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# Re: st: Inefficiency measures greater than one for frontier commands

 From Federico Belotti To statalist@hsphsun2.harvard.edu Subject Re: st: Inefficiency measures greater than one for frontier commands Date Fri, 26 Apr 2013 17:11:56 +0200

Dear Aljar,

the definition used in the Kumbhakar and Lovell's book is the only theoretical definition of cost efficiency. It is always bounded in the unit interval and its empirical counterpart cannot range between 1 and \infty. You can find the same definition at pag.53 of the Coelli, Rao, O'Donnel and Battese book "Introduction to efficiency and productivity analysis", and I think in any other book on this topic. So, my point is that the empirical measure of cost efficiency given by -frontier- and -xtfrontier- is not theoretically coherent and should be corrected. One expects (as Reut expected) that cost efficiency cannot be greater than one.

Best,
Federico

On Apr 26, 2013, at 3:42 PM, Aljar Meesters wrote:

> Dear Federico,
>
> Thank you for pointing out that the Kumbhakar and Lovell book
> (Stochastic Frontier Analysis) is using another definition than that
> is used by Stata, I didn't know that. I think that this stresses the
> impartance that you at least know the intuition behind a definition.
> Best,
>
> Aljar
>
>
> 2013/4/24 Federico Belotti <f.belotti@gmail.com>:
>> Dear Aljar and Reut,
>>
>> As reported in the Kumbhakar book "Stochastic Frontier Analysis", cost efficiency is a measure of the ratio between the minimum feasible cost and the observed expenditure. Hence, CE is by construction bounded between 0 and 1. Accordingly, a measure of CE in the SF framework is always provided by
>>
>> CE = exp{-E(u|e)},
>>
>> where E(u|e) is the (post-)estimate of cost inefficiency obtained through the Jondrow et al. (1982) estimator. In the case of a cross-sectional normal-half normal cost frontier, this estimator corresponds to the equation 4.2.12 of Kumbhakar book. Equivalently, another estimator (the estimator implemented in the post estimation command of both -frontier- and -xtfrontier-) can be obtained using the Battese and Coelli (1988) approach
>>
>> CE = E(exp{-u}|e),
>>
>> that it is still bounded in the unit interval (in the case of a cross-sectional normal-half normal cost frontier this estimator is reported in equation 4.2.14 of Kumbhakar book).
>>
>> Thanks to the Reut's post, I realized that both the -frontier- and -xtfrontier- commands show a "strange" behaviour (as well as the FRONTIER 4.1 Fortran routine by Tim Coelli).
>> Indeed, if you run the following commands
>>
>> webuse frontier2, clear
>> frontier lncost lnout lnp_l lnp_k, cost d(hn)
>> predict ce, te
>>
>> you will get point estimates of cost efficiency that range from 1.53 to 1152.92. The same results can be obtained by running a cross-sectional normal-half normal cost frontier using FRONTIER 4.1 on the same data.
>>
>> My guess is that the issue is in the formula implemented behind the post-estimation -frontier- (and -xtfrontier-) command. Indeed, the Stata manual reports for the -frontier- case the following equations
>>
>> CE = normal(-scost'*sigma1+z)/normal(z) * exp(-scost'*mu1+1/2*sigma1^2),
>>
>> where
>>        z = mu1/sigma1,
>>        mu1 = - scost'* eps * sigma^2_u / sigma^2,
>>        sigma1 = sigma_u*sigma_v / sigma^2,
>>
>> with  scost' = 1 for production and scost' = -1 for cost frontiers.
>>
>> In my view (and given equation 4.2.14 in Kumbhakar book) the correct formula should be the following
>>
>> CE = normal(-sigma1+z)/normal(z) * exp(-mu1+1/2*sigma1^2).
>>
>> In other words, the only sign change needed to adapt the Battese & Coelli (1988) estimator to the case of cost efficiency is limited to mu1 (since a cost frontier is characterized by a compounded error term with positive skewness, eps = v + u).
>>
>> For some odd reason, both Tim Coelli and Stata developers used CE = E(exp{u}|e) instead of CE = E(exp{-u}|e).
>> So, a strategy to avoid the problem is to take the reciprocal of what the -frontier- (or -xtfrontier-) command is giving you in order to get approximated Battese & Coelli (1988) point estimates of cost efficiency
>>
>> predict ce, te
>> replace ce = 1/ce
>>
>> An alternative strategy is to use the Jondrow et al. (1982) approximation through
>>
>> predict u, u
>> gen ce = exp(-u)
>>
>> hope that helps,
>> Federico
>>
>>
>>
>> On Apr 23, 2013, at 11:24 PM, Aljar Meesters wrote:
>>
>>> Your understanding about - predict var, te - is correct. Your
>>> conceptual question needs some elaboration. A score of one indeed
>>> represents a fully efficient bank, you can call this 100% efficient.
>>> If you find a score of say 1.2 you can say that that particular bank
>>> makes 20% more costs than a fully efficient bank would make. I think
>>> you can say that the bank is 20% inefficient. Although opinions on
>>> this may differ, it is at least clear what the 20% means. If you
>>> prefer to have a score between zero and one (one is fully efficient),
>>> you can calculate a new score by one over the old score, yet, in this
>>> case there is no clear interpretation, as far as I know. So, if you
>>> find that bank Y has a score of 0.8 after the rescaling and call this
>>> bank 80% efficient, I don't know what this 80% exactly means. However,
>>> you do find cost efficiencies in the literature that are scaled
>>> between zero and one, so, it is not uncommon.  As a side note, if you
>>> rescale the efficiency score by one over the old score, you will
