime }P_{X}{\tilde{e}}-N({\hat{\mu }}_{2}^{\prime })^{2}]=\frac{1}{N}{\tilde{e}}^{\prime }P_{X}{\tilde{e}}-({\hat{\mu }}_{2}^{\prime })^{2}, \end{aligned}$$

where $P_{X}\iota =\iota$ and ${\tilde{e}}^{\prime }\iota =N{\hat{\mu }} _{2}^{\prime }$. We normalize ${\tilde{e}}$ by dividing it by its average ${\hat{\mu }}_{2}^{\prime }$, i.e., $\mathring{e}={\tilde{e}}/{\hat{\mu }} _{2}^{\prime }$ , such that $\mathring{e}^{\prime }\iota ={\tilde{e}}^{\prime }\iota /{\hat{\mu }}_{2}^{\prime }=N$. Thus,

$$\begin{aligned}&\frac{3}{\Pi ^{2}}\frac{{\hat{\sigma }}_{\varepsilon }^{2}}{({\hat{\mu }} _{2}^{\prime })^{3}}\ln N\cdot [ R(X^{\prime }X)^{-1}R^{\prime }] \\&\quad =\frac{3}{\Pi ^{2}}\frac{{\hat{\sigma }}_{\varepsilon }^{2}}{({\hat{\mu }} _{2}^{\prime })^{3}}\ln N\cdot \frac{1}{N^{2}}{\tilde{e}}^{\prime }M_{0}X(X^{\prime }X)^{-1}X^{\prime }M_{0}{\tilde{e}}] \\&\quad =\frac{3}{\Pi ^{2}}\frac{\ln N}{N}\cdot \frac{1}{N}\mathring{e}^{\prime }M_{0}X(X^{\prime }X)^{-1}X^{\prime }M_{0}\mathring{e}]. \end{aligned}$$

Since for a large N,

$$\begin{aligned} \mathring{e}^{\prime }M_{0}X(X^{\prime }X)^{-1}X^{\prime }M_{0}\mathring{e}= \frac{1}{N}\mathring{e}^{\prime }M_{0}X\left( \frac{X^{\prime }X}{N} \right) ^{-1}X^{\prime }M_{0}\mathring{e}=O_{p}(1), \end{aligned}$$

we obtain,

$$\begin{aligned} \text {RHS}=\frac{3}{\Pi ^{2}}\frac{\ln N}{N}\cdot \frac{O_{p}(1)}{N}. \end{aligned}$$

For a relatively large sample size N, RHS falls into the unity interval, implying the existence of ${\hat{k}}_{0}$ as the solution to $\min _{k_{0}\in [ 0,1)}C(k_{0})$.

Uniqueness of ${\hat{k}}_{0}$ is guaranteed by the second-order condition. The second-order derivative of $C(k_{0})$ is

$$\begin{aligned} \frac{d^{2}C(k_{0})}{dk_{0}^{2}}= & {} \frac{1}{N}\frac{d^{2}SSR_{r}(k_{0})}{ dk_{0}^{2}} \\= & {} \frac{\Pi ^{2}({\hat{\mu }}_{2}^{\prime })^{3}}{3N[R(X^{\prime }X)^{-1}R^{\prime }]}\frac{d\left[ \frac{1}{\Pi }\frac{{\hat{\mu }}_{3}^{\prime }}{( {\hat{\mu }}_{2}^{\prime })^{3/2}}k_{0}^{1/2}+k_{0}^{2}\right] }{dk_{0}} \\= & {} \frac{\Pi ^{2}({\hat{\mu }}_{2}^{\prime })^{3}}{3N[R(X^{\prime }X)^{-1}R^{\prime }]}\left[ \frac{1}{2\Pi }\frac{{\hat{\mu }}_{3}^{\prime }}{(\hat{\mu }_{2}^{\prime })^{3/2}}k_{0}^{-1/2}+2k_{0}\right] >0 \end{aligned}$$

for any $0<k_{0}<1$ since OLS residual skewness $\frac{{\hat{\mu }}_{3}^{\prime }}{({\hat{\mu }}_{2}^{\prime })^{3/2}}>0$ in the presence of wrong skewness.