**Modified ordinary least squares (MOLS) — Transcript & Summary | SozAI**
Source: https://sozai.app/transcript/modified-ordinary-least-squares-mols/

Introduction to Modified Ordinary Least Squares (MOLS) as a bridge between corrected OLS and stochastic frontier analysis in production efficiency modeling.

## Key Takeaways

- MOLS modifies OLS by using the expected value of error terms instead of the maximum error term for correction.
- It requires assuming a one-sided distribution for error terms, commonly half-normal or exponential.
- MOLS is rarely used in practice due to its computational complexity and distributional assumptions.
- The model conceptually bridges simple parametric approaches and advanced stochastic frontier analysis.
- Understanding MOLS helps in grasping the theoretical foundation leading to stochastic frontier analysis.

## What the video covers

- The session introduces Modified Ordinary Least Squares (MOLS) as an extension of corrected OLS for measuring technical efficiency in production analysis.
- MOLS replaces the maximum error term correction with the expected value of error terms, requiring distributional assumptions.
- The model was independently proposed by Aigner and Richmont in the early 1970s but sees limited empirical application due to computational complexity.
- MOLS serves as a theoretical bridge between basic parametric models like corrected OLS and advanced models like stochastic frontier analysis (SFA).
- The intercept in the production function is adjusted using the expected value of the one-sided error term, ensuring all observations lie below the production frontier.
- One-sided distributions such as half-normal and exponential are used to model the error term in MOLS.
- The half-normal distribution is a truncated normal distribution with zero mean, considering only the positive side.
- The exponential distribution is another one-sided distribution used to estimate expected error values in MOLS.
- MOLS involves more statistical and probabilistic concepts compared to corrected OLS, making it computationally more involved.
- The session concludes by motivating the transition from MOLS to stochastic frontier analysis, which offers a more advanced framework for technical efficiency estimation.

## Chapters

1. 00:00 Introduction and Recap of Corrected OLS and MAD Approaches
2. 02:04 Historical Background and Proposal of MOLS
3. 03:19 Estimation of Production Function Coefficients Using OLS
4. 04:40 Replacing Maximum Error with Expected Value in MOLS
5. 06:15 Concept of One-sided Deviations in Production Function
6. 07:37 One-sided Distributions: Half-normal and Exponential
7. 09:29 Mathematical Details of Half-normal Distribution
8. 11:29 Expected Value Estimation and Model Adjustments in MOLS
9. 14:29 Technical Efficiency Measurement under MOLS
10. 16:56 Limitations of MOLS and Transition to Stochastic Frontier Analysis

Answers

## Questions about this video

What is the main difference between corrected OLS and modified OLS?

Corrected OLS adjusts the intercept using the maximum error term, while modified OLS uses the expected value of the error terms based on a one-sided distribution.

Why is MOLS rarely used in empirical applications?

MOLS requires strong distributional assumptions and involves more complex computations, making it less practical compared to other models like stochastic frontier analysis.

What types of distributions are used in MOLS for error term modeling?

One-sided distributions such as the half-normal and exponential distributions are used to model the error term in MOLS.

## Full Transcript — Download SRT & Markdown

00:05

Speaker A

[music] [music] Welcome back to the course on layer production analysis using MATLAB. In the last session, we discussed corrected OLS and corrected mean absolute deviation approaches as two frameworks that come under the parametric approaches for measuring technical efficiency.

00:36

Speaker A

In today's session, we see one extension of this same model, but in place of the maximum of the error term, here we'll be considering the mean or average value of, or expected value of, our error terms for making such corrections.

00:53

Speaker A

So, we name it as modified ordinary least squares or MOLS model. And also, along with that, I'll be explaining or I'll be giving you a motivation for having a model that is basically the stochastic frontier analysis model, which is the most advanced framework in

01:13

Speaker A

the context of the parametric approach. So, in this session, we will see the appropriateness or the pre-estimation approaches to see whether at all we need to go for an advanced model like stochastic frontier analysis in the parametric framework.

01:30

Speaker A

So, modified ordinary least squares, as the name suggests, modifies the inner step such a way that the line that you, or the regression line that you estimated using OLS, can be adjusted such a way that it follows the

01:48

Speaker A

properties of our production function. But, at the end, we are going to see that as a criticism, we will not be able to represent this line or this model in a very full-fledged panel that represents our production function. We'll come to

02:04

Speaker A

that point at the latest stage. It was proposed by Aigner and Richmont in 1972 and 1974 independently.

