Now in this section, what we are going to discuss is how do we choose the weights. You can see this in the next two slides, what we offered you there, a broad menu of different options that have been usually employed in the literature. Just to try to help you a bit in the understanding on the whole process, I would distinguish between three different broad ways of choosing weights. One would be data driven techniques. What do I mean by this? We have the partial indicators. We have already the data. And with this data, we are going to select the weights. Second option. We have a completely opposite option. That would be the weights are selected by some experts, by some committee, by some individuals, but it's completely outside from data. So, data are there, they are helpful to elaborate the composite indicator but weights are defined in another external weight different from the data. And of course, there is a third way out that needs some solution, say in between. On one side, we use the data to compute these weights but at the same time, we have some external information, some off sample information, that is already used to decide the weights. So now, as you can see, also in the slide, within what we call data driven techniques, we can distinguish between several methods. First, we have what we call the frequency base weights. What do we mean by that? We take the empirical distribution of the data for one dimension, and what we do is, once we have this empirical distribution, we assign weights to individuals according whether this partial indicator is very present or not for this individual. We can take one over the density or we can take directly the density as the weight. We have also what we call the statistical weights and statistical weights are a broad set of techniques that basically consists in estimating the weights through some multi-varied analysis techniques. For example, Principle Component Analysis, Factor Analysis, and so on. Then, within also the class of data driven techniques, we can find what we call most favorable weights, or what is also called in this terminology, benefit of the doubt approach. Within this setting, the most frequently used technique is what we call the Data Envelopment Analysis. What about normative weights? Normative weights, as I told you before, are weights that are defined of sample information. We distinguish, basically, three different groups of techniques. One is what we call the equal or arbitrary weights and it's,of course, as you can understand based on subjective choices, then we have what we call expert opinion weights. We will distinguish in this area basically, two techniques; what we call budget allocation techniques and analytic hierarchy processes. And then, there is a third group of techniques within this class of normative weights that we call price-based weights. Of course, in this direction the explanation is clear. And finally, what about the hybrid approach that have normative choosing weights but at the same time, with some data driven contamination? Then in this case, we distinguish between two different techniques. What we call stated preference weights, which are techniques really regression based, and then what we call hedonic weights. Hedonic weights are also based on regression but they are basically, based on prices or pressure with related to the market and this type of buying. Now, in the next slides, what we are going to do is we will discuss all of these techniques. But let me just tell you that at some point, we will give you in the material in the slides some mathematical developments that we will not discuss here in the talk. The reason is that it's definitely self-contained and for those who have already an statistical foundations, it will be easy to follow and for those who are not, we'll give you the right references to follow up. So in the next slide, we will give you an example of equal and arbitrary weights. It's basically, as you can see, a very fairly easy choice, wj will be just equal to 1 over n. So, you remember that j was the number of dimension, so basically, it means an equal weight. It's in the human development in the index of economic well-being in the previous example we gave you, it will be this .25. That would be the equal weight. In the next slide, we find out what we called frequency base weights. The expressions we have the slides are just clear for you instead, probably this f of xj function we introduced here. This is just the density function related to the xj partial indicator. As you can see there, the weight can be directly related to this expression wj equals f of xj or can be inversely related, that is 1 over f of xj. Why do we use that, or when do we use the data, or the inverse relationship? Well, it depends on whether we want to weight a lot of individual in which we do not find this characteristic very frequently, or we want to weight an individual where we do find this characteristic very frequently. For example, in an index related or the composite index designed for environmental reasons, we may want to give a good weight or a larger weight to, for example, individuals who are driving old cars. In this case, we will use the data density. If we want to weight in the inverse way, we would use 1 over f, that's standard procedure. But these types of weights are very common and broadly used in the literature of composite indices. If we go to the next slide. Not only the next slide, but the next eight slides are related to the composed principal component analysis and factor analysis techniques to estimate the weights. As I told you before, these two are statistical techniques are used to compute the weights and they were framed into what we call the data driven techniques to find the weights. In this case, it's clear that most part of the slides explain what the principal components are and that statistical techniques or principal components. I'm not going to explain it here, and I'm going to go straight to the slide. This is concretely slide number 21 when we develop the weights selection, the main principal. So, how do we choose the weights by using principal component analysis? If it would be the question, the answer are the following is steps. First of all, we design or we write down, we estimate the correlation matrix of the data of partial indicators. Second, once we have done this, we