Abstract
This chapter describes the economic insights of the unifying framework known as Stochastic semi-Nonparametric Envelopment of Data (StoNED), which combines the virtues of the widely used neoclassic production models, Data Envelopment Analysis (DEA), and Stochastic Frontier Analysis (SFA). Like DEA, StoNED is able to estimate an axiomatic production function relaxing the functional form specification required in most implementations of SFA. However, StoNED is also consistent with the econometric models of noise, providing a distinct advantage over standard DEA models. Further, StoNED allows for the possibility that systematic inefficiency is negligible consistent with neoclassical theory, thus providing a unifying framework. StoNED is implemented by estimating a conditional mean using convex nonparametric least squares (CNLS) followed by using standard SFA techniques to estimate the average efficiency and decompose the residual. Detailed descriptions of General Algebraic Modeling System (GAMS) and matrix laboratory (MATLAB) code will aid readers in implementing the StoNED estimator.
The authors would like to gratefully acknowledge both the support from the Aalto Energy Initiative, as part of the Sustainable Transition of European Energy Markets—STEEM project and the Finnish Energy Market Authority for providing the data on the performance of electricity distributors in Finland. We are also indebted to Abolfazl Keshvari for his helpful comments and his assistance in developing Fig. 3.8. Additional codes and materials are available at http://www.andyjohnson.guru
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Notes
- 1.
We have found CVX, an additional toolbox that must be downloaded separately, for MATLAB performs well. Also our experience is, CPlex, Minos, XA are solvers for GAMS that perform well. However, because the computational optimization algorithms differ between software, often slight differences in the results exist for both QP and NLP problems.
- 2.
For extensions to the general multi-input multi-output setting, see Kuosmanen et al. (2014).
- 3.
See Sect. 2.3.2 of the Chapter by Ray and Chen in this book for a more detailed description of the assumptions regarding the production possibility set.
- 4.
Modeling heteroskedastic inefficiency and noise is discussed in Kuosmanen et al. (2014), Sect. 8.
- 5.
Our discussion centers on estimators based on ordinary least squares. The attempts of Banker and Maindiratta (1992) to combine axiomatic estimation with standard models of noise in a maximum likelihood framework should also be recognized. However, to the best of our knowledge no applications of this maximum likelihood approach exist do to computational challenges.
- 6.
We follow the terminology of Chen (2007), who provides the following intuitive definition: “An econometric model is termed ‘parametric’ if all of its parameters are in finite dimensional parameter spaces; a model is ‘nonparametric’ if all of its parameters are in infinite-dimensional parameter spaces; a model is ‘semiparametric’ if its parameters of interests are in finite-dimensional spaces but its nuisance parameters are in infinite-dimensional spaces; a model is ‘semi-nonparametric’ if it contains both finite-dimensional and infinite-dimensional unknown parameters of interests” Chen (2007, p 5552, footnote 1).
- 7.
The Finnish Energy Market Authority measures CAPEX as the replacement value of the capital stock owned by the distributor depreciated by a constant depreciation rate. Thus, CAPEX is directly proportional to the total capital stock.
- 8.
The only distinction between parameters and variables in GAMS is variables are determined as the results of an optimization problem, whereas parameters are assigned values via calculations or assign statements.
- 9.
When entering data, be sure to use good practices regarding significant figures. If you include data with many significant figures, this will increase computational time significantly.
- 10.
Note the path should be adjusted to point to the location where the data file is saved.
- 11.
- 12.
From this point forward, we will refer to convex regression, recognizing that concave regression can be achieved through reversing an inequality, discussed in Sect. 3.3.2.
- 13.
In a power curve or s-shape single-input production function, the inflection point in the input value at which the second derivative changes sign or in other words where the production function changes from being a convex function to a concave function.
- 14.
- 15.
Note in our notation, \({\varvec{\upbeta}}_{i}^{\prime } {\mathbf{x}}_{i} = \beta_{i1} x_{i1} + \beta_{i2} x_{i2} + \cdots + \beta_{im} x_{im} .\) Further, this formulation is intended to show the relationship to other mathematical models, i.e., classic OLS regression and the Afriat inequalities. For computational purposes, the problem may be reformulated to reduce the number of variables and/or constraints as discussed in Sect. 3.4.1.
- 16.
For those familiar with DEA, the parameters \(\alpha_{i}\) and \({\varvec{\upbeta}}_{i}\) are analogous to \(u_{0}\) and \({\mathbf{u}}\) in the multiplier formulation of DEA.
- 17.
The linear program used to calculate the lower bound function \(\hat{f}_{\hbox{min} }^{{\text{CNLS}}}\) is equivalent to the DEA estimator under the assumption of variables returns to scale and replacing the observed output levels with the estimated output level \(\hat{f}^{{\text{CNLS}}} ({\mathbf{x}}_{i} )\) coming from (3.5).