02:14

Speaker A

To be honest or to be very frank, you don't see much of the paper that uses MOLS in empirical applications.

02:25

Speaker A

It is basically a model in between our COLS and stochastic frontier analysis. So the main objective of this session is to give you a philosophical framework or the theoretical framework that gives our, or that makes our transition from a

02:45

Speaker A

COLS or a simple OLS model to the advanced models like stochastic frontier analysis.

02:53

Speaker A

So, though you don't see a lot of applications or any MATLAB codes or dedicated software packages for estimating MOLS, we do respect this model because this model can be considered as a bridge between the very basic models that

03:10

Speaker A

come under the framework of the parametric approach and the advanced models like stochastic frontier analysis.

03:19

Speaker A

So, here what we do as a researcher, we estimate the slope and intercept coefficient of our production function using OLS.

03:34

Speaker A

But, the intercept we cannot use as it is because that does not, or the regression line that you're estimating using OLS does not represent a production function because few of the observations will be lying above the line

03:48

Speaker A

or of course few of them will be lying below the line. So, since few of them are lying above the line, we cannot call the line that you estimated using OLS as the production function.

03:59

Speaker A

So, as we saw in the case of corrected OLS, what we do? We estimate the coefficient using OLS.

04:08

Speaker A

Then we do such a correction that takes the maximum of the error term. Then that we add to the intercept, and accordingly each observation will have a new error term. We have already seen that.

04:22

Speaker A

Then that new intercept we corrected by using the maximum of our error term will serve as the intercept for our production function, where the technology remains the same, or the slope coefficient remains the same.

04:40

Speaker A

So, here in place of maximum of UIs, what we do? We take an expected value of UI. So, this is basically going to be the expected value of UIs in place of the maximum of UI.

04:59

Speaker A

So, as most of you know, what does it mean by expected value? This is basically a probability-weighted value for each outcome from a probability point of view, or if from a statistics point of view. So, we can

05:12

Speaker A

call it an average also, but it has a meaning beyond the simple average that we calculate in a general framework.

05:22

Speaker A

So, as I mentioned, it involves a bit more statistics or probability as compared to our corrected OLS.

05:31

Speaker A

So, here we need to get the expected values of the error term. Okay? So, or the expected values of the deviation, the estimated error terms. So, in this case, what we do, we need to conceptualize that expected value by using some

05:58

Speaker A

distribution. So here as I mentioned, this deviation should be always such a way that the observations fall below the line. So, all the deviations should be negative or at least one or few observations can have a

06:15

Speaker A

deviation of zero. There is no positive deviation we can conceptualize in the framework of the production function.

06:22

Speaker A

So, for getting this one-sided value, we can conceptualize several distributions or we can bring in several one-sided distributions that you use in the context of statistics for getting this expected value. As I mentioned, you don't see a lot of

06:42

Speaker A

empirical application using MOLS, mainly because you need to bring in distributional assumptions and based on that you need to estimate an expected value of UI, so and so.

06:57

Speaker A

It's a bit more computationally involved. You will see the same framework in the context of stochastic frontier analysis itself.

07:03

Speaker A

So, what we do, we assume a distributional or one-sided distribution. So, here, exponential or half-normal follow such one-sided distribution. So, half-normal is like, we can have something of this sort.

07:20

Speaker A

Here you can see this one is basically an exponential function that I have given over here. Half-normal looks like, so, this is basically the normal distribution with zero mean and sigma square variance.

07:37

Speaker A

Or, so, here, what we do, we need to conceptualize this as a one-sided distribution. For that only this portion comes in the context of our the new error term that you're going to conceptualize.

07:57

Speaker A

How to conceptualize such a production such a functional form? So, here we know the f of x or f of z of our normal distribution. This can be modified.

08:12

Speaker A

Once we have an f of z, you know the formula for that, the bell-shaped function that I'm talking about. So, ideally, if you integrate it between minus infinity to infinity dz, it should be equal to one.

08:29

Speaker A

So, at the moment we are using one-sided half, the truncated one-sided distribution which has a truncation at zero. So, this will become our integral will become zero to infinity and then it will become two times f of z. And here z is basically a standard

08:47

Speaker A

normal notation that I'm using, but not necessarily it has a one variance. It can have a different value of variance also other than one.

08:56

Speaker A

So, here the distribution changes in this way. We'll revisit this concept when we are dealing with such distribution in detail, particularly when we are talking about the stochastic frontier analysis.

