Stochastic Frontier Analysis: Introduction

[music] [music] Hi, welcome back to the course Applied Production Analysis Using MATLAB. In the last session, we saw one model that is basically modified OLS, which was theoretically sound and statistically involving approach for estimating the production function, and with reference to that, you can estimate the technical efficiency. Also, we saw some tests that we can do to see the characteristics of our composite error term or even the OLS error term that we can do before going for any advanced models like stochastic analysis.

technical efficiency. Also we saw some tests that we can do to see the characteristics of our composite error term or the even the oilless error term that we can do before going for any advanced models like stoastic analysis.

Moving forward, in this session, I'll give you a brief outline of stochastic analysis. A simple disclaimer: this may become or this may sound very technical for at least a few of you, particularly if you have a very basic background in statistics, but I'll try to make it as simple as possible. My objective is to make you understand the procedure involved or the theoretical or the philosophical foundation involved in stochastic frontier analysis. If I keep going and seeing each and every derivation and everything, it is going to be equivalent to a one-week session we need to spend for stochastic analysis. So I'll give you an outline or I'll give you an overall picture why we need a model like stochastic analysis in the context of efficiency and later in productivity analysis. So the motivation is, till now, whether it is in the case of corrected OLS or modified OLS, we conceptualize the production function accordingly.

background in statistics but I'll try to make it as simple as possible. My objective is to make you understand the procedure involved or the theoretical or the philosophical foundation involved in stoastic from analysis. If I keep going and uh seeing each and every derivations

Then whatever deviation is happening for the actual outcome, we conceptualize it as a result of inefficiency. But in reality, sometimes some factors beyond input can have an influence on the potential output. We call them the random component or the noise component or the stochastic component in the way that we use in the context of statistical framework. So in some sense, your potential outcome may become higher or lower than the average outcome or the deterministic outcome, which we call the stochastic component.

efficiency and later in the productivity analysis. So the motivation is till now whether it is in the case of corrected oilless or modified oilers we conceptualize the production function accordingly.

So it can be positive or negative. We'll see such examples. And before that, just to clarify, stochastic frontier analysis comes under the broader umbrella parametric approach. So we start with the functional form, whether it is Cobb-Douglas or translog, based on the context that you are dealing with. So here I can give you a generalized functional form that is basically yi as a function of xi beta. Beta is the coefficient associated with the inputs including the intercept plus epsilon i, which I have already referred to in the last session.

But in reality sometime some factors beyond input can have an influence on the potential output.

So here this epsilon is basically a composite error term. I can call it a composite error which consists of vi and ui, and vi is basically the error term that we see in the context of econometric modeling or we can call it as the random noise, which will be normally distributed with zero in the context of our production framework. Also, the new variable that I'm bringing is basically the ui, and ui is basically here we have already put a negative, so it is always a one-sided distribution as we saw in the context of MS and the non-negative disturbance, which accounts for our technical inefficiency in this context. As I mentioned, ui is asymmetrically distributed, and here in this functional form, we can conceptualize say one input one output case. So I can take the log of that: log of qi equal to beta 0 plus beta 1 log xi plus vi minus ui. So this is basically the epsilon I was referring to.

So in some sense your potential outcome may become higher or lower than the average outcome or the deterministic outcome which we call as the stoastic uh component.

Then we, or else, can take the exponential form. It will be qi equal to exponential of the entire component. Okay. Alternatively, we can have the log form which we have already mentioned over here. My point was the moment we have stochastic frontier analysis, the first component that is basically exponential of beta 0 plus beta 1 log xi, this is basically the deterministic component, and the vi will capture the noise component and minus of ui will capture the inefficiency component involved in the production process. So each observation will be getting one vi and one minus of ui, and the vi is basically the random noise involved in the production function and the ui is basically the inefficiency component. In a simple diagrammatic manner, we are considering one input one output case.

So we start with the functional form whether it is scoped the glass or translog based on the context that you are dealing with. So here I can give you a generalized functional form that is basically yi as a function of x i beta.

So say we have firm A. It's an optimistic case. We are referring to firm A has an output QA in this functional form exponential of beta 0 plus beta 1 log XA plus VA minus UA. This is the actual outcome or the actual output the firm A could have achieved.

So here this epsilon is basically a composite error term. I can call it as a composite error which consist of vi and ui and vi is basically the error term that we see in the context of econometric modeling or we can call it

As we have it in the context of our COS or CM, we have a deterministic frontier which is free from any noise. This is the deterministic frontier, and here I'm bringing in a new concept that is basically the noise effect. So here noise effect consists of an impact of any factors other than input or some environmental factors.

it is always a one-sided distribution as we saw in the context of MS and the non- negative disturbance uh which account for our technical inefficiency in this content. As I mentioned UI is symmetrically distributed and here this functional form we can conceptualize

Environmental not necessarily the environment related. It is firm environment related. Some environmental factors or market conditions that create some influence on the potential output outcome, and here in this context, potential output which makes them deviate from the deterministic counterpart. So let's consider the case of firm A. Say here it is an optimistic case, as I mentioned. It's like a case due to a very favorable market condition or very favorable climate condition or monsoon in the context of agricultural sector.

Then we or else we can take the exponential form. It will be qi equal to exponential of the entire component.

