Convex Nonparametric Least Squares Research Articles

Clustering and meta-envelopment in data envelopment analysis

We propose techniques of classification of a potentially heterogeneous data set into groups in a way that is consistent with the intended purpose of the clustering, which is Data Envelopment Analysis (DEA). Using standard clustering techniques and then applying DEA is shown to be sub-optimal in many instances of empirical relevance. Our methods are based on a novel interpretation and implementation of convex nonparametric least squares (CNLS) which allows not only classification into different clusters but also finding the number of clusters from the data. Moreover, we provide techniques for model validation in CNLS regarding the allocation into groups using efficiency criteria. We provide a prior designed to minimize variation within groups and maximize variation across groups. The new techniques are examined using Monte Carlo experiments and they are applied to a data set of large U.S. banks. Additionally, we propose new techniques for meta-envelopment or meta-frontier formulations in efficiency analysis.

A two-stage regression-based benchmarking approach to evaluate school's energy efficiency in different tariff regions

In this paper, we study the recent energy use intensity (EUI) in kWh/m2/year of Hong Kong's secondary schools. We found a 22% increase of EUI from, 2003 (86 kWh/m2/year) to 2017 (105 kWh/m2/year) approximately. It may be due to the increase in the number of air-conditioned classrooms. We then use a novel two-stage regression-based approach to analyse the most important set of statistically significant energy performance factors. Then, the energy use intensity (EUI) is normalized with these energy performance factors to find the benchmarking scores used to assess whether the electricity tariff affects schools' energy efficiency. In the first stage of the approach, we first conduct three methods of fitting regression models, backward, forward, and stepwise (OLS), to find a set of normalized factors and construct the benchmarking model. The first stage provides the normalized EUI range, approximately between 50 and 163 kWh/m2/year. In the second stage, if better goodness of fit is desired in the second stage, convex nonparametric least squares (CNLS) regression analyses are employed with the same set of normalized factors. This two-stage regression-based approach is called the OLS-CNLS approach. Using simple EUI (kWh/m2/year) results, we find that high tariff schools' energy efficiency is better than low tariff schools. However, the benchmarking scores (normalized EUIs) of the OLS-CNLS approach show that the null hypothesis cannot be rejected. The energy efficiency of high tariff schools is not significantly better than that of low tariff schools. It implies that the tariff level is not a crucial factor associated with secondary schools' energy efficiency in Hong Kong. As the OLS-CNLS approach allows us to take all the significant energy performance factors into consideration to address the electricity tariff concern, the funding/budget policies probably motivate the school management to look for better EUI.

A study of energy consumption of secondary school buildings in Hong Kong

Junior and senior secondary classes in Hong Kong are conducted in the same secondary school premise, which consists of a multistorey main (interlocking) building and an assembly hall. According to our survey, the building complex does not use a central cooling or heating system. Instead, air-conditioning is commonly provided using a mix of split- and window-type air conditioners and ceiling fans. Lighting services are provided using fluorescent lamps (T5 and T8), and light-emitting diode (LED) systems are rarely used. The school buildings have no dormitory or canteen; thus, electricity constitute the primary energy consumption. The average energy consumption and energy use intensity per school are 529,925 kWh and 105.61 kWh/m2/year, respectively.On the basis of the characteristics of the management practices of school building, regression analyses involving energy performance factors (independent variables) that can be managed by the school management are conducted. In the perspective of the school management, a regression model with manageable energy performance factors is desirable. We conducted a backward stepwise (BS) regression analysis to achieve this objective. However, because the resulting BS regression model does not provide a satisfactory goodness of fit and only one manageable factor, we performed convex non-parametric least squares (CNLS) to obtain a non-parametric regression model with improved variables and goodness of fit. The obtained CNLS regression model can be used to forecast energy consumption, as confirmed by an illustrative application.

Convex nonparametric least squares and stochastic semi-nonparametric frontier to estimate the shadow prices of PM2.5 and NOx for Taiwan’s transportation modes

With the growing environmental awareness, people are concerned about the air pollution issues because these issues not only cause adverse environmental effects but also damage human’s health conditions. Most of previous studies estimating the shadow prices of emissions mainly focused on stationary sources (e.g. coal-fired power plants) rather than mobile sources; however, traffic is one of critical emission sources. This paper uses convex nonparametric least squares (CNLS) and stochastic semi-nonparametric envelopment of data (StoNED) to estimate the shadow prices of PM2.5 and NOx for eleven transportation modes in seven air quality areas in Taiwan in 2013. We find that their shadow prices for two-stroke and four-stroke scooters are indifferent, and infer that it is unnecessary to develop different emissions policies for these transportation modes. However, the shadow prices for gas-fuelled vehicles which are relatively high and thus we suggest mandating advanced emissions technology or retrofits, or subsidies for electric vehicle replacements of this transportation mode. The significant distinctions in the corresponding shadow prices of PM2.5 and NOx we find for eleven transportation modes in Taiwan’s seven air quality areas indicate that policy-makers and regulators consider a variety of mitigation approaches.