09:09

Speaker A

This is one set of distribution which can be conceptualized in the conduct of one-sided distribution that is basically the half normal. So, the concept of half normal is coming from the fact that we have a distribution normally normal

09:24

Speaker A

distribution not necessarily a standard normal distribution with one variance. So, normal distribution with where we are doing the zero mean and we are doing the truncation at exactly at the value zero and only the right-hand side of the

09:40

Speaker A

component is or right-hand side of the distribution is being considered. And that is basically called the half normal distribution.

09:48

Speaker A

Another set of distribution that we can conceptualize in the context of modified OLS or even in the context of stochastic frontier analysis we'll be using the same.

09:59

Speaker A

That is basically the exponential function. Here you can see exponential function basically it has a one-sided distribution and it always takes a positive value.

10:10

Speaker A

And then here, based on the parameter that is the variance sigma u, the stretch changes. So here, higher the variance, the higher will be the spread, as you know. And this can be used for getting an

10:29

Speaker A

expected value of our UI hats that we are going to use for our model. So just to repeat, like so here y

10:44

Speaker A

not necessarily half normal or exponential. Sometimes we can have a truncated normal distribution. Say you have a distribution like this where you have zero over here. Say you mu is the mean. So here we can see we the moment you

11:04

Speaker A

truncate here, this is not a standard normal or it has a non-zero mu. This the moment you truncate it over here, this is going to be the distribution. So this also comes very similar to half normal distribution, but

11:18

Speaker A

uh here the truncation is happening at zero, but not at the mean of the distribution.

11:24

Speaker A

In the half normal case, it was coming as mu equal to uh zero. That is not the case over here.

11:30

Speaker A

So So is happening the truncation at zero, which is not the mean of the distribution. So this entire portion which falls toward the right side of the distribution, we are considering at the uh distribution for our model. And this is basically the

11:44

Speaker A

truncated normal distribution, which can also be used for our ML S estimation. Moving ahead, uh this is the half-normal distribution I was referring to based on the values of sigma use. You can see the spread is changing, but the mean or the

12:06

Speaker A

peak of the distribution remain same, that is basically zero. And what we do with this distribution?

12:14

Speaker A

Once we have this distribution, we'll be using these one-sided distribution for getting the expected values. So before that, as we did for our corrected OLS in the first stage, using the input-output data, we estimate the beta zero and

12:33

Speaker A

the technology component on the coefficient associated with the in- coefficient associated with the input, say beta in input variable that you are having. So we'll be estimating beta zero and beta in that is basically called as beta zero

12:49

Speaker A

hat and beta in using OLS. As we mentioned, uh here beta in we consider as a consistent estimate, but beta zero is not fitting in the context because we know that it it it get biased in the sense that uh it

13:09

Speaker A

does not capture the inefficiency component. So there is a systematic uh issue happening over here, and which needs to be adjusted in order to use it as a production function.

13:20

Speaker A

So what we do, the moment we get uh beta zero hat and beta in, we are not going to modify anything with the beta ends or the coefficient associated with inputs, we modify our beta zero hat such a way

13:34

Speaker A

that we get the beta zero double star, which is considered as beta zero hat plus expected value of UI hat.

13:44

Speaker A

And then the moment you adjust our intercept in this manner, so this is basically like this.

13:53

Speaker A

So, let's say OLS beta zero hat, and then you say you get beta zero hat plus expected value of UI hat.

14:07

Speaker A

So, you may get a slight confusion. In econometry, we say expected value of UI to be zero, but not necessarily then the moment you estimate it using OLS, you get expected value of UI hat to be zero. That is basically an

14:20

Speaker A

asymptotic property that we are imposing. So, as I mentioned, this was the UIs that we were talking.

14:29

Speaker A

But the moment you adjust our UIs, this is going to be the UI star, say it will be estimated again.

14:37

Speaker A

This point. Let's say if you had an observation here, this is going to be the UIs.

14:47

Speaker A

I'll come to this point, so here there are some issue that we are going to face in this context.

14:53

Speaker A

So, this is going to be the new UI uh star or we call it as UI hat double star, and then the moment we do such adjustment, our UIs double star is going to be UI hat minus expected value of UI

15:11

Speaker A

hat. And the moment you have UI hat double star, which is the negative uh it's always a negative value.

15:21

Speaker A

Not necessarily here. For this observation, we are not going to get a negative value. We come to that point.

15:26

Speaker A

So, negative of or exponential of minus UI uh hat double star will give you a measure of technical efficiency. This is going to be the technical efficiency measure that we are going to get under MOLS.

