Speaker A
[music] [music] Hi. Welcome back to the course Applied Production Analysis Using MATLAB. So, in the last two sessions, we discussed extensions of the DEA model. I specifically called them extensions of the DEA model.
Introduction to super-efficiency DEA models, their advantages over conventional DEA, and applications in efficiency ranking and outlier detection.
Key Takeaways
- Super-efficiency DEA models allow efficiency scores to exceed 1, solving ranking issues in conventional DEA.
- Excluding the evaluated firm from the reference set is key to constructing the super-efficiency frontier.
- Super-efficiency models can face feasibility problems, especially under output-oriented approaches.
- These models are useful for both ranking firms and detecting outliers but require careful handling of data and assumptions.
- Understanding the theoretical and practical limitations is crucial for applying super-efficiency DEA in empirical studies.
What the video covers
- The video introduces advanced DEA models focusing on super-efficiency to overcome limitations of conventional DEA models.
- Conventional DEA models produce bounded efficiency scores between 0 and 1, causing ranking difficulties when multiple firms score 1.
- Super-efficiency DEA excludes the evaluated firm from the reference set, allowing efficiency scores greater than 1 and better ranking discrimination.
- The video explains the construction of super-efficiency frontiers and discusses output-oriented and input-oriented approaches.
- Limitations of super-efficiency models are highlighted, including infeasibility issues for certain firms under output orientation.
- Mathematical formulation of the super-efficiency DEA model is presented, including constraints and linear programming setup.
- Differences between variable returns to scale (VRS) and constant returns to scale (CRS) models in the super-efficiency context are discussed.
- Applications of super-efficiency models include ranking firms more accurately and detecting outliers in efficiency analysis.
- Practical challenges such as rule-of-thumb thresholds and data size effects on super-efficiency scores are addressed.
- The video emphasizes revisiting econometric and policy implications of efficiency scores and the importance of careful interpretation.
Chapters
- 00:00Introduction and Review of DEA Extensions
- 01:15Motivation for Super-efficiency DEA Models
- 02:04Limitations of Conventional DEA Models
- 04:21Ranking Issues with Bounded Efficiency Scores
- 07:12Constructing Super-efficiency Frontiers
- 09:24Feasibility Challenges in Super-efficiency Models
- 10:17Summary of Super-efficiency Model Concept
- 11:37Mathematical Formulation of Super-efficiency DEA
- 12:47Linear Programming and Model Constraints
- 15:35Applications and Practical Considerations
Full Transcript — Download SRT & Markdown
Speaker A
Basically, we worked upon only the data point where the reference set was defined as their observation from the current period and the past, which gives us the sequential frontier.
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In case of group frontier, we consider only observation from a particular group. That gives us a group frontier model or their observation from both the groups or the groups that come under a particular industry,
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that gives us the metafrontier. Moving ahead, we see some advanced DEA models. I claim it as advanced DEA model. It's not a simple extension.
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Basically, here also we are doing with the data. Basically, we are changing the reference sets, but still the DEA framework itself is getting modified slightly.
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So, in this session, we motivate ourselves with the background of what are the limitations of the conventional DEA model.
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And we see the specific limitation. One is the bounded nature. How the super efficiency model overcomes the bounded nature of our conventional DEA model.
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And we formally introduce our super efficiency model. And also we see what are the applications of the super efficiency model.
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And then we see what are the limitations that we are going to encounter when you are using a super efficiency model.
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Super efficiency model is basically an efficiency estimation where you are not including the observation, say K, when you're estimating efficiency of firm K or DMU K.
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You don't include that particular observation in the reference set or while forming the technology. That is like in a single line.
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Before going into the details, we see what are the limitations of the simple DEA model starting with a simple example of one output and one input case.
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Say these are the observations. So on and so on. So now if you construct the frontier, we satisfy convexity, free disposability of input and free disposability of output, and we are imposing the VRS assumption over here.
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So here you can see, say, three observations A, B, and C from input-oriented or output-oriented manner, all three of them will be getting technical efficiency value one.
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And the later cases C, D, E, F, G, they will get a technical efficiency lower than one because they are standing away from the frontier.
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So, in an earlier discussion on our DEA model, the very first CCR model, our objective was to get a standardized score such a way that it comes within the bound of zero to one, and anyone without understanding the
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units under consideration or even without understanding the complexity of estimating technical efficiency should be able to understand
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or should be able to interpret the technical efficiency score. But now we are revisiting that bounded nature of our technical efficiency score.