- 18.
Our experiments with GAMS were performed on a personal computer with an Intel Core i7 CPU 1.60 GHz and 8-GB RAM. The optimization problems were solved in GAMS 23.3 using the CPLEX 12.0 Quadratically Constrained Program (QCP) solver. Our experiments with MATLAB were performed on a laptop computer with an Intel Core i5 CPU 2.50 GHz and 4-GB RAM.
- 19.
- 20.
Approximation algorithms are also possible strategies, but we focus on calculating the exact solution to the CNLS formulation.
- 21.
Lee et al. found that if there were more than 100 observations, the group strategy for adding constraints was always preferred to other methods tested and that the sweet spot strategy’s threshold value could be adjusted based on the number of observations and the dimensionality of the data. In the experiments of Lee et al., they generate input data uniformly and do not correlate the inputs. However, when input variables are correlated CNLS becomes easier to solve. Thus, in observed data where the inputs are typically highly correlated, the computational improvement will allow problems even larger than 1,000 observations to be solved.
- 22.
- 23.
Some preliminary test indicates that XA is very effective for solving CNLS problems.
- 24.
For a more extensive summary, see Kuosmanen et al. (2014).
- 25.
Also called “the no free lunch” axiom, it states that the production of positive output is impossible without the use of at least one input.
- 26.
They limit their computational time to 5 h and use a GAMS/CPlex implementation.
- 27.
- 28.
The DEA literature defines nonincreasing returns to scale and nondecreasing returns to scale production functions. Within CNLS, similar production functions can be estimated by imposing restrictions on the coefficients \(\alpha_{i}\)
-
Nonincreasing returns to scale (NIRS): impose \(\alpha_{i} \ge 0\,\forall \,i\)
-
Nondecreasing returns to scale (NDRS): impose \(\alpha_{i} \le 0\,\forall \,i\)
.
-
- 29.
Here, we construct a vector call it r, such that \(r_{i} = \ln y_{i} - \ln (\hat{\phi }_{i} )\), then r is regressed on z without an intercept term.
- 30.
We do not advocate this solution to limited data. We see deterministic estimators as useful when the deterministic assumption is likely to hold.
- 31.
In (3.18), since all of the \(\varepsilon_{i}^{\text{CNLS - }}\) are nonpositive, squaring the objective is simply a monotonic transformation, and thus, it is not necessary.
- 32.
However, Stochastic semi-Nonparametric Envelopment of Data (StoNED) can be used.
- 33.
The average value, \(\mu\), is typically a function of the parameters of the distribution of u. For example, if u is distributed half-normally, then \(E(u) = \sqrt {2/2\pi } \sigma_{u}\) where \(\sigma_{u}\) is the pretruncated standard deviation of u. More discussion related to this point is provided in Sect. 3.7.2.
- 34.
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Appendices
Appendix 1–Codes
GAMSCodeStandard Formulation
GAMSCodeCNLS+G
Matlab Code–Standard Formulation
Matlab Code–CNLS+G Formulation
Appendix 2
Data
OPEX | CAPEX | TOTEX | Energy | Length | Customers | PerUndGr | |
|---|---|---|---|---|---|---|---|
1 | 681 | 729 | 1612 | 75 | 878 | 4933 | 0.11 |
