**Extensions of Basic DEA Models — Transcript & Summary | SozAI**
Source: https://sozai.app/transcript/extensions-basic-dea-models/

Explore extensions of basic DEA models including meta frontier, group frontier, and technology closeness ratio using MATLAB for efficiency analysis.

## Key Takeaways

- Meta frontier analysis provides a broad efficiency benchmark across an entire sector.
- Group frontier analysis offers a more realistic efficiency measure within specific subsectors allowing for technological heterogeneity.
- Technology Closeness Ratio helps identify which subsectors are closer to the overall sector's technology frontier.
- Sequential frontier analysis is useful for panel data to track efficiency changes over time.
- DEA models can be complemented or replaced by parametric methods like stochastic frontier or corrected OLS.

## What the video covers

- Introduction to extensions of basic Data Envelopment Analysis (DEA) models focusing on meta frontier and group frontier concepts.
- Explanation of how meta frontier includes all observations from a broader sector, while group frontier focuses on specific subsectors or groups.
- Discussion on the use of parametric approaches like stochastic frontier and corrected OLS as alternatives to DEA for frontier estimation.
- Illustration of the meta frontier and group frontier concepts using a food processing industry example with sugar and vegetable oil subsectors.
- Introduction of the Technology Closeness Ratio (TCR) as a measure comparing technical efficiency under meta frontier and group frontier.
- Mathematical formulation of technical efficiency averages using geometric means for both group and meta frontiers.
- Interpretation of TCR as an indicator of how close a subsector's technology is to the overall sector's technology, allowing for heterogeneity.
- Proof that TCR is always less than or equal to one, as the meta frontier envelops the group frontier.
- Introduction to sequential frontier analysis when dealing with panel data to capture technology changes over time.
- Use of linear programming formulations to estimate efficiencies and frontiers in both input-oriented and output-oriented frameworks.

## Chapters

1. 00:00 Introduction to DEA and course overview
2. 01:43 Parametric approaches vs DEA for frontier estimation
3. 03:45 Example: Meta frontier and group frontier in food processing
4. 05:19 Importance of group frontier for policy and performance comparison
5. 07:18 Defining technical efficiency and Technology Closeness Ratio (TCR)
6. 09:03 Mathematical formulation of efficiency averages
7. 11:04 Interpretation and properties of Technology Closeness Ratio
8. 14:07 Sequential frontier analysis for panel data
9. 17:11 Linear programming formulation for sequential frontier
10. 18:10 Summary and concluding remarks on DEA extensions

Answers

## Questions about this video

What is the difference between meta frontier and group frontier in DEA?

Meta frontier is constructed using all observations from the entire sector, representing the broadest technology frontier. Group frontier is constructed using observations from a specific subsector or group, allowing for heterogeneity in technology within the sector.

How does the Technology Closeness Ratio (TCR) help in efficiency analysis?

TCR compares the technical efficiency of a subsector estimated under the meta frontier to that estimated under the group frontier, indicating how close the subsector's technology is to the overall sector's technology.

Can DEA be replaced by other methods for frontier estimation?

Yes, parametric approaches such as stochastic frontier analysis or corrected ordinary least squares (OLS) can also be used to estimate meta frontiers or group frontiers instead of DEA.

## Full Transcript — Download SRT & Markdown

00:05

Speaker A

[music] [music] Hi, welcome back to the course of player production analysis using MATLAB. In the last session, we familiarized ourselves with one of the most popular tools used in the field of efficiency and productivity analysis, that is data envelopment analysis.

00:31

Speaker A

We also saw how to use MATLAB for estimating technical efficiency using DEA. In this session, I'll take you through extensions of the basic DEA model. When it comes to extensions, there are a lot of extensions.

00:54

Speaker A

But today's session, we'll see two sets of extensions and the derived outcomes of that, especially from the data point of view.

01:06

Speaker A

The moment you give data to the DEA model, say you have one sector within that there are two product groups. If you give the entire sector, that will become the meta frontier.

01:26

Speaker A

And suppose you want to give the model only the data of a particular sector or a group, we can call it the group frontier. We see how to construct the group frontier and meta frontier using DEA models. Not necessarily you

01:43

Speaker A

need to use DEA for estimating meta frontier or group frontier. You can use even parametric approaches, say stochastic frontier or even corrected OLS.