The potential outcome itself has improved. So I'm referring to something which has happened outside the system which is not being part of the input or the entire production process. We are referring to it as an external factor that influences the potential outcome, which in this case has become a positive outcome, and even with that, actually the firm couldn't tap the benefit of this positive outcome or the positive noise has happened in the production framework and it is producing somewhere over here. So we will be considering this under deviation happening for this firm A against the potential output that is basically QA star as the inefficiency component, and this deviation, as I mentioned, this is happening because of the deviation from the deterministic frontier and for the potential outcome that is we are referring to, that is basically your QA star. This deviation is happening because of some noise component, which we acknowledge as an external factor or the stochastic factor influencing the potential outcome of our individual observations we are considering. As against that, say we can consider firm B or the decision-making B. Its outcome is basically here QB, and you can see in this context there are some negative factors that are impacting the potential output which is deviating from the deterministic frontier. This is basically the deterministic frontier.

My point was the moment we have uh stoastic frontier analysis the first component that is basically exponential of beta 0 plus beta 1 dog x i this is basically the deterministic component and the ui or vi will capture the noise

Don't get confused. This is not the stochastic frontier. Even if I say that it's a graphical illustration of stochastic frontier analysis, this point that I'm, this line that I draw here is not it. It is basically a deterministic frontier, and for getting a stochastic frontier, we'll have to connect all the potential outcome points, which is going to be a bit more conceptually or technically tricky in this context. So this diagram that we are having is basically the deterministic frontier. Coming to observation B, this is basically the actual outcome, and due to some unfavorable market condition or unfavorable monsoon or even policies, your potential outcome itself has come down. So it has a negative noise effect, and inefficiency we calculate again this potential outcome that is basically the deviation from potential outcome and for the individual observation that we are referring to.

basically the inefficiency component in a simple diagrammatic manner we considering one input one output case.

So just to summarize, the stochastic frontier analysis accounts for the noise component along with the inefficiency component, and here whatever deviation that is happening for individual outcome against the deterministic, we are not considering completely as an inefficiency component. We are giving room for the stochastic component like a clim...

This is the actual outcome or the actual output the firm A could have achieved.

as we have it in the context of our COS or CM we have a deterministic frontier which is free from any noise this is the deterministic frontier and here I'm bringing in a new concept that is basically the noise effect so

here noise effect consist of a impact of any factors other than input or some environmental factors.

Environmental not necessarily the environment related. It is it is firms environment related. Some environmental factors or market conditions that create some influence on the potential output outcome and here in this context potential output which makes them to deviate from the deterministic

counterpart. So let's consider the case of firm A. Say here it is an optimistic case as I mentioned it's like a case due to a very favorable market condition or very favorable climate condition or monsoon in the context of agricultural sector

the potential outcome itself has improved. So I'm referring to something which has happened outside the system which is not being part of the input or the entire production process. we are referring to it is an external factor that influence the potential outcome

which has in this case it has become a positive outcome and even with that actually the firm couldn't tap the benefit of this positive outcome or the positive noise has happened in the production framework and it is producing

somewhere over here. So we will be considering this endear deviation happening for this firm A against the potential output that is basically Q A star as the inefficiency component and this deviation as I mentioned this is happening because of the this

this deviation that is deviation from the deterministic frontier and for the uh potential outcome that is we are referring 2 that is basically your QA star. This deviation is happening because of some noise component and which we acknowledge as a

external factor or the stoastic factor influencing the potential outcome of our individual observations we are considering as against that say we can consider firm B or the decision making B its outcome is basically here QB and you can See in this context there

are some negative factor that is impacting the potential output which is deviating from the deterministic frontier. This is basically the deterministic frontier.

Don't get confused. This is not the stoastic frontier. Even if I say that it's a graphical illustration of stoastic frontier analysis. this uh point that I'm this line that I draw here is not it is basically a deterministic frontier and for getting a

stoastic front we'll have to connect all the potential outcome points which is going to be bit more conceptually or technically tricky in this context so this diagram that we are having is basically the deterministic frontier coming to observation B this is

basically the actual out outcome and due to some unfavorable market condition or unfavorable monsoon or even policies your potential outcome itself has come down. So it has a negative noise effect and inefficiency we calculate again this potential outcome that is basically the

deviation from potential outcome and our for the individual observation that we are referring to.

So just to summarize uh the stoastic frontier analysis account for the uh noise component along with the inefficiency component and here whatever deviation that is happening for individual outcome against the determinist we are not considering completely as an inefficiency component.

We are giving a room for the stoastic component like a climate or market condition or even policies as I mentioned to have an impact on the potential outcome and after taking out such impacts or extracting out such impacts we are estimating the

inefficiency which is more realistic in or it is more fair for the decision-m units to conceptualize their inefficiency or efficiency.

in the framework that we are considering. So here the moment we have qi that is basically the actual outcome and the denominator it will become the potential outcome that is basically exponential of x i beta which is in the matrix form. So

we take x i inverse beta plus vi. And here you can see that the qi is conceptualized as that component with our noise as well as the inefficiency. So this is basically x i beta plus vi minus ui. So this is basically individual

observation deviating from the uh the deterministic component with their noise as well as the inefficiency. And the moment we take this as a ratio of exponential of x i -ash beta plus vi minus ui divided by exponential of x i -

beta plus ui we get exponential of minus ui as the remaining component which can be conceptualized as a technically efficiency machine that's how we are going to proceed in the context of stoastic frontier analysis.

So the main reference we are going to have a guide to frontier version 4.1.

The frontier is basically a package which is basically a windows based package for estimating technical efficiency following stoastic frontier analysis from center for efficiency and productivity analysis kinsland university. So I would encourage all of you go and see this

package and uh get an idea. It is basically you can download it's an open-source packet that you can download with the download from the website of Ca Queensland University and uh it comes along with the documentation which can

be used as reference material for our uh SFA also we are referring to the Kumbager and Novel book on stoastic front analysis which is basically the source for the theoretical understanding of our stoastic frontier analysis model.

Thank you. [music] [music]