15:46

Speaker A

What are the problem? Here, you can see under MOLS the correction that you are doing or the modification that you are doing is done by using the expected value.

15:58

Speaker A

So, any correction to the intercept that we have done using the expected value does not ensure that the entire observation that you are getting, this is basically the production function, the entire observation we are getting against the production function or the

16:15

Speaker A

deviation against the production function are negative. That will not be the case. So, few observations you may get a positive value for our UI hat double star, which will be very difficult for you to interpret as a researcher from a

16:30

Speaker A

theoretical point of view. That is the main limitation that we face in the context of modified OLS.

16:37

Speaker A

But as against our COLS or the variant CMADI, we'll not be having very sensitive outcome in terms of outliers as compared to COLS. Our MOLS will be less sensitive to outlier, but it has a problem that it does not uh

16:56

Speaker A

ensure that entire observations are falling below the frontier. Also, we carry forward an issue that we faced in the case of COLS.

17:07

Speaker A

COLS has a criticism that it does not fit the or it does not envelop the observation as closely as possible, which will be the case here as well.

17:16

Speaker A

Also, here what we are doing we are simply shifting the OLS curve or OLS line simply by adjusting the intercept but not doing anything with the our slope coefficient. So, that gives us an impression that the technology remains

17:35

Speaker A

the same or it's a parallel shift we are doing. Then, the technology remains the same only the intercept is being changed which can also be questioned in strength.

17:45

Speaker A

So, as I mentioned this model it's a very uh similar to the philosophy of our COLS but it bridges a gap between the very basic model and our stochastic frontier analysis by bringing in the distributional assumption uh that you

18:06

Speaker A

conceptualize about our error term. That is basically the inefficiency we have a distributional assumption that we invoke.

18:15

Speaker A

So, in upcoming session we'll not be dealing with this EMOLS. We'll be dealing with the advanced model that is basically the stochastic frontier analysis which basically an advancement over or which is being developed by bringing in the philosophical foundation of

18:30

Speaker A

EMOL. So, before going for our discussion on our stochastic frontier analysis, we see how to ensure or how to decide whether we need to go for our COLS model or whether we need to go for our stochastic frontier analysis or not.

18:53

Speaker A

So, here in stochastic frontier analysis what we do we conceptualize our error term that is basically the epsilon that I'm calling. It is basically a error term which has VI minus UI.

19:12

Speaker A

It is going to be the compositor that we are going to have. So we have in as again the UI that we had it in the context of our COLS or CMED that is basically we are attributing the entire deviation

19:28

Speaker A

happening for individual observations against the frontier as a result of inefficiency. But the stochastic frontier model is going to be a model that talks of composite error term which has VI. VI will follow the standard distribution that we have it in

19:46

Speaker A

the context of error term in our classical linear regression model and UI is basically a one-sided distribution as we have it in COLS and CMED which represent the inefficiency.

20:00

Speaker A

So before doing our stochastic frontier analysis, what we can do we do the estimation using OLS and based on the distribution of our OLS we get an idea whether we are at all going for any stochastic frontier analysis or not.

20:19

Speaker A

So it is a very similar to I can simply say example. So this can be like a before going to the technical details of this test, what we can do we can conceptualize this as an example where say you have

20:33

Speaker A

two container. Simple example, not a very theoretically sound example but for you to get an idea how it works.

20:44

Speaker A

We need to see whether at all need to go for a stochastic frontier analysis by using the simple example. It is a school level example I would say. So say here you have two containers and here say you have some liquid.

21:02

Speaker A

Say one is in what one is filled with water and say other one is filled with uh uh lemon. Lemon syrup or whatever.

21:13

Speaker A

Okay? So, now what we are doing, we are mixing these two uh liquid here. Not a very theoretically sound example.

21:24

Speaker A

So, say you are getting one mix of this thing. Or a composite of this.

21:37

Speaker A

So, I call this as a V and this as U and this as the epsilon.

21:50

Speaker A

Okay. So, in this example, you know uh what happened? The characteristics of epsilon will be purely depending on the amount of V and U that you are adding into the uh the third container. Say you're uh U is very less.

22:14

Speaker A

Very less. So, why did I bring this example here? We are talking about E which has several characteristics.

22:24

Speaker A

And that characteristics will be depending on the amount of V and U that we are adding into the the mix or the composite.

22:36

Speaker A

So, before doing a stochastic frontier analysis from a computational point of view, it's very theoretically involving.

22:44

Speaker A

Over and above we should not use SFA in a model or in a data set which does not require an SFA, stochastic frontier analysis.