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With this bound, from an econometric point of view, it has a problem. When you have several observations with value one, then few of them with less than one, and then the moment you include that in the regression model, it creates a
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problem. We will revisit that at a later stage. But here, from a policy perspective or from an outcome perspective of the technical efficiency estimation, I want you to see which firm is performing well in this context, whether firm A or
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B or C. So, we are talking about ranking. So, the moment you take this efficiency score, all three of them will be getting value one, and then from a ranking point of view, you'll have to share the rank one among
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all of them, which will be very difficult from our DEA perspective or policy perspective.
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So, this bounded nature, as I mentioned, is an econometric problem or it is coming from the fact that it is more of a parsimonious, like it is not spurious, not parsimonious. It's a spurious outcome.
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Because you don't have the entire sample. Because of not having the entire sample, sometimes if you have the entire population, the frontier would have been different and ABC might not have been the technically efficient firm. That we'll revisit as I
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That's the spuriously efficient case that we revisit at a later stage. So, now we see this ranking perspective, especially when you are using this efficiency score for the ranking purpose.
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So, how to overcome this limitation? In the first instance, what we can do when estimating the technical efficiency of firm B, you can avoid firm B in the sample.
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So, that becomes a case where your frontier will be this way. Okay? So, this is basically the frontier of super efficiency that we are constructing for firm B.
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And now you can see firm B is lying above the frontier. And this will become the potential output under the new frontier that you constructed, and this will become the actual output.
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And then here, once you take the Y by Y star of firm B, it gives you a ratio greater than one. It does not have a very theoretical explanation, but that gives us a score which is greater than one, which allows us to cross the bound of
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our zero and one, specifically. And in case of firm C, what we do is we construct the frontier in this manner without considering firm C.
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So, but by the virtue of this model or the data, it again passes through C, and C will get an efficiency score of one.
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And in case of firm A, it has a problem. In case of firm A, the frontier will be something without observation A.
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So, you can see D. Yeah. So, what is happening here? This technology frontier is something here.
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And this technology set in this context is defined as the set of all feasible output from a theoretical point of view, but we are not including A as a point in the reference set or the point in the
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So, here what you see, the algorithm will search for an optimal value or a value under the optimal value that comes within the technology set.
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Initially, it tried to look in the upper direction, but it will not find. It may search for the lower direction also, it will not find.
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That means for firm A, especially when you are not, especially specifically when you are following the output-oriented approach, it will not get a feasible solution
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under the super efficiency model. That is a limitation that we are going to, oh, limitation that we are going to encounter.
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But at the same case, if you are estimating, we'll see why it is happening at a later stage.
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The same case when you are estimating an input-oriented technical efficiency, this firm may get a super efficiency or efficiency score greater than one because the actual input that you are using is less than the point on the reference set
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that is being constructed under the super efficiency framework. So, just to summarize what we did so far, this reference set is framed without including the particular observation in our particular observation in our DEA model.
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And then we see where the potential outcome from an output-oriented manner, it will be output or from input-oriented manner, input lies, and then we see.
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So, that formulation allows us to have an efficiency score greater than one without any bound of zero, zero to one bound that we are having in the context of convention.
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So, how to formalize it? So, here we can define our variable that is basically, I'm sorry.
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Super efficiency DEA model. We're following output-oriented. Specifically, we are, so since it is output-oriented, it will be a maximization problem.
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Sum of, we are considering one into one case, one output, one input. Lambda J Y J should be greater than or equal to Y K fee over here.
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Case VK star will be the maximum value that you are considering. And while estimating the technical efficiency of firm K under the super efficiency framework, we should not include our K in the observation. So, it will come J not equal to
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K. Okay? And the N is the number of observations we are having. And the input constraint will become lambda J X J should be less than or equal to X K J one up to N, but
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K. And lambda J sum of lambda J should be equal to one for VRS.
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And here also J not equal to K. So, here firm K will not uh arise as its own peer.
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That is basically lambda J should be greater than or equal to zero. And for lambda K, it will take a value zero. So, we need not um avoid the case that J takes a value zero. So, we can have an input-oriented
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counterpart also. At the end, you'll be getting fee K star as the efficiency. So, one by fee K Sorry, fee K star as the optimal value will become the technical efficiency of firm K under output oriented under VRS, but this is
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going to be the super efficiency framework. This is going to be the linear programming problem.
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So, revisiting the uh case of our firm A which was not going to get any feasible solution under uh super efficiency model. So, what happened?
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Here in output oriented approach if the output of any observation, say XA is less than the minimum of XJ, J not equal to A.
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In that case, actually it will not have a feasible solution under input oriented VRS.
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But that is not going to be the case if you are running a CRS model.