Benchmarking by convex non-parametric least squares with application on the energy performance of office buildings

Regression analysis can be used to develop benchmarking systems for the energy performance of office buildings. A linear regression model can be developed using ordinary least squares (OLS) regression analysis to normalize the factors that affect the energy consumption performance of office buildings and develop the benchmarking model. Poor model fit and the assumption of linearity of OLS are the limitations in developing a reliable benchmarking model. In this study, we introduce and discuss the use of convex non-parametric least squares (CNLS) to develop a benchmarking model using the resulting hyperplanes. CNLS is advantageous in that (i) it is a non-parametric regression method, (ii) does not specify the functional form a priori, and (iii) is used to estimate monotonic increasing and convex functions. The resulting benchmarking model can be enhanced with a good model fit using the three advantages. An illustrative application to office buildings is also provided.

Stochastic non-convex envelopment of data: Applying isotonic regression to frontier estimation

Isotonic nonparametric least squares (INLS) is a regression method for estimating a monotonic function by fitting a step function to data. In the literature of frontier estimation, the free disposal hull (FDH) method is similarly based on the minimal assumption of monotonicity. In this paper, we link these two separately developed nonparametric methods by showing that FDH is a sign-constrained variant of INLS. We also discuss the connections to related methods such as data envelopment analysis (DEA) and convex nonparametric least squares (CNLS). Further, we examine alternative ways of applying isotonic regression to frontier estimation, analogous to corrected and modified ordinary least squares (COLS/MOLS) methods known in the parametric stream of frontier literature. We find that INLS is a useful extension to the toolbox of frontier estimation both in the deterministic and stochastic settings. In the absence of noise, the corrected INLS (CINLS) has a higher discriminating power than FDH. In the case of noisy data, we propose to apply the method of non-convex stochastic envelopment of data (non-convex StoNED), which disentangles inefficiency from noise based on the skewness of the INLS residuals. The proposed methods are illustrated by means of simulated examples.

Multi-Product Marginal Productivity of Efficient Frontiers in Data Envelopment Analysis

Differential characteristics of the production function represent elasticity measures and marginal rates of production technologies. We extend the study by Podinovski and Forsund (2010) and develop multi-product marginal productivity (MP) by showing a consistent estimation of MP generated by data envelopment analysis (DEA), convex nonparametric least squares (CNLS), and directional distance function (DDF). Based on multi-product MP, a meta-DEA approach is developed to address finding the improving direction of the efficient firm on the frontier towards the allocatively efficient benchmarks for marginal profit maximization. This approach, which emphasizes “planning” over “evaluation” and focus on “margin” than “level”, forms the basis for transforming a typical “ex-post” DEA into a novel “ex-ante” DEA study. Two case studies show that the proposed model provides an explicit span of multi-product MP for productivity improvement via a tradeoff between distinct directions.

A more efficient algorithm for Convex Nonparametric Least Squares

Convex Nonparametric Least Squares (CNLSs) is a nonparametric regression method that does not require a priori specification of the functional form. The CNLS problem is solved by mathematical programming techniques; however, since the CNLS problem size grows quadratically as a function of the number of observations, standard quadratic programming (QP) and Nonlinear Programming (NLP) algorithms are inadequate for handling large samples, and the computational burdens become significant even for relatively small samples. This study proposes a generic algorithm that improves the computational performance in small samples and is able to solve problems that are currently unattainable. A Monte Carlo simulation is performed to evaluate the performance of six variants of the proposed algorithm. These experimental results indicate that the most effective variant can be identified given the sample size and the dimensionality. The computational benefits of the new algorithm are demonstrated by an empirical application that proved insurmountable for the standard QP and NLP algorithms.

Estimating the shadow prices of SO2 and NOx for U.S. coal power plants: A convex nonparametric least squares approach

Weak disposability between outputs and pollutants, defined as a simultaneous proportional reduction of both outputs and pollutants, assumes that pollutants are byproducts of the output generation process and that a firm can “freely dispose” of both by scaling down production levels, leaving some inputs idle. Based on the production axioms of monotonicity, convexity and weak disposability, we formulate a convex nonparametric least squares (CNLS) quadratic optimization problem to estimate a frontier production function assuming either a deterministic disturbance term consisting only of inefficiency, or a composite disturbance term composed of both inefficiency and noise. The suggested methodology extends the stochastic semi-nonparametric envelopment of data (StoNED) described in Kuosmanen and Kortelainen (2011). Applying the method to estimate the shadow prices of SO2 and NOx generated by U.S. coal power plants, we conclude that the weak disposability StoNED method provides more consistent estimates of market prices.