identify a small number of principal components. Basically, as you probably know, we tried to find out the number of principal components are at least fairly enough to explain 80 percent of variation of the whole sample. Then, afterwards, what we do is rotate the factors. Some varimax approach can be useful for that, in order to obtain the maximum correlation of these factors with respect to a linear combination of these special indicators. And then, once we have identified this principal component, what we do is we create subgroups of partial indicators. We create a group of partial indicators. Let me point out something here. When we were discussing about how to elaborate or how to define the different dimensions in their composite indexes, we said it was important to have an economic and a statistical framework because this was basically what he was going to decide, the number of dimensions and the criteria to assign any of these partial indicators within one dimension. Here, when we are talking about principal component analysis, as you probably realize, the approaches are slightly different because the dimensions, the number of dimensions are being assigned, and have been found out directly from data. This is an advantage because somehow it leaves to the researcher much less space to arrange or to manipulate data. But on the other side is rather dangerous because it could be or it could appear as a sort of black box model in the sense that we are computing the dimensions or we are defining the dimensions, we are computing the weights, but indeed we have no prior information about what the model is. So, as you can see there we have some drawbacks also to the use of principal components analysis. Okay, but in any case, let me come back to the steps we were discussing to. And then in this fourth step as you can see in the slide, we can already find a weight for any of these new dimensions we have defined. You can see that yes, the weight is centered magnetization of the factor loadings in the principal components analysis. Just to know what the factor loading is, go back to the slides, to the previous one that are already there for your help. Once we have computed these previous weights that are associated or related to the different dimensions, we can, as a fifth step renormalize again and this WJ would be the final weight. We are going to assign to the Jth dimension in the principal components analysis setting. Okay, in the next two slides, the principal component analysis approach is applied to compute the weights for the index of economic well-being. We give you the tables and we believe you will be able, following carefully our explanations to calculate them by hand. This is a very important or a very interesting example. I suggest you to do for the sake of the learning or the elaboration of the composites indices. Okay, in the next slide, we present a new technique which is what we call the benefit of the doubt approach. It's again a data driven technique. As I told you before, it's based on what we call data envelopment analysis technique. What is a data envelopment analysis technique? This technique is borrowed from the efficiency analysis, and it said basically, linear programming technique that measures relative performance of observational units. In this case, the relative performance of individuals or countries. How the data envelopment analysis techniques relate this or explain this efficiency concept or this relative performance concept? What the data envelopment analysis technique does is it creates a frontier of efficiency or frontier of performance. And then, it takes any of the observational units we have any of the countries are interrelated with this, with respect to this frontier. This is the measurement of inefficiency of the data envelopment analysis. In the next slide, you have a nice graphic where you can say how this technique works. So for example, let me consider the point A that is just thinking a country, country A that is characterized by these two values for the partial indicators X1 and X2. Let me take ray A that passes through the value, through the origin and through the point A and that index X intersects the border or the frontier, which is the note or which is represented by the blue line. Then, if you see the intersection of this frontier with respect to the ray within red line, you will or we will find out the point B. So, the distance between the point B and the point A, is what we denote by inefficiency of the unit, A with respect to the frontier. That is the measurement of inefficiency that reflects that. So, how is then the indicator calculated? You can see it in the next slide. It's of course a mathematical expression that, I guess with the explanations you have received and with the material you have there, you will be able to understand, but in any case, you can just realize that what we are doing is just in this dimension is taking the ratio between [inaudible] A and [inaudible] B. This is basically the measure of inefficiency and this is what is going to be the weight for every of the countries, or every of the individuals in this app. Okay, in the next slide, let me just give you some ideas about the advantages of using this technique of data envelopment analysis. First of all, these techniques is hard because it's sensitive to national priorities. So, somehow, the policy makers can design a system of weights that fits somehow to the national interests, to the national ideas. Second, it's also important the fact that it's not based on theoretical bonds, is data driven. So, this technique just less picks the data and fix the weights. And finally, it's useful for policy considerations because actually, it's true that there is this sort of inefficiency message that somehow encourages or might encourages policymakers to undertake the reforms or to undertake policies in order to diminish these efficiency. So, somehow it gives incentives to, for example, inefficiency countries. Which are the disadvantages of this technique? First of all, we can't find too many countries or too many individuals that are fully efficient, that get weights one. So, this is sometimes not very nice. Also, since it is a linear programming technique, we can get sometimes indeterminacy of the weights and this is not also very convenient. Finally, it rewards somehow the status quo.