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Now in this section,

Â what we are going to discuss is how do we choose the weights.

Â You can see this in the next two slides,

Â what we offered you there,

Â a broad menu of different options that have been usually employed in the literature.

Â Just to try to help you a bit in the understanding on the whole process,

Â I would distinguish between three different broad ways of choosing weights.

Â One would be data driven techniques.

Â What do I mean by this?

Â We have the partial indicators.

Â We have already the data.

Â And with this data,

Â we are going to select the weights.

Â Second option.

Â We have a completely opposite option.

Â That would be the weights are selected by some experts,

Â by some committee, by some individuals,

Â but it's completely outside from data.

Â So, data are there,

Â they are helpful to elaborate the composite indicator but weights

Â are defined in another external weight different from the data.

Â And of course, there is a third way out that needs some solution, say in between.

Â On one side, we use the data to compute these weights but at the same time,

Â we have some external information,

Â some off sample information,

Â that is already used to decide the weights.

Â So now, as you can see, also in the slide,

Â within what we call data driven techniques,

Â we can distinguish between several methods.

Â First, we have what we call the frequency base weights. What do we mean by that?

Â We take the empirical distribution of the data for one dimension,

Â and what we do is, once we have this empirical distribution,

Â we assign weights to individuals according whether

Â this partial indicator is very present or not for this individual.

Â We can take one over the density or we can take directly the density as the weight.

Â We have also what we call the statistical weights and statistical weights are

Â a broad set of techniques that basically

Â consists in estimating the weights through some multi-varied analysis techniques.

Â For example, Principle Component Analysis,

Â Factor Analysis, and so on.

Â Then, within also the class of data driven techniques,

Â we can find what we call most favorable weights,

Â or what is also called in this terminology,

Â benefit of the doubt approach.

Â Within this setting,

Â the most frequently used technique is what we call the Data Envelopment Analysis.

Â What about normative weights?

Â Normative weights, as I told you before,

Â are weights that are defined of sample information.

Â We distinguish, basically, three different groups of techniques.

Â One is what we call the equal or arbitrary weights and it's,of course,

Â as you can understand based on subjective choices,

Â then we have what we call expert opinion weights.

Â We will distinguish in this area basically, two techniques;

Â what we call budget allocation techniques and analytic hierarchy processes.

Â And then, there is a third group of techniques within

Â this class of normative weights that we call price-based weights.

Â Of course, in this direction the explanation is clear.

Â And finally, what about

Â the hybrid approach that have normative choosing weights but at the same time,

Â with some data driven contamination?

Â Then in this case, we distinguish between two different techniques.

Â What we call stated preference weights,

Â which are techniques really regression based,

Â and then what we call hedonic weights.

Â Hedonic weights are also based on regression but they are basically,

Â based on prices or pressure with related to the market and this type of buying.

Â Now, in the next slides,

Â what we are going to do is we will discuss all of these techniques.

Â But let me just tell you that at some point,

Â we will give you in the material in the slides

Â some mathematical developments that we will not discuss here in the talk.

Â The reason is that it's definitely self-contained

Â and for those who have already an statistical foundations,

Â it will be easy to follow and for those who are not,

Â we'll give you the right references to follow up.

Â So in the next slide,

Â we will give you an example of equal and arbitrary weights.

Â It's basically, as you can see,

Â a very fairly easy choice,

Â wj will be just equal to 1 over n. So,

Â you remember that j was the number of dimension,

Â so basically, it means an equal weight.

Â It's in the human development in

Â the index of economic well-being in the previous example we gave you,

Â it will be this .25.

Â That would be the equal weight.

Â In the next slide, we find out what we called frequency base weights.

Â The expressions we have the slides are just clear for you instead,

Â probably this f of xj function we introduced here.

Â This is just the density function related to the xj partial indicator.

Â As you can see there,

Â the weight can be directly related to this expression

Â wj equals f of xj or can be inversely related,

Â that is 1 over f of xj.

Â Why do we use that,

Â or when do we use the data,

Â or the inverse relationship?

Â Well, it depends on whether we want to weight a lot of individual

Â in which we do not find this characteristic very frequently,

Â or we want to weight an individual where we do find this characteristic very frequently.

Â For example, in an index related or

Â the composite index designed for environmental reasons,

Â we may want to give a good weight or a larger weight to,

Â for example, individuals who are driving old cars.

Â In this case, we will use the data density.

Â If we want to weight in the inverse way,

Â we would use 1 over f, that's standard procedure.

Â But these types of weights are very common and

Â broadly used in the literature of composite indices.

Â If we go to the next slide.

Â Not only the next slide,

Â but the next eight slides are related to

Â the composed principal component analysis and factor analysis techniques

Â to estimate the weights.

Â As I told you before,

Â these two are statistical techniques are used to compute

Â the weights and they were framed into what we call

Â the data driven techniques to find the weights.

Â In this case, it's clear that most part of the slides explain

Â what the principal components are and that

Â statistical techniques or principal components.

Â I'm not going to explain it here,

Â and I'm going to go straight to the slide.

Â This is concretely slide number 21

Â when we develop the weights selection, the main principal.

Â So, how do we choose the weights by using principal component analysis?

Â If it would be the question,

Â the answer are the following is steps.

Â First of all, we design or we write down,

Â we estimate the correlation matrix of the data of partial indicators.

Â Second, once we have done this,

Â we identify a small number of principal components.

