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Empirical Economics

Erschienen in:

08.01.2021

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

verfasst von: Jun Cai, Qu Feng, William C. Horrace, Guiying Laura Wu

Erschienen in: Empirical Economics | Ausgabe 6/2021

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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.

Vorheriger Artikel Aggregate efficiency of industry and its groups: the case of Queensland public hospitals

Nächster Artikel Managing power supply interruptions: a bottom-up spatial (frontier) model with an application to a Spanish electricity network

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The skewness of the composed error is positive in the stochastic frontier cost function model. We use the terminology COLS following Olson et al. (1980). COLS is also called MOLS. See Greene (2007).

Greene (2007, p.131) claims “In this instance, the OLS results are the MLEs, and consequently, one must estimate the one-sided terms as 0.”

For example, estimating the variance parameters in COLS is invalid in this case.

It may also be a consequence of a misspecified model, but that is not our focus here.

Waldman (1982, p. 278) also suggests that $(b,0,s^{2})$ may be a global maximum. There are two roots in this normal-half normal model: OLS $ (b,0,s^{2})$ and one at the MLE with positive $\lambda $. When the residual skewness is positive, the first is superior to the second (Greene 2007, note 28).

Kumbhakar et al. (2013) propose a stochastic frontier model to accommodate the presence of both efficient and inefficient firms in the sample.

Waldman (1982, p.278) notes that for $\sigma _{u}>0$ “as the sample size increases the probability that $\sum e_{t}^{3}>0$ and hence that $ (b,0,s^{2})$ locates a local maximum goes to zero.”

Badunenko et al. (2012) find that the estimation of efficiency scores depends on the estimated ratio of the variation in efficiency to the variation in noise. As discussed by Kim et al. (2007) and Feng and Horrace (2012) in fixed effects stochastic frontier models, small signal-to-noise ratio leads to inaccurate inference.

As pointed out by Simar and Wilson (2010, p.72), this problem could happen in other one-sided specifications. In a previous version of this paper, our Monte Carlo experiments suggest that wrong skewness could also occur with high probability in exponential and binomial SPF models, when the signal-to-noise ratio is small.

This stems from the fact that Waldman (1982) shows that OLS is local maximum in the parameter space of MLE when the OLS residuals are positively skewed. In fact, the non-positivity constraint will bind globally (when the OLS residuals are positively skewed), if OLS is a global maximum, as the Monte Carlo studies of Olsen et al. (1980) suggest.

It is worth noting that (10) is not a direct linearization of (7). Alternatively, a full linearization of (7) can be similarly obtained by replacing $R=\frac{1}{N}{\tilde{e}}^{\prime }M_{0}X$ with $R=\frac{1}{N}({\tilde{e}}^{\prime }M_{0}-\sqrt{{\hat{\mu }}_{2}^{\prime }} {\hat{\mu }}_{3}^{\prime }e^{\prime })X$. The additional term $-\frac{1}{N} \sqrt{{\hat{\mu }}_{2}^{\prime }}{\hat{\mu }}_{3}^{\prime }e^{\prime }X$ is from the effect of the denominator of the constraint in (7). Monte Carlo simulations suggest that the estimation results are robust to this choice. Details are available upon request.

In the empirical example below, the command Frontier in Stata, which allows for a linear constraint, is employed.

Table 1 in Simar and Wilson (2010) provides some guidance. On the one hand, when $\lambda ^{2}\le 0.1$ (i.e., $k=1/(\frac{\pi }{\pi -2}\frac{1}{\lambda ^{2}}+1)<0.035)$ for samples with size less than 200, the proportion of wrong skewness is close to $50\%$, implying that the inefficiency term is hard to distinguish from noise. On the other hand, when $\lambda ^{2}\ge 1$ ($k\ge 0.267$), the wrong skewness probability decreases dramatically. For example, only $6\%$ of samples display wrong residual skewness for $\lambda ^{2}=2$ ($k=0.421$) and $N=200$. We have a similar finding both for Simar and Wilson’s design and the design in Sect. 5 of this paper. Results are available upon request.