2 | 559 | 673 | 1659 | 62 | 964 | 6149 | 0.21 |
3 | 836 | 851 | 1708 | 78 | 676 | 6098 | 0.75 |
4 | 7559 | 8384 | 18918 | 683 | 12522 | 55226 | 0.13 |
5 | 424 | 562 | 1167 | 27 | 697 | 1670 | 0.03 |
6 | 1483 | 1587 | 3395 | 295 | 953 | 22949 | 0.65 |
7 | 658 | 570 | 1333 | 44 | 917 | 3599 | 0.11 |
8 | 1433 | 1311 | 3518 | 171 | 1580 | 11081 | 0.16 |
9 | 850 | 564 | 1415 | 98 | 116 | 377 | 1.00 |
10 | 1155 | 1108 | 2469 | 203 | 740 | 10134 | 0.64 |
11 | 14235 | 11594 | 28750 | 2203 | 7007 | 167239 | 0.61 |
12 | 44481 | 50321 | 117554 | 6600 | 67611 | 420473 | 0.23 |
13 | 1116 | 766 | 1925 | 117 | 436 | 7176 | 0.61 |
14 | 1604 | 946 | 2747 | 135 | 902 | 8614 | 0.46 |
15 | 27723 | 19818 | 48605 | 3601 | 6007 | 334757 | 0.92 |
16 | 2480 | 2420 | 5486 | 409 | 2773 | 14953 | 0.19 |
17 | 494 | 476 | 1091 | 43 | 506 | 3156 | 0.32 |
18 | 801 | 466 | 1297 | 61 | 541 | 4296 | 0.05 |
19 | 875 | 555 | 1691 | 62 | 1081 | 6044 | 0.07 |
20 | 2133 | 1913 | 4605 | 256 | 2540 | 23361 | 0.31 |
21 | 1139 | 1635 | 3102 | 197 | 1817 | 6071 | 0.05 |
22 | 907 | 1127 | 2260 | 200 | 1106 | 14936 | 0.49 |
23 | 120 | 106 | 341 | 17 | 133 | 772 | 0.06 |
24 | 3454 | 2428 | 6100 | 489 | 1312 | 44594 | 0.87 |
25 | 535 | 479 | 1440 | 53 | 789 | 3391 | 0.05 |
26 | 974 | 754 | 1958 | 95 | 971 | 6806 | 0.37 |
27 | 929 | 853 | 1976 | 75 | 869 | 5165 | 0.24 |
28 | 9842 | 13925 | 29722 | 985 | 25611 | 95367 | 0.09 |
29 | 548 | 412 | 1254 | 123 | 51 | 24 | 0.44 |
30 | 1456 | 1136 | 2665 | 165 | 875 | 14646 | 0.71 |
31 | 725 | 569 | 1376 | 73 | 716 | 5069 | 0.39 |
32 | 2525 | 388 | 3121 | 540 | 70 | 58 | 0.31 |
33 | 2002 | 1442 | 3864 | 300 | 1301 | 20325 | 0.47 |
34 | 1846 | 1112 | 3221 | 207 | 429 | 16878 | 0.73 |
35 | 982 | 1094 | 2561 | 99 | 1618 | 8566 | 0.20 |
36 | 2727 | 2151 | 5779 | 164 | 3330 | 12231 | 0.12 |
37 | 1799 | 2073 | 4380 | 171 | 3736 | 15217 | 0.06 |
38 | 604 | 675 | 1423 | 73 | 989 | 5711 | 0.25 |
39 | 400 | 430 | 907 | 40 | 646 | 2968 | 0.20 |
40 | 4092 | 3173 | 7915 | 482 | 3294 | 42952 | 0.40 |
41 | 3362 | 3078 | 6639 | 456 | 1375 | 48140 | 0.66 |
42 | 390 | 438 | 868 | 23 | 589 | 2227 | 0.18 |
43 | 10852 | 9366 | 25556 | 1233 | 12512 | 98650 | 0.13 |
44 | 688 | 700 | 1540 | 85 | 866 | 6022 | 0.36 |
45 | 761 | 701 | 1564 | 100 | 800 | 7193 | 0.40 |
46 | 453 | 576 | 1229 | 25 | 1078 | 3342 | 0.04 |
47 | 4076 | 4007 | 9807 | 494 | 4696 | 43911 | 0.29 |
48 | 308 | 297 | 669 | 17 | 432 | 1752 | 0.03 |
49 | 2746 | 2529 | 6097 | 315 | 4042 | 26265 | 0.20 |
50 | 5614 | 5509 | 12154 | 1042 | 4296 | 75870 | 0.60 |
51 | 400 | 519 | 1186 | 39 | 614 | 2211 | 0.01 |
52 | 1821 | 1753 | 4020 | 223 | 2117 | 12945 | 0.09 |
53 | 794 | 747 | 1589 | 98 | 418 | 5146 | 0.66 |
54 | 2269 | 2795 | 6414 | 348 | 2127 | 21072 | 0.37 |
55 | 711 | 556 | 1515 | 77 | 762 | 4513 | 0.16 |
56 | 4609 | 5342 | 10600 | 993 | 3205 | 80702 | 0.70 |
57 | 1766 | 2338 | 5431 | 402 | 3207 | 25994 | 0.13 |
58 | 813 | 666 | 1872 | 130 | 905 | 5394 | 0.19 |
59 | 884 | 1104 | 2206 | 138 | 1423 | 9015 | 0.26 |
60 | 1662 | 1358 | 3767 | 117 | 2532 | 9930 | 0.24 |
61 | 81 | 106 | 268 | 22 | 133 | 1467 | 0.22 |
62 | 11776 | 11864 | 28295 | 988 | 20934 | 84445 | 0.04 |
63 | 4021 | 3767 | 9689 | 749 | 3225 | 47572 | 0.53 |
