[MUSIC] As we have already pointed out in other sections of this course, the success that a composite indicator have experience during this last decade in many different fields of society has been mainly because they have been able to sensitize, to summarize. All in for huge data sets of information and they have put or they have got it in very small pieces of easy to understand information for individuals and for the society in general. However, even though this success experience, we could mention some drawbacks that have experienced these composite indicators. Basically, the methodology that has been used in this type of on the building of these composite indicators has been sometimes discussed. And also it's true that at some point, there is a risk of manipulation of these indicators. These, on these grounds, there have been some discussion about their accuracy and their reliability. Furthermore, as you probably know already, there have been some adult procedures that have somehow caused some problems with their robustness of their results of this composite indicators. For example, the choice of their aggregation procedure, the choice of the weights, there are also some problems with missing data. All these drawbacks in general have caused at some point a glum performance of the composite indicators in terms of the results. So basically at some point, we might get results for some countries or for some individuals in terms of the composite indicator that are hard to explain. This is a very dangerous point because if at some political level, public policies are oriented toward the improve of this composite indicators. They might be misleading the political measures, and they had no targeted because basically they are not targeted to the right instruments. Okay, so basically, sorry to tell you about this bad news because we would like to give you a whole core of well based principles or guide to construct compassing indicators. But this is not going to be the easy. What we are going to have in many cases are compass indicators or families of compassing indicators targeted to measure one quantity or one objective. But we need or we would like to have at some point, a set of rules that would help us to decide which one to take or which not. Okay, let us go a bit further. I would like to tell you that basically this exists, but it's very related to the type of Target we are going to establish for the composite indicator. Just to make a bit more clear about that, let us go to the second slide, and have a look please to the definition, to a very general definition of a composite indicator. As you can see in this slide, we have basically that an composite indicator is an index function that maps a set of explanatory variables into a real number. Of course, this is very interesting because since we are getting only a real number from all these process, it's possible to order, for example, countries off individual from a smaller to larger. Then inside this definition as you can see we obsereved, on one side transformation of the partial indicators. We see also a parameter beta that we will explain later in the course. And we will see also what we call w1, w2, wn, which are the weights. And then let us go a bit to the slide number three, where we would like to point out basically that a composite indicator is just a weighted average of the transform indicators of order beta. This is just an composite indicator, a weighted average in terms of that. So in principle as you might see all these slides are tending or are showing that in the way we are looking at composite indicators. We have only a question or name of order individuals or order in societies, or order in countries, good. What are the reasons for their transformation, you might ask about this transformation we find this capital I in the expression. There are mainly two reason why which is to make this transformation. The first reason is that variable usually are measure in different units. And at some point, of course, to make this composite indicator inaggregate, we need to make homogeneous all these measurements. Second reason why we have this transformation, is the fact that there might be outliers, there might be missing values. And all that are going to somehow disturb our analysis. And this is why somehow we decide to make these transformations. Okay, if we go the next slide, you will see fairly nice definition of what is preferred or what is indifferent between two countries. We say country i is preferred to country k, if the value that takes partial indicator for country i is larger than the values this indicator takes for country k. On the other side, we will say both countries are indifferent whether the value that the indicators i takes for country. Then this indicator or this partial indicator takes for country i is equal to the value that this pattern indicator takes for country k. So what this basically, let us go to the next slide then, we will see there basically what is the problem of defining of elaborating a composite indicator. The problem is just basically to chose the weights, to choose the transformation and to chose the beta. And then once we have done this, try to order the results in terms of individuals, in terms of countries or whatever it is that we are trying to measure. So if you remember, we said before that the principles we were going to establish in terms of the composite indicators, well for a good, or we said good criterias. For composite indicators, we're going to depend on the target, we wanted to aim to. In this case, our target is how to rank? How to rank countries or individuals according to the value these composite indicators are taking? Okay, of course, we can pass now to the next slide. Of course, what we can realize is that if we are trying to order individuals or we are trying to order countries, we have a basic problem, right? The problem is how to aggregate individual preferences in a collective decision framework. This is a very as you probably might understand this is a very old problem in terms of economics. And it rises here, ae believe it's really relevant here. In terms of the choice, and basically of the choice of the weights of the composite indicators, but also in terms of the aggregation. So if our target is this, if our target is ranking the different individuals of the different countries. Then there are some principles that need to be followed when elaborating or when composing a composite indicators. The first golden rule, let me put it in this way, would be reduce the sources of uncertainty, that would be the first. We need to have data, and we need to write models that somehow reduce the amount of uncertainty in the system. Because this amount of uncertainty is going to effect for sure to the rank countries and individuals in the resulting composite indicator. A second interesting point that we need also to point out is that whenever it is possible, we need to avoid ad hoc parameters, why? Because ad hoc parameters usually create problems of robustness in the construction of composite indicators. So basically, we don't want to have them affecting very several, or in small, or in few very cases. Then let us go now to the next slide. You can see a very important principle here or so, is which the avoid complete compensability, avoid complete compensability. What does it mean? This basically means that there cannot be full substitution between two partial indicators in a composite indicator. For example, you would have sustainability composite indicator, and this indicator is a function of some economic variables and some environmental variables. The indicator cannot be the same whether we replace GDP by pollution. That cannot be possible, that the value of the composite indicator is the same by exchanging fully one by the other by the other variable. This is what we call in other fields, cardinality. And this can be avoided just by using ordinality relevance, and using ordinal meanings. That's the basic suggestion we make here, and that's one of the main criteria. We will also try to following constructive composing when constructing composite indicators. Of course, introducing ordinality, it generates by itself, two different additional problems. One of the first problems, is really well known since long time, which is called Condorcet paradox. And this basically means that, or by the way the Condorcet paradox is explained in the next slide. So I'm not going to spend a lot of time on this now. But let me just tell you that a Condorcet voting system creates what we call cycles. What is a cycle in term of voting paradox? A cycle is that if we have three different sets of preferences, A, B, and C. If A is preferred to B and B is preferred to C, there is no transitivity between A and C, C might be preferred to A. This is what we call the cycle. And these cycles appear in a very natural way in composite indicators when we introduce the ordinality condition. A second problem that is detailed in the next slide Is what we call the Arrow's impossibility theorem. So Arrow's impossibility theory mainly points out that in a group of individuals with different preferences, it does not exist a voting is key that can transfer this individual preference into a collective set of preferences. [MUSIC]