The constraint $k\ge k_{0}$ is always binding in the neighborhood of OLS. And a restriction on k is equivalent on $\lambda $, which is a monotonic increasing function of k in the half-normal model,

$$\begin{aligned} \lambda =\sqrt{\frac{\sigma _{u}^{2}}{\sigma _{v}^{2}}}=\sqrt{\frac{ Var(u_{i})}{\frac{\pi -2}{\pi }\sigma _{v}^{2}}}=\sqrt{\frac{k}{\frac{\pi -2 }{\pi }(1-k)}}=\sqrt{\frac{\pi }{\pi -2}\frac{1}{(1/k-1)}}. \end{aligned}$$

Per a referee’s advice, we experimented with selecting $k_0$ by minimizing the mean integrated squared error of the difference between the constrained and unconstrained residual densities, but the selection performed poorly in terms of the RMSE of the estimated coefficients in finite samples.

In this sense, our approach is different from the literature on models with moment conditions, e.g., Moon and Schorfheide (2009).

When N is large, the wrong skewness problem is less likely to occur unless the inefficiency variance ratio is very small. When it does occur in this setting, alternative approaches including respecification may be considered.

Strictly speaking, restricting $\lambda $ as a constraint yields a different result from constraint (7). Though the population skewness is equal to $g(k_{0})$ and thus a monotonic function of $\lambda $, the sample skewness is not a function of $\lambda $. However, the insights derived here on the effect of the chosen value of $k_{0}$ on estimation still apply.

As pointed out by a referee, the second term on the right-hand side of (17) is a constant in X. See Papadopoulos (2018), pp. 338–339. This is due to the fact we only take a first-order Taylor expansion of $\phi (\frac{\lambda }{\sigma }\varepsilon _{i})/ [1-\Phi (\frac{\lambda }{\sigma }\varepsilon _{i})]$. With higher-order terms included, constrained MLE and COLS involve additional terms.

For a small value of $k_{0}$, e.g., $k_{0}\in [0.1,0.3]$, $\lambda $ lies in the interval [0.5530, 1.0860].

Coelli (1995) also uses this signal-to-noise ratio measure, denoted by $ \gamma ^{*}$, in his Monte Carlo experiments.

Due to misspecification and small sample, the variance estimates with proposed method do not show a consistency property and may yield extreme estimates in some cases, for instance $N=200$ with $k=0.5$.

With the exception of perhaps Green and Mayes (1991), Mester (1997), and Parmeter and Racine (2012), there appear to be very few empirical studies with wrong skewness in the literature. As in Greene (2007, Table 2.11), we use this panel data example as a cross-sectional one only for the purpose of illustration.

Inconsistent with the statements of Waldman (1982) and Greene (2007), the MLE with positive $\lambda $ achieves a slightly bigger value of log-likelihood than OLS for this dataset. Similarly, the inconsistency between OLS and MLE in the presence of positive OLS residual skewness by using FRONTIER is discussed by Simar and Wilson (2010). Greene (2007, p. 202) notes: “... for this data set, and more generally, when the OLS residuals are positively skewed, then there is a second maximizer of the log-likelihood, OLS, that may be superior to the stochastic frontier.”

This property can be obtained by the equation (3) in Waldman (1982, p.278):

$$\begin{aligned} \Delta l=\frac{\mu ^{3}}{6s^{3}}\sqrt{\frac{2}{\pi }}\frac{\pi -4}{\pi } \sum \limits _{i=1}^{N}e_{i}^{3} \end{aligned}$$

where $\mu $ can be regarded as $\lambda $ changing from 0 as in the analysis in Sect. 3.1. Since $\pi -4<0$, in the presence wrong skewness $ (\sum \nolimits _{i=1}^{N}e_{i}^{3}>0$), the log-likelihood decreases with the imposed value of $\lambda $ (and $k_{0}$).

The constant term is calculated by OLS intercept plus $\sqrt{2{\hat{\sigma }} _{u}^{2}/\pi }$. The standard errors formula of the COLS estimators of constant term, $\sigma ^{2}$ and $\gamma $ (not $\lambda $) can be found in Coelli (1995).

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Titel: Wrong skewness and finite sample correction in the normal-half normal stochastic frontier model
verfasst von: Jun Cai
Qu Feng
William C. Horrace
Guiying Laura Wu
Publikationsdatum: 08.01.2021
Verlag: Springer Berlin Heidelberg
Erschienen in: Empirical Economics / Ausgabe 6/2021
Print ISSN: 0377-7332
Elektronische ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-020-01988-z

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