64 | 2597 | 3224 | 7226 | 378 | 3567 | 30801 | 0.21 |
65 | 995 | 848 | 1871 | 95 | 340 | 7812 | 0.89 |
66 | 548 | 587 | 1280 | 43 | 977 | 4272 | 0.02 |
67 | 1573 | 1780 | 3539 | 237 | 882 | 19455 | 0.52 |
68 | 4129 | 4001 | 9853 | 440 | 6330 | 26798 | 0.22 |
69 | 2151 | 1450 | 3758 | 266 | 772 | 21662 | 0.83 |
70 | 2438 | 2496 | 5499 | 316 | 4117 | 22313 | 0.28 |
71 | 14064 | 15175 | 37368 | 1601 | 24485 | 106336 | 0.08 |
72 | 2058 | 1521 | 3735 | 268 | 928 | 19899 | 0.74 |
73 | 8643 | 6819 | 16141 | 1654 | 3567 | 124661 | 0.59 |
74 | 483 | 367 | 987 | 37 | 730 | 2611 | 0.08 |
75 | 1018 | 939 | 2067 | 158 | 822 | 10537 | 0.56 |
76 | 1593 | 2326 | 5105 | 196 | 3470 | 13391 | 0.09 |
77 | 7501 | 4734 | 12687 | 1141 | 2360 | 67456 | 0.62 |
78 | 305 | 411 | 861 | 19 | 520 | 1207 | 0.17 |
79 | 5426 | 6446 | 12831 | 787 | 5808 | 60239 | 0.41 |
80 | 2618 | 2795 | 6055 | 293 | 3741 | 23446 | 0.20 |
81 | 1033 | 951 | 2156 | 137 | 902 | 11654 | 0.39 |
82 | 6786 | 6638 | 13794 | 1281 | 3009 | 93769 | 0.75 |
83 | 2169 | 2172 | 5054 | 210 | 3693 | 17129 | 0.16 |
84 | 40787 | 45434 | 108310 | 4825 | 60659 | 378089 | 0.18 |
85 | 2741 | 2475 | 6162 | 310 | 3381 | 19059 | 0.16 |
86 | 307 | 225 | 594 | 28 | 351 | 2078 | 0.07 |
87 | 321 | 281 | 672 | 30 | 338 | 2008 | 0.32 |
88 | 300 | 289 | 616 | 15 | 318 | 1364 | 0.01 |
89 | 891 | 693 | 1776 | 105 | 575 | 9084 | 0.59 |
Estimated residuals
Resid | |
|---|---|
1 | −2.93 |
2 | 1.30 |
3 | −22.31 |
4 | −350.89 |
5 | −13.46 |
6 | 100.99 |
7 | −29.04 |
8 | −14.06 |
9 | −1.00 |
10 | 56.88 |
11 | 285.52 |
12 | 679.39 |
13 | −20.31 |
14 | −70.03 |
15 | 10.52 |
16 | 74.58 |
17 | −6.73 |
18 | −30.25 |
19 | −40.29 |
20 | −27.80 |
21 | 48.75 |
22 | 87.00 |
23 | 26.07 |
24 | 23.93 |
25 | −2.33 |
26 | −22.94 |
27 | −37.94 |
28 | −351.05 |
29 | 66.65 |
30 | −21.72 |
31 | −9.11 |
32 | 215.96 |
33 | 37.57 |
34 | −31.84 |
35 | −23.79 |
36 | −201.37 |
37 | −69.31 |
38 | 6.13 |
39 | 3.56 |
40 | −74.62 |
41 | −1.99 |
42 | −12.15 |
43 | −236.73 |
44 | 6.44 |
45 | 11.53 |
46 | −19.80 |
47 | −67.22 |
48 | −5.47 |
49 | −55.97 |
50 | 265.53 |
51 | 2.17 |
52 | −17.68 |
53 | 4.54 |
54 | 38.84 |
55 | −3.07 |
56 | 349.55 |
57 | 163.97 |
58 | 34.87 |
59 | 28.38 |
60 | −99.16 |
61 | 33.36 |
62 | −604.02 |
63 | 197.10 |
64 | 21.41 |
65 | −26.83 |
66 | −15.29 |
67 | 29.36 |
68 | −128.27 |
69 | −16.48 |
70 | −13.41 |
71 | −293.84 |
72 | −2.58 |
73 | 476.64 |
74 | −10.02 |
75 | 32.04 |
76 | −17.26 |
77 | 114.79 |
78 | −3.54 |
79 | 35.42 |
80 | −62.95 |
81 | 8.88 |
82 | 349.42 |
83 | −80.72 |
84 | −604.75 |
85 | −59.86 |
86 | 6.47 |
87 | 5.94 |
88 | −6.29 |
89 | −1.01 |
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Johnson, A.L., Kuosmanen, T. (2015). An Introduction to CNLS and StoNED Methods for Efficiency Analysis: Economic Insights and Computational Aspects. In: Ray, S., Kumbhakar, S., Dua, P. (eds) Benchmarking for Performance Evaluation. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2253-8_3
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Publisher Name: Springer, New Delhi
Print ISBN: 978-81-322-2252-1
Online ISBN: 978-81-322-2253-8
eBook Packages: Business and EconomicsBusiness and Management (R0)