Speaker A
[music] [music] Hi, welcome back to the course Applied Production Analysis Using MATLAB. In today's session, we'll see how to integrate the bootstrap DEA in the context of TF estimation.
This video explains bootstrap-based bias correction for the Malmquist Productivity Index using DEA models to improve TFP estimation accuracy.
Key Takeaways
- Bootstrap bias correction improves the accuracy and statistical validity of Malmquist Productivity Index estimates.
- CRS frontiers are more sensitive to sample bias than VRS frontiers.
- Simar and Wilson’s algorithms provide a framework to correct bias and serial correlation in efficiency scores.
- Bootstrap-corrected MPI allows construction of confidence intervals and inferential testing for TFP changes.
- The method is computationally intensive and requires careful data preparation including environmental variables.
What the video covers
- Introduction to bootstrap DEA integration in total factor (TF) productivity estimation.
- Discussion of three TF decomposition variants: FGLR, FGNZ, and Randling based on DEA distance functions.
- Explanation of bias in efficiency scores due to sample-based frontier estimation versus population frontier.
- Comparison of CRS and VRS frontiers sensitivity to new observations and sample bias.
- Presentation of Simar and Wilson's bootstrap bias correction algorithms for DEA efficiency scores.
- Application of bias-corrected efficiency scores in Malmquist Productivity Index (MPI) to obtain statistically valid TFP estimates.
- Advantages of bootstrap-corrected MPI include bias correction, confidence intervals, and inferential statistics.
- Limitations include high computational complexity, infeasibility issues, and data requirements (environmental variables).
- Trade-offs between sample size and availability of environmental variables for bias correction.
- Encouragement to consult Simar and Wilson’s 1999 paper for detailed methodology.
Chapters
- 00:00Introduction to Bootstrap DEA and TF Estimation
- 01:28CRS and VRS Frontiers in TF Decomposition
- 02:37Impact of New Observations on Frontiers and Bias
- 03:51Bias in Efficiency Scores and Distance Functions
- 05:03Bias Correction of Efficiency Scores Using Bootstrap
- 06:31Sensitivity of CRS vs VRS Frontiers to Sample Changes
- 07:43Simar and Wilson Algorithms for Bias and Serial Correlation
- 09:05Advantages of Bootstrap-Corrected Malmquist Productivity Index
- 10:29Statistical Inference and Confidence Intervals in TFP Estimation
- 11:28Limitations and Data Requirements for Bootstrap Correction
Full Transcript — Download SRT & Markdown
Speaker A
We have already covered three variants of TF decompositions: FGLR, FGNZ, and Randling. As you remember, all these decompositions are based on distance functions, which are basically estimated using a DEA model.
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So in this session, we see how the bias in the estimation of efficiency scores in the context of DEA has an implication on the TFP score that you are going to estimate using the same DEA or distance function scores.
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So the context is that we have inputs and outputs. So here you can see inputs are given over here and outputs are given over here. So using this input-output data, what we did, we constructed the frontier. So here you
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can see, in the context of Ray and decomposition, we need to construct both CRS and VRS frontiers. In case of FGLR, we need only CRS frontier, but in case of FGNZ, we need VRS and CRS frontiers.
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So consider the case, the same example I'm referring to, we had A, B, C, D as the observation.
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Suppose this A, B, C, D are sampled, but you have another observation, say E, which was not included in this data point. So say, for example, E0 is somewhere, say E0 is somewhere here and E1 is
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somewhere here. Then what are the implications in that case? CRS frontier will also slightly shift for the period zero. You can see here this way, and even the VRS frontier should have shifted this manner. CRS of period one will not get affected
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because it's already over and above the point over here, but the VRS frontier can get affected, and say data point was here, it will get affected in this manner. As a result, having a new observation or going towards the population, our fronts are
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shifting. As a result, the efficiency or the distance function that we are estimating in this context will also get a bias. So these efficiency scores are always a bit biased because the reference point that you are having is
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a subset of the population, which is basically the estimated frontier. It will always be lying below or away from the frontier, or actual frontier, in the manner that it is always below or it is a subset of the
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actual frontier. So as a result, whether it is the TC or TC or SCF that you are estimating, it will get affected. All these components will also get affected by this bias in the efficiency estimation or the bias in the estimation of
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distance functions which are being used for estimating or decomposing the productivity in the context of Malcus productivity index.
