This section of the course is about to study the robustness and sensitivity of our results to changes in the different components in the composite index. As you already know, composite indices depend basically in three different issues. The type of transformation we have proposed for the partial indicators, the value of the elasticity of substitution of the beta, the so-called beta parameter we previously defined, and finally, the value of their weights. As you probably remember also in another section of this course, we established some simplifying assumptions that basically would mean that we could express the exchangeability relation between the different partial indicators as a function of the relative ratio of the relative weights of the partial indicators. If this is the case, then please note that in most part of cases, it is of great interest to change, to vary the weights in the composite indicator and to analyze the results we are obtaining. This is important in terms of the conclusions we can obtain or we can withdraw from a composite indicator. Because otherwise if we don't make a sensitivity analysis and we don't study its robustness, we may be ranking or we may be out-performing or down-performing, I guess some characteristic of a country or an individual because there has been some problem or some outlying observations or some wrong choice of some of the weights. So please note that this is a very important part in the making process of a composite indicator. In the next slide, you are going to find some examples that have been taken from Section Seven from the already mentioned interesting work of Sharpe and Andrews, 2012. So basically, if you note in this table and you realize in this table, the ordinal rankings for all weighting methods of the different countries do vary significantly. This is the case, for example, when we take equal weights with respect to them, the first alternative in selecting their weights is already designing in the previous slide. We can see that Norway is still at the top and Spain is still or remains at the bottom. But however, we can see that in between there are some countries that change their position. So somehow, of course, these tables provide us with a very interesting information about rankings and about the impact of changes in their weight in terms of the ranking of countries. But it is also true that it give us a too wide picture, it is hard to find really significant information from all these tables. What do we usually do them? Basically if you go to the next slide, you will find what we call the dominance criteria. This dominance criteria, which is rather simplified here, basically determines under which conditions a country X strongly dominates a country Y or a country X weakly dominates a country Y. You can find in this slide the definitions given by these two relationships and they are based always, of course, in the values of the composite indicator for each country and for every value of the partial indicators in this country. So we will say, country X strongly dominates country Y whether the composite index evaluated for this country takes a larger value than the composite indicator for the other country. And we say it weakly X weakly dominates Y, if we are allow for the possibility that both composite indices are equal. These are the two difference of dominance criteria we are going to use. Once we have established these dominance criteria, we go now to the next slide where it has been computed that different relations in terms of dominance between or among the different countries. Then we can see if we have a look at through rows, we will see for example that Finland gets a zero with respect to all other countries but Spain where he gets a one. This means that Finland dominates Spain with respect to the criteria we have studied in the choice of the weights. For example, take another like Norway, you can see in the rows that Norway takes a lot of ones, this mean that Norway dominates these countries where we get the ones with respect to these different criteria. These would be, the first table in this slide refers about the strong dominance and the second table in the next slide refers about weak dominance. With these tables we are already able to decide which type or weights are more robust to the outcomes in the composite indicator. Let me, just to finish, let me point out here that robustness is something necessary to get nice properties for composite indicators. It's very interesting to show the audience and to show public opinion that our results are robust to the choice of weights and somehow this robustness is going to reinforce the conclusions we are getting from the composite indicators. Because otherwise, as you probably understand, anyone could tell us that okay, we have chosen or we have obtained these results. But if the weights would have been another, then the results probably would have changed also. And this is something we don't want to have.