One-stage and two-stage DEA estimation of the effects of contextual variables

Two-stage data envelopment analysis (2-DEA) is commonly used in productive efficiency analysis to estimate the effects of operational conditions and practices on performance. In this method the DEA efficiency estimates are regressed on contextual variables representing the operational conditions. We re-examine the statistical properties of the 2-DEA estimator, and find that it is statistically consistent under more general conditions than earlier studies assume. We further show that the finite sample bias of DEA in the first stage carries over to the second stage regression, causing bias in the estimated coefficients of the contextual variables. This bias is particularly severe when the contextual variables are correlated with inputs. To address this shortcoming, we apply the result that DEA can be formulated as a constrained special case of the convex nonparametric least squares (CNLS) regression. Applying the CNLS formulation, we develop a new semi-nonparametric one-stage estimator for the coefficients of the contextual variables that directly incorporates contextual variables to the standard DEA problem. The proposed method is hence referred to as one-stage DEA (1-DEA). Evidence from Monte Carlo simulations suggests that the new 1-DEA estimator performs systematically better than the conventional 2-DEA estimator both in deterministic and noisy scenarios.

A More Efficient Algorithm for Convex Nonparametric Least Squares

Convex Nonparametric Least Squares (CNLS) is a nonparametric regression method that does not require a priori specification of the functional form. The CNLS problem is solved by mathematical programming techniques; however, since the CNLS problem size grows quadratically as a function of the number of observations, standard Quadratic Programming (QP) and Nonlinear Programming (NLP) algorithms are inadequate for handling large samples, and the computational burdens become significant even for relatively small samples. This study proposes a generic algorithm that improves the computational performance in small samples and is able to solve problems that are currently unattainable. A Monte Carlo simulation is performed to evaluate the performance of six variants of the proposed algorithm. These experimental results indicate that a particular variant is most efficient given the sample size and the dimensionality. The computational benefits of the new algorithm are demonstrated by an empirical application that proved insurmountable for the standard QP and NLP algorithms.

Stochastic non-smooth envelopment of data: semi-parametric frontier estimation subject to shape constraints

The field of productive efficiency analysis is currently divided between two main paradigms: the deterministic, nonparametric Data Envelopment Analysis (DEA) and the parametric Stochastic Frontier Analysis (SFA). This paper examines an encompassing semiparametric frontier model that combines the DEA-type nonparametric frontier, which satisfies monotonicity and concavity, with the SFA-style stochastic homoskedastic composite error term. To estimate this model, a new two-stage method is proposed, referred to as Stochastic Non-smooth Envelopment of Data (StoNED). The first stage of the StoNED method applies convex nonparametric least squares (CNLS) to estimate the shape of the frontier without any assumptions about its functional form or smoothness. In the second stage, the conditional expectations of inefficiency are estimated based on the CNLS residuals, using the method of moments or pseudolikelihood techniques. Although in a cross-sectional setting distinguishing inefficiency from noise in general requires distributional assumptions, we also show how these can be relaxed in our approach if panel data are available. Performance of the StoNED method is examined using Monte Carlo simulations.

How Operational Conditions and Practices Affect Productive Performance? Efficient Semi-Parametric One-Stage Estimators

Understanding the effects of operational conditions and practices on productive efficiency can provide insights that lead to improved managerial strategies. Previous studies have employed a two-stage data envelopment analysis (DEA) method where the DEA efficiency estimates are regressed on contextual variables representing the operational conditions. We re-examine the statistical properties of the two-stage DEA estimator, and find that it is statistically consistent under more general conditions than earlier studies assume. However, the finite sample bias of DEA in the first stage carries over to the second stage regression, causing bias in the estimated coefficients of the contextual variables. This bias is particularly severe when the contextual variables are correlated with inputs. To address this shortcoming, we apply insights from Kuosmanen and Johnson (2009), who showed that DEA can be formulated as a constrained special case of the Convex Nonparametric Least Squares (CNLS) regression. Utilizing this link between DEA and the regression analysis, we develop two new semi-parametric one-stage estimators for the coefficients of the contextual variables. The first estimator directly incorporates contextual variables to the standard DEA model, and is hence referred to as one-stage DEA. The second estimator introduces contextual variables to the CNLS regression, relaxing the sign-constraint of DEA. We show that both one-stage estimators are statistically consistent, but the CNLS-estimator requires less restrictive assumptions. The CNLS-estimator for the contextual variables is shown to be unbiased, asymptotically efficient, asymptotically normally distributed, and converge at the standard parametric rate of order n-1/2. Therefore, the conventional methods of statistical inference can be applied to the CNLS-estimator. Finite sample performance of the proposed estimators is examined through Monte Carlo simulations and demonstrates benefits over two-stage DEA.