01:55

Speaker A

And then the derived outcome, we can take the ratio of these two efficiency scores estimated again meta frontier and group frontier, which gives you an idea about technology closeness ratio.

02:11

Speaker A

Alternatively, when you are having panel data, we can conceptualize something called sequential that we are going to discuss in detail in this session.

02:26

Speaker A

Meta frontier and group frontier. Let's consider a very simple case: one input, one output again, and assume you now have one industry. Say, for the simple example, you can consider food processing, and within that, we can have two segments,

03:12

Speaker A

say sugar and say the other one vegetable oil. Both of them come under the broader classification of food processing. Say you have one input and one output, and you are having observation from both the subsegments or the groups I would

03:45

Speaker A

call hereafter. See, these are the observations from the sugar industry, and the green dots are the observations from the vegetable oil sector.

04:06

Speaker A

So following the framework what we had so far, say you want to estimate the efficiency of these sectors which come under the broader segment of food processing.

04:20

Speaker A

What we did so far, we constructed a frontier which is going to be the point that connects the entire observations, and here I would call this the meta frontier.

04:45

Speaker A

So in case of food processing with two subsegments, sugar and vegetable oil, we construct the meta frontier by considering the entire observation that comes under the broader classification.

05:00

Speaker A

But as you can see in this framework, if you compare the sugar producing unit, which also comes under the food processing sector, against this frontier, it's very unfair or it's very unrealistic in some sense.

05:19

Speaker A

That is the fact number one. Alternatively, as a policymaker, you'll be interested to see which subsegment of the particular industry is performing better or lagging behind as compared to the other segment or other group.

05:42

Speaker A

So in that case, we introduce something called group frontier, which is basically a frontier constructed particularly for the group by considering the observation that comes under the particular group. In this case, the group frontier of vegetable oil, that is marked as

06:18

Speaker A

green dots. It remains the same frontier as against that the group frontier of sugar industry is going to be something from this point, this point, this way.

06:44

Speaker A

So what is the interpretation over here? Here you can see vegetable oil as a subsector. It is lying on the boundary of the production possibility set, but it is constructing the food processing sector's production frontier, or sugar as a

07:03

Speaker A

sector that is lying behind. These concepts are from Hayami and Trend, which are 1969-72, that you can see several works by Hayami and Trend which gives you the theoretical foundation of this thing.

07:18

Speaker A

So our objective is to see whether the construction of meta frontier and group frontier we can extend to make a bit more logical reasoning or get a very objective measure that is basically the technology closeness ratio.

07:39

Speaker A

So what we can do, we can have technical efficiency of group G, the subscript G is basically the sector, the subsector that I'm referring to. It can be estimated against—sorry, this G is basically the estimated again group frontier, and this within

08:14

Speaker A

bracket we are mentioning that of group G. It can be defined as a—since it is a ratio for an average, that for getting the average value it is better to take a geometric mean. So we take a geometric

08:32

Speaker A

mean by taking the product of k=1 up to NG. NG is basically the number of observations that come under the particular group.

08:43

Speaker A

Then we can have technical efficiency again group frontier, and we take 1 by NG. So that gives you the nth root of the entire product that you are getting.

09:03

Speaker A

So this will be basically the average efficiency of group G estimated against group frontier,

09:35

Speaker A

which is basically estimated against the group frontier. Alternatively, you can have technical efficiency of the same group but estimated again meta frontier, which is going to be the product of k=1 up to NG again where NG is the number of

10:00

Speaker A

observations in group G into TE of k but estimated again meta frontier into NG square root.

10:19

Speaker A

So this is basically the average estimated against group frontier, and this is basically the average estimated again meta frontier.

10:32

Speaker A

Now we introduce a new concept that is basically TCR, technology closeness ratio. That is basically technical efficiency of group G estimated under meta frontier to technical efficiency of the same group estimated against group frontier.

11:04

Speaker A

What is the intuitive explanation of this ratio? This ratio tells how close each subsector or subsegment's frontier is to the frontier constructed for the entire sector. Also, this group frontier analysis is a bit more realistic because by constructing a meta frontier

11:28

Speaker A

we are imposing the same technology throughout the subsector. But the moment you are going for a group frontier analysis, that allows for heterogeneity in terms of technology when it comes to the subsectors, and this ratio, technology closeness ratio,

11:48

Speaker A

it tells which sector is close to that or by looking at technology closeness ratio we'll be able to see which sector is performing better in terms of efficiency as compared to other subsegments in the sector or

12:04

Speaker A

which sector is lagging behind its comparable counterparts in the particular sector. So here by formulation you can see meta frontier is a framework that considers entire observation and group frontier is something that considers only the observation from the particular group.