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So in 1999, you can see here Simar and Wilson in their EJ paper proposed that the same framework of bias correction can be extended in the context of our TFV estimation also. How do we do that? For that,
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TFV estimation involved technical efficiency of firms under CRS and technical efficiency of firms under VRS, and those two we estimate against T period, against T period reference period, T minus one reference period, or T plus [laughter] reference period, and not go into the
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technicalities of that, which we already covered in the earlier session. Same will happen for VRS also.
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So what we can do, these efficiency scores that you are estimating for against T or T plus 1 or T minus one period technology, these things also you can do a bit of a bias correction. So this in place of these scores I'll be
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using TE CS BC, which is basically a bias-corrected version. Same way, we can have T_VRS BC, which is also a bias-corrected version of CRS or VRS from here.
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So quickly, one clarification: we have both CRS and VRS from here. Which one is going to get affected or which is more sensitive to the population and sample counterpart? So here you can see the moment we have a
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new observation, even a slight change in CRS frontier is affecting almost all the observations, but VRS actually will not have much of the mutation. VRS frontier is more of a not a radial expansion or contraction. So having a
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new observation will not have a significant impact on all the observations that we are considering in this case. So here I would like to claim that CRS frontiers are more sensitive to the sample when you have population and
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taking a sample out of that and estimating a frontier. The CRS will be more biased as compared to your population counterpart of the frontier, that is the true frontier.
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So just to continue our discussion on our bootstrap correction. So what we can do, these efficiency scores you can correct by using the bootstrapping algorithm. It can be a simple jackknife approach, but that might not be very sufficient in the context of TFP
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calculation. Also, the same way what we claimed in the context of DEA model, this might be an approach but not the appropriate approach. Then we do have two algorithms of Simar and Wilson. So algorithm one is basically correcting only the
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serial correlation, and we have algorithm two with two loops. In loop one, it corrects for bias, and loop two, it corrects for the serial correlation.
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I'm referring to Simar and Wilson 2007 algorithm. So from a TFP point of view, what you can do, you can take loop one of Simar and Wilson algorithm 2. You can run the efficiency score estimation with this
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bias correction for period T technology, T minus one technology, and T plus 1 technology and get the T CRSBC and T VRSBC and plug them in and get the bias-corrected efficiency scores or distance function and get it plugged
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in the TFP estimation. So what are the advantages of that? So advantages of bootstrap-corrected market productivity index:
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The biggest advantage is it gives you a bias-corrected TFP and decomposition. Also, you can use these scores to get a confidence interval because the conventional MPI scores are lacking any statistical properties. You cannot use this conventional MPI score for getting
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the confidence interval. But the moment you do bootstrapping, you'll be able to get a confidence interval also. Also, in conventional manner, you get an efficiency score 1.2; it shows that improvement in the TFP, but sometimes in a
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statistical point of view, it might not be a statistically significant improvement. Say that will become even complicated when you have 1.02.
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So here the conventional framework will claim it as an improvement. But the moment you have a bootstrap TF score, you can see an inference also, inference.
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You can do inferential statistics also in the context of TFP estimation. You can see whether the change in TFP being claimed by the model is statistically significant or not.
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What are the limitations? Of course, bootstrap-corrected MPI comes with a bundle of advantages or it is more of a refined framework, but it has a lot of limitations. It is a very computationally involving problem because you need to estimate TE
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again T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, or distance function against the
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T period technology, then T minus one period technology, T plus 1 period technology. So that will complicate the entire algorithm and it is a very computationally involving process.