Â Basically, as you probably know,

Â we tried to find out the number of principal components are at least fairly

Â enough to explain 80 percent of variation of the whole sample.

Â Then, afterwards, what we do is rotate the factors.

Â Some varimax approach can be useful for that,

Â in order to obtain the maximum correlation of

Â these factors with respect to a linear combination of these special indicators.

Â And then, once we have identified this principal component,

Â what we do is we create subgroups of partial indicators.

Â We create a group of partial indicators.

Â Let me point out something here.

Â When we were discussing about how to elaborate or how to

Â define the different dimensions in their composite indexes,

Â we said it was important to have an economic and

Â a statistical framework because this was basically what he was going to decide,

Â the number of dimensions and the criteria to

Â assign any of these partial indicators within one dimension.

Â Here, when we are talking about principal component analysis, as you probably realize,

Â the approaches are slightly different because the dimensions,

Â the number of dimensions are being assigned,

Â and have been found out directly from data.

Â This is an advantage because somehow it leaves

Â to the researcher much less space to arrange or to manipulate data.

Â But on the other side is rather dangerous

Â because it could be or it could appear as a sort of

Â black box model in the sense that we are

Â computing the dimensions or we are defining the dimensions,

Â we are computing the weights,

Â but indeed we have no prior information about what the model is.

Â So, as you can see there we have some drawbacks also to

Â the use of principal components analysis.

Â Okay, but in any case,

Â let me come back to the steps we were discussing to.

Â And then in this fourth step as you can see in the slide,

Â we can already find a weight for any of these new dimensions we have defined.

Â You can see that yes,

Â the weight is centered magnetization of

Â the factor loadings in the principal components analysis.

Â Just to know what the factor loading is,

Â go back to the slides,

Â to the previous one that are already there for your help.

Â Once we have computed

Â these previous weights that are associated or related to the different dimensions,

Â we can, as a fifth step renormalize again and this WJ would be the final weight.

Â We are going to assign to the Jth dimension in the principal components analysis setting.

Â Okay, in the next two slides,

Â the principal component analysis approach is

Â applied to compute the weights for the index of economic well-being.

Â We give you the tables and we believe you will be able,

Â following carefully our explanations to calculate them by hand.

Â This is a very important or a very interesting example.

Â I suggest you to do for the sake

Â of the learning or the elaboration of the composites indices.

Â Okay, in the next slide,

Â we present a new technique which is what we call the benefit of the doubt approach.

Â It's again a data driven technique.

Â As I told you before,

Â it's based on what we call data envelopment analysis technique.

Â What is a data envelopment analysis technique?

Â This technique is borrowed from the efficiency analysis,

Â and it said basically,

Â linear programming technique that measures relative performance of observational units.

Â In this case, the relative performance of individuals or countries.

Â How the data envelopment analysis techniques relate

Â this or explain this efficiency concept or this relative performance concept?

Â What the data envelopment analysis technique does is it creates

Â a frontier of efficiency or frontier of performance.

Â And then, it takes any of the observational units

Â we have any of the countries are interrelated with this,

Â with respect to this frontier.

Â This is the measurement of inefficiency of the data envelopment analysis.

Â In the next slide,

Â you have a nice graphic where you can say how this technique works.

Â So for example, let me consider the point A that is just thinking a country,

Â country A that is characterized by these two values for the partial indicators X1 and X2.

Â Let me take ray A that passes through the value,

Â through the origin and through the point A and that index

Â X intersects the border or the frontier,

Â which is the note or which is represented by the blue line.

Â Then, if you see the intersection of this frontier with respect

Â to the ray within red line,

Â you will or we will find out the point B.

Â So, the distance between the point B and the point A,

Â is what we denote by inefficiency of the unit,

Â A with respect to the frontier.

Â That is the measurement of inefficiency that reflects that.

Â So, how is then the indicator calculated?

Â You can see it in the next slide.

Â It's of course a mathematical expression that,

Â I guess with the explanations you have received and with the material you have there,

Â you will be able to understand,

Â but in any case, you can just realize that what we are doing is just in

Â this dimension is taking the ratio between [inaudible] A and [inaudible] B.

Â This is basically the measure of inefficiency and this is what is

Â going to be the weight for every of the countries,

Â or every of the individuals in this app.

Â Okay, in the next slide,

Â let me just give you some ideas about the advantages of

Â using this technique of data envelopment analysis.

Â First of all, these techniques is hard because it's sensitive to national priorities.

Â So, somehow, the policy makers can design

Â a system of weights that fits somehow to the national interests,

Â to the national ideas.

Â Second, it's also important the fact that it's not

Â based on theoretical bonds, is data driven.

Â So, this technique just less picks the data and fix the weights.

Â And finally, it's useful for policy considerations because actually,

Â it's true that there is this sort of

Â inefficiency message that somehow encourages or might encourages

Â policymakers to undertake the reforms

Â or to undertake policies in order to diminish these efficiency.

Â So, somehow it gives incentives to,

Â for example, inefficiency countries.

Â Which are the disadvantages of this technique?

Â First of all, we can't find

Â too many countries or too many individuals that are fully efficient,

Â that get weights one.

Â So, this is sometimes not very nice.

Â Also, since it is a linear programming technique,

Â we can get sometimes indeterminacy of the weights and this is not also very convenient.

Â Finally, it rewards somehow the status quo.

Â