12:31

Speaker A

So with that formulation, the technology closeness ratio will be always greater than or equal to one.

12:37

Speaker A

Sorry, less than or equal to one because in meta you have the entire observation and the group frontier you have only the particular observation from the particular group, that is only NG observation. Say N is the entire observation and NG is basically the

12:54

Speaker A

observation that comes under the particular group. So going into the details, what guarantees that the ratio will always be less than or equal to one? From a diagrammatic point of view, the meta frontier will always lie above the group frontier or group

13:15

Speaker A

frontier is a subset of meta frontier. As a result, the distance to move to hit on the meta frontier will be larger than that of a group frontier.

13:30

Speaker A

From a formulation point of view or a linear programming problem point of view, meta frontier consists of more observations or group frontier consists of less observations.

13:47

Speaker A

Having said that, technical efficiency will always be a non-increasing function of the number of observations that you are including into the formulation or the linear programming problem. The moment you have a new observation, it is not going to pull down your

14:08

Speaker A

frontier. And as a result, the moment you have a new observation added to the data set, it never increases the efficiency of the observation being estimated earlier before that adding a new observation. It always reduces the efficiency.

14:31

Speaker A

frontier above. Never pull the frontier below. That is the concept. So conducting that you can see that techn the average technical efficiency that you estimated again meta frontier will always be less than the average technical efficiency that you estimated

14:48

Speaker A

again the group frontier as a result this technology crossness ratio will lie between zero and one of these.

14:58

Speaker A

So, so far whatever discussed it comes under the framework of cross-section data. But the moment you have panel data, we can have a model called sequential frontier model.

15:25

Speaker A

This also not necessarily under the DEA framework. You can conceptualize a sequential frontier even in the case of parametric approaches.

15:36

Speaker A

What does sequential frontier do? Let's consider a simple case of X and Y. But we have year also and ID over here.

15:53

Speaker A

Simple one input one output case. So you have a b c a b c for year 1 one one and for your p two 2 and the corresponding input output value.

16:21

Speaker A

So can we pull the entire data into one uh matrix and do an efficiency analysis?

16:30

Speaker A

No. Because the moment you pull the entire data into one data for the both the period it will create a bit more unrealistic picture for period one because especially from a DEA point of view we have assumption one

16:50

Speaker A

all actually observed input output bundle are feasible. The moment you combine both the data coming from period 1 and two into one data set, it is basically questioning the assumption one whether in period 1 the outcome being observed in period 2

17:11

Speaker A

or or outcome being observed in period 2 were feasible or not. So we cannot uh combine both the data.

17:21

Speaker A

If you are having a panel data, you need to be bit more careful while constructing the front.

17:29

Speaker A

Asking the same question from a period 2 perspective, can I say that period 1 observation was feasible in terms of period 2?

17:46

Speaker A

We can logically say yes because whatever technology available in period 1 should remain in period 2 also. From a technological perspective the input output bundle that you realize in the context of period one should be feasible in the period 2 also condition

18:05

Speaker A

that the technology is not regressive which is rarely happen in the context of upload analysis.

18:12

Speaker A

So now having said that what you can do it doesn't make any problem or a theoretical inconsistencies even if you include observation from period 1 to period 2's analysis. So what are the advantages the moment you include period 1 and two. So

18:33

Speaker A

here also you can see we have very less number of observation. So if you have very less number of observation we will run into a problem called curs of dimensionalities. Ideally for a DA efficiency analysis not necessarily DA

18:48

Speaker A

for parametric approach also you need to run a model with a minimum number of observation. So we call it as 3 into n + m. If you have the rule of thumb, if you have m output and n input, ideally we

19:04

Speaker A

should have m 3 into m plus n observation. Otherwise, you'll not be able to conceptualize. You can conceptual but you will not be able to get a reliable result from a linear programming problem point of.