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Over and able decompositions like R&D decomposition, that is randomly decomposition, already has an infeasibility issue, and the same infeasibility issue may get aggravated when you are using the bootstrap mus productivity index. This will get even more problematic. You can write it as
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it will become a serious concern because when you're doing bootstrapping, you may get a bootstrap failed bootstrapping because sometimes it may not converge, sometimes it may give you an infeasible solution, and it complicates the entire process I'm referring to.
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Moreover, there are data limitations. So I can give you in a simple framework: say you have X, Y. For implementing the Simar and Wilson algorithm, you need environmental variables.
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Simar Wilson algorithm requires Z variable also. So along with X, Y, and Z, along with X input and output variable, you need at least one environmental variable that needs to be plugged in for the
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So what will happen this environmental variable that I'm referring to might not be available for almost all the observation. Say you initially you have six observation. It's a hypothetical case I'm considering with six observation you'll not be able to
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estimate DEA but I'm considering six observation. So for that X and Y are available for all period but uh zed is available only for four observation.
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The moment you try to implement Simar and Wilson algorithm, you'll not be able to include the first one and the last one observation which does not have a Z variable. So already the problem that we are having it is our data is sampled or
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data is small. The sample is very small as compared to our population. And in order to get rid of the limitations of the small sample, we are doing the bootstrapping. But the moment you plug in a new variable that is
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basically the Z variable into the picture you may not get more observation your observation may go even down. So then there is a uh trade-off between having the retaining the same observation and estimating the frontier and getting the or keeping the
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conventional efficiency score for our TF estimation or alternatively do a bootstrapping which has a cost of another two observation in this context or another few observation in the general context and whether to at all do the bootstrapping or not. So it is a
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very tricky question to have and this has to be answered by looking at the amount of data set that you are having the number of data set that you are having and also say you are doing a state level analysis say due to some
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reason you are getting data for 25 states and few states you are not getting in that case actually say three or four states for which you don't have the data you should not go for a bootstrapping because this 25 itself or
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the data that you are having itself is somewhere close to the population and there is no very serious bias that you need to be corrected in this content. So what I intend to say whether to go for a
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conventional TFP and decomposition or whether to go for a bootstrap based TFP and decomposition it is very highly depending on the computational infrastructure that you are having and over and above the number of observation that you are having and the number of
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missing data that you will encounter when you are including a new explanatory variable or zed variable which we call as the environmental variable into the uh bootstrapping framework.
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So just to summarize as I mentioned very similar to the DEA efficiency scores which are being biased in the context of sample the TFP scores also which are bas basically estimated using these efficiency score will also get affected
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of the uh sample sensitivity. So in order to solve that problem what you can do in place of the conventional efficiency scores used for measuring the productivity we can use the bias corrected efficiency score which will give you a bias corrected TFP
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and decomposition of the same and with that you will be able to make inference about the efficient productivity change or change in the components of productivity along with the confidence interval. But it comes with a complication that is basically it is
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computationally involving also it can create infeasibility particularly in the context of rand composition and sometime the solution that you are suggesting to overcome the limitation of sample in terms of bootstrapping may even reduce your number of observation particularly
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in the context of simar algorithm which require a environmental variable for which you may have few missing observation or missing data point and the moment you are missing one data point it is not period suppose this was period one and you have another period 2
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but for period 2 this data is there but actually that is of no use unless you have a continuous panel in the context of TF estimation so I encourage you to go through the paper by Simarin Wilson 1999 which is published in EJ estimating
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the bootstrapping ing multis index to get a better idea about this framework and implement it in your research and learning activities. Thank you.
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Topics:Malmquist Productivity Indexbootstrap bias correctionData Envelopment AnalysisTotal Factor ProductivityCRS frontierVRS frontierSimar and Wilson algorithmefficiency scoresTFP decompositionApplied Production Analysis