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So, if we had a CRS model what it does pick for firm A it would have picked firm B as the observation with maximum uh average productivity and then it passes through this way and you get a DEA frontier
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and of super efficiency of firm A. So, in that case, actually this firm will have a super efficiency which is greater than one in that context. Okay.
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So, what is happening here? The moment you have an output-oriented approach and then any input of the reference firm is less than the minimum of input of the remaining firm.
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The linear programming problem will not be able to satisfy sum of lambda j equal to 1 or sum of lambda j x j less than or equal to x k.
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So, both of them will not be able to satisfy at the same time because whichever observation that you're having in the reference point you put without giving a weightage greater than 1 or less than 1 uh in the less than 1 in the earlier
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case, you'll not be able to get a convex combination. Similarly, when it comes to input-oriented approach, say here we had a uh fee instead of that will become minimization of fee minimization of theta.
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So, in case of in case of our uh firm A, that constraint will get adjusted in such a way that it keeps the value sum of lambda j equal to 1, but theta get adjusted accordingly. So, here we were getting a theta
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greater than 1 or we were getting an efficiency score greater than 1 for that firm.
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So, in input-oriented approach case, if output of observation that firm A or firm K is greater than the maximum of output of y j j not equal to A.
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In this case you get a case of not getting any uh infeasible solution. So, that would have been the case if we had an observation over here. So, for that the frontier will become somewhere below this uh point and the moment you
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try to do input oriented, you will not give any you will not get any point to hit on the frontier or reference point uh on the benchmark point and you get an infeasible solution.
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So, we will revisit this approach. Uh we will revisit uh what are the complication that we get when you are estimating the model.
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Now we first one use or the application of super efficiency model not not uh scale efficiency super efficiency one for ranking.
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The second usage that we can conceptualize is basically for the detection of outliers. Say this was the observation and we have some observation over here.
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So, in the first instance we This is going to be the This is going to be the frontier. This is something we uh encounter when you are doing a uh analysis for SBI and uh other banks in the same data set when you consider SBI
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as a bank, it comes with a lot of or huge amount of output uh because of the brand that the number of branches, so and so it has.
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That is something we encounter. So now we know this observation is not something that comes in the technology because this is contaminating the entire efficiency uh estimation. For that for identifying or detecting outliers in your sample, you can estimate the super
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efficiency model. So what we do, we estimate the super efficiency model for this observation, super efficiency model we estimate it uh using this way.
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For other observation also, so we estimate the super efficiency model. Other than uh unfortunately for the observation this thing, super efficiency will be the same uh what we estimated in the model with uh outlier.
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So what happened? Once you estimate super efficiency we can have a rule of thumb, some literature for 1.26 as a rule of thumb, not necessarily the same for our uh data. So you can have a rule of thumb.
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And any observation that comes with a value greater than that rule of thumb, you can mark them as a outlier.
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And anything less than or value getting one even in the super efficiency estimation or whatever, we can consider it as an observation without any uh claim for outliers.
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So we can use uh super efficiency for ranking purpose and we can use the super efficiency model for the outlier purpose.
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As I pointed, not necessarily you have to follow the rule of thumb. Sometimes you will not be able to follow the rule of thumb, especially when you are having a smaller data.
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And you need to consider two two cases. One is the contamination that the particular observation creates for the entire efficiency estimation.
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Here, you know this observation that we are considering, it is going to contaminate the entire efficiency estimation.
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Then the second case is basically loss of information. Sometime excluding one particular observation, especially say the SBI that I'm considering, in a sample of one particular analysis may cost you a lot over and above the contamination that happens. So, these are basically the
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contradicting cases. The moment you try to reduce the contamination, you're going to likely to create lot of loss of information uh in your case. We will revisit these cases when you are doing the analysis for an empirical example using the same Korean
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electricity utility data. So, just to summarize, in this session we were motivating ourselves uh with the limitations of conventional DEA, there's a need for a new model that is basically the super efficiency model.
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In one word or one sentence, super efficiency model is something that estimate the frontier without considering the particular observation uh in the reference set.
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And it helps us to have a score greater than one or it avoid the bounded nature of efficiency score, so we get more uh observation with greater than one or uh not many will get a value one when
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you're estimating the super efficiency score. Also, the same framework you can use for uh detecting outliers, but it comes with come with a limitation that the moment you estimate an efficiency score under super efficiency framework, it's likely that you may get an
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infeasible solution. Uh we saw the case of infeasible solution in the context of input oriented and output oriented. Thank you.
Speaker A
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Topics:Data Envelopment AnalysisDEASuper-efficiency DEAEfficiency rankingInput-oriented DEAOutput-oriented DEAVRS modelCRS modelOutlier detectionApplied production analysis