19:21

Speaker A

So now we have observation six observation. So it is always better to have as many as observation possible but from a theoretically sound manner. So for period one we consider only period one data and for period two we consider period 1

19:49

Speaker A

plus period 2. This is the concept of sequential fronting. So it is basically data for period one as it is for period 2 you stack stack the data for period 2 over the period one data so and so. So you

20:08

Speaker A

get a dynamically evolving uh frontier and that frontier is being called as sequential frontier. We will see application of sequential frontier in several context. The main advantage of sequential frontier it is like it gives you more number of observation and the

20:31

Speaker A

more more the number of observation you get more degrees of freedom or your estimation become bit more realistic from a nonparametric point of view. The same the case for a parametric approach also.

20:47

Speaker A

How to construct the linear programming problem over here? So as you remember in a simple case we have maximization of fe subject to by changing the values of lambda and p subject to say you have input one input and one output. So we had sum

21:15

Speaker A

of lambda j xj should be less than or equal to x k. We are estimating the efficiency of form K.

21:28

Speaker A

Sum of lambda J YJ should be greater than or equal to YK that should be F \* K and sum of lambda J = 1 and lambda J take the value greater than or equal to zero or one up to

21:49

Speaker A

N. This was the uh framework we followed for a one input one output that two at a single period we call it as the contemporaneous framework. So now we extend this into a sequential frontier framework.

22:12

Speaker A

Before that here in this case star is basically the maximum of P and 1 by P star will give you the technical efficiency of B. Okay.

22:26

Speaker A

Extending the same framework for any sequential frontal model objective function remain the same maximum of fee and here now we are adding a new subscript that is basically XJ T. XJT is basically the capital T. XJT is basically the input

22:50

Speaker A

output B input bundle of J firm J for the period T and YJ T is basically the output bundle of the form J for the period T. So now sum of lambda j t.

23:23

Speaker A

So now we add a double sum over here. J 1 up to n t is basically one of capital t.

23:36

Speaker A

Okay. And this should be less than or equal to x k for the period t.

23:48

Speaker A

Okay. So suppose you are considering the period one it will be only one period right. So it become the contemporaneous from here. So for period 2 it will take the value sum of lambda j1.

24:06

Speaker A

Sorry we have xj t. So it will become double the uh data point if we are having a balance spanner and lambda weights also will change in the sense that uh each observation will be considered as an individual observation.

24:29

Speaker A

So you not necessarily that the peers or the referral point of a particular observation will come from the same period it can change.

24:40

Speaker A

Similarly, you'll have the output constraint J1 up to N and T 1 up to capital T. And we are referring to firm K for the period t lambda j t y j t should be greater than or equal to y

25:05

Speaker A

k free time and sum of j 1 up to n t 1 up to capital t lambda JT should be equal to 1. If your case of V RS and lambda J can take the value greater than or equal to zero for J1 up

25:29

Speaker A

to N T 1 up to T. So this is how we construct a uh sequential frontier. So as I mentioned for period one you take the entire observation from particular period only.

25:49

Speaker A

For period 2, you take the observation from period 1, then period 2, or period 3, period 1, 2, 3.

25:58

Speaker A

In literature, especially this thing come from uh Tulken's contribution mostly and in literature sometime you may see something called window analysis.

26:10

Speaker A

And here in this sequential frontier analysis, what was the fundamental assumption? Whatever technology being observed in a period before the particular period it was remaining but sometime there might be a technology that may keep changing but with a lapse

26:26

Speaker A

of period. So in that case actually you may consider period 1 to three as one window. So for period one one observ period one observation for period 2 you take 1 and two for period 3 you take 1 2

26:39

Speaker A

3. Alternatively for period 3 four you consider the same window but in with that actually period one may lose out from the observation or being removed from the observation. So for period four it will come 2 3 4 so and so that

26:58

Speaker A

becomes a window and it's up to the researcher to identify what is the span of window and to do the analysis according. So that becomes a window analysis case where technology is remaining but not for a infinite point

27:12

Speaker A

of time. So these are the very first uh two extension that you can conceptually but these extensions are basically from the data point of view the way we fire the data to the DEA model the formulation remain more or less same you're not

27:31

Speaker A

using the linear programming problem formulation you can conceptualize the same thing in the context of input oriented also so and so the very first one we discussed was the meta frontier two frontier. The second one is basically the

27:47

Speaker A

sequential frontier analysis and uh window analysis as a extension of that. Thank you. [music] [music]

Topics: Data Envelopment Analysis DEA extensions Meta frontier Group frontier Technology Closeness Ratio Efficiency analysis Productivity analysis Sequential frontier MATLAB Stochastic frontier

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