22:55

Speaker A

So, as I mentioned it's in a very simple manner. V is basically normally distributed.

23:06

Speaker A

Zero with zero mean. And U is basically one side distributed. And then before doing our SFA analysis, we look at the distribution of epsilon that the composite error term and see whether at all we have uh it has a one-sided distribution or

23:28

Speaker A

whether at all it has a symmetric distribution. If the distribution is symmetric, that means it is basically a normally distributed error term.

23:38

Speaker A

That is basically the stochastic or the random component dominating in determining the epsilon. And if it has a one-sided distribution, uh it is basically dominated by our one-sided component that is basically U, which has only negative value, so it is

23:55

Speaker A

negative side distributed. If you have a distribution when you are doing um the test, I will be seeing the test, and you are seeing a one-sided distribution in the positive side, that is a bit bit alarming case where you need to go

24:11

Speaker A

back and see the way you estimated or the way you estimated the coefficients are right or not. So, there is a misspecification in the model.

24:19

Speaker A

So, here in this context to see the characteristics of uh the composite error term before starting our stochastic frontier analysis, there are two tests being proposed. One is by Schmidt and Lee and in the year 1984, which is basically a sample moment-based

24:37

Speaker A

uh estimate. We have M3 divided by M2 into square root of M2. And here M2 is basically the second moment, that is basically I one up to N sum of I one up to N Xi minus X bar squared

24:52

Speaker A

divided by N. That is basically the second moment. Similarly, M3 is defined as sum of xi minus x bar cube divided by n. So, this second moment and third moment can be used to see the skewness of the distribution.

25:08

Speaker A

And then Schmidt and Lin conceptualized square root of beta as the uh test for that. And if M3 is uh less than zero, that implies the oil less residuals are negatively skewed.

25:22

Speaker A

Which is giving you a hint that our technical inefficiencies dominating or it is present in the uh the composite error term, which says that there is a scope for efficiency analysis.

25:37

Speaker A

And uh but there is a problem. This square root of B, uh which is conceptualized as M3 divided by M2 square root of M2, which does not have any uh theoretical distribution. It is has to be calculated separately or this has to this cannot be

25:55

Speaker A

approximated to any theoretical existing theoretical distribution. So, in order to overcome that limitation, Coelli in the year 1995 came up with a very similar but slightly modified test for seeing the skewness of the composite error term, that is basically M3t, that

26:12

Speaker A

is basically M3 divided by square root of 6M2 cube divided by n as the new distribution. And it asymptotically it follows normal distribution and you can see the standard normal distribution for getting the theoretical values of this test.

26:28

Speaker A

And then based on that, we can see whether our composite error term is skewed or symmetrically distributed. And if it is squarely distributed, that means uh as I mentioned, that means there is like a presence of inefficiency in the model. And that

26:46

Speaker A

gives us a hint that, okay, uh there is a scope for uh efficiency analysis. And then we can go for a model like stochastic frontier analysis.

26:59

Speaker A

So, these references uh I have mainly used Kumbhakar and Lovell stochastic frontier analysis for getting the material related to MOLS and these two tests that I have referred over here.

27:17

Speaker A

So, just to summarize in this this session, we were seeing a model in between corrected OLS and stochastic frontier analysis.

27:26

Speaker A

As again, taking the maximum of UI hat for getting the correction of our intercept, what we are doing we are taking the expected value of UI hat, and then our intercepts were modified by taking this expected value.

27:41

Speaker A

Then, our production function was conceptualized in that manner with a shift in our intercept.

27:49

Speaker A

But unfortunately still after correction or even after conceptualizing our production function, some observations were lying above the frontier, which has a theoretically unsound uh context where you will not be able to make an interpretation, but still this model is less sensitive to

28:11

Speaker A

outlier. Also, before starting our discussion on stochastic frontier analysis, which we'll be discussing in detail in the upcoming session, we so, what are the pre-estimation tests that we can do for seeing the appropriateness of stochastic frontier analysis, which are mainly based on

28:31

Speaker A

skewness or the M3, the third moment base uh estimates or the test, which we have already discussed in the session, and which we'll be doing which you should be doing before going for in stochastic frontier analysis to see the appropriateness of stochastic

28:49

Speaker A

frontier analysis. Thank you.

Topics: Modified Ordinary Least Squares MOLS Corrected OLS Stochastic Frontier Analysis Technical Efficiency Production Function One-sided Distribution Half-normal Distribution Exponential Distribution Parametric Approach

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