>>> ignore Jensen's inequality (E[f(x)] != f(E[x])). Whether you find this
>>> problematic or not is up to you.
>>> Best,
>>>
>>> Aljar
>>>
>>> 2013/4/23 Reut Levi <rlevi2@student.gsu.edu>:
>>>> Thank you!
>>>>
>>>> To clarify and make sure I understand.  The syntax: predict VariableName, te would give me inefficiency scores that range from 1 to infinity (for cost functions), right?
>>>>
>>>> In addition, here is a conceptual question. The frontier represents 100% efficiency. According to the inefficiency scores described above, banks that receive a score of one are 100% present efficient. Therefore, scores above 1 would represent banks that are operating above the cost frontier and therefore less efficient.  Now, how can I interpret those inefficiency scores above one? Is there an accepted way to normalize them to range from 0 to 100%, so I will be able to make a statement such as "bank Y is X% efficient/inefficient"?
>>>>
>>>> Thank you for your help and inputs,
>>>> Reut
>>>>
>>>>
>>>>
>>>> ________________________________________
>>>> From: owner-statalist@hsphsun2.harvard.edu [owner-statalist@hsphsun2.harvard.edu] on behalf of Federico Belotti [f.belotti@gmail.com]
>>>> Sent: Tuesday, April 23, 2013 12:55 PM
>>>> To: statalist@hsphsun2.harvard.edu
>>>> Subject: Re: st: Inefficiency measures greater than one for frontier commands
>>>>
>>>> If you are using the -xtfrontier- command the syntax is
>>>>
>>>> predict te, te
>>>>
>>>> In this way you obtain an estimate of efficiency scores through the Jondrow et al. (1982) formula.
>>>>
>>>> Federico
>>>>
>>>> On Apr 23, 2013, at 5:35 PM, Reut Levi wrote:
>>>>
>>>>> Thank you Federico!
>>>>>
>>>>> Do you happen to know if there is a way to predict efficiency scores in STATA, instead of inefficiency scores?
>>>>> If there is, can you please specify the command syntax?
>>>>> If there isn't, how should I go about converting the inefficiency scores predicted to represent efficiency levels?
>>>>>
>>>>> Thank you very much,
>>>>> Reut
>>>>>
>>>>> ________________________________________
>>>>> From: owner-statalist@hsphsun2.harvard.edu [owner-statalist@hsphsun2.harvard.edu] on behalf of Federico Belotti [f.belotti@gmail.com]
>>>>> Sent: Monday, April 22, 2013 5:54 AM
>>>>> To: statalist@hsphsun2.harvard.edu
>>>>> Subject: Re: st: Inefficiency measures greater than one for frontier commands
>>>>>
>>>>> Dear Reut,
>>>>>
>>>>> in the stochastic frontier framework, "inefficiency" scores ranges from 0 to infinity, while "efficiency" scores are restricted between 0 and 1 by construction since
>>>>>
>>>>> TE = exp{-E[su|e]}  following  Jondrow et al., 1982,
>>>>> or,
>>>>> TE = E{exp(s*u)|e}  following Battese and Coelli, 1988,
>>>>>
>>>>> where s = 1 (s = -1) in the cost frontier (production frontier) case.
>>>>>
>>>>> Hope this helps.
>>>>> Federico
>>>>>
>>>>> On Apr 21, 2013, at 2:28 AM, Reut Levi wrote:
>>>>>
>>>>>> Dear Statalist members,
>>>>>>
>>>>>> I am using the xtfrontier command to estimate inefficiency levels for the U.S banking industry. My data comprised of information from the FFIEC Call Report for the year 2012. It is a large data set with over 29,000 observations. I broke it down by asset size in order to reduce the number of observation and also because the literature suggests that asset size peer group will produce more appropriate inefficiency measures. After breaking down the dataset, the average number of banks in each peer group data set is 650, with observations for 4 quarters, totaling in 2700 data points. All of my variable are in natural logs.
>>>>>>
>>>>>> I am using the xtfrontier command with the options ti and cost. I then predict the inefficiency measures using predict with the option u, but some of my inefficiency predications are greater than one. How is it possible? The manual says that the inefficiency measures are restricted to be between 0 and 1. Am I doing something wrong? Or what could explain those measures greater than 1?
>>>>>>
>>>>>> I am relatively new to STATA so please take it into consideration in your response.
>>>>>> Thank you very much,
>>>>>> Reut
>>>>>>
>>>>>>
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>>>>>
>>>>> --
>>>>> Federico Belotti, PhD
>>>>> Research Fellow
>>>>> Centre for Economics and International Studies
>>>>> University of Rome Tor Vergata
>>>>> tel/fax: +39 06 7259 5627
>>>>> e-mail: federico.belotti@uniroma2.it
>>>>> web: http://www.econometrics.it
>>>>>
>>>>>
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>>>>>
>>>>>
>>>>>
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>>>>
>>>> --
>>>> Federico Belotti, PhD
>>>> Research Fellow
>>>> Centre for Economics and International Studies
>>>> University of Rome Tor Vergata
>>>> tel/fax: +39 06 7259 5627
>>>> e-mail: federico.belotti@uniroma2.it
>>>> web: http://www.econometrics.it
>>>>
>>>>
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>>>>
>>>>
>>>>
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>>>
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>>
>> --
>> Federico Belotti, PhD
>> Research Fellow
>> Centre for Economics and International Studies
>> University of Rome Tor Vergata
>> tel/fax: +39 06 7259 5627
>> e-mail: federico.belotti@uniroma2.it
>> web: http://www.econometrics.it
>>
>>
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--
Federico Belotti, PhD
Research Fellow
Centre for Economics and International Studies
University of Rome Tor Vergata
tel/fax: +39 06 7259 5627
e-mail: federico.belotti@uniroma2.it
web: http://www.econometrics.it

*
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