Please can you explain me concordant and discordant values in English..? I mean how to look for those values.. In the meisener reading you have represented it mathematically and I am not able to understand that.
You are referring to the Kendall's tau?
If so, then do the Spearman Rank correlation's first step of Ranking the Returns, once we Rank the Returns of a set of Assets say A and B , the values of the Rank constitute the x and y values
Look carefully at the magnitude ( the values )of the Rank, we get pairs of numbers for each pair of Assets. The example in Meissner uses 5 pairs of Returns corresponding to 5 years. According to Combinatorics, we can choose two pairs in exactly 5C2 ways which is 10 ways, so write down the pairs of numbers, this is important as this is the value of t and t* that I will explain shortly
To make it clearer, look at the Table below: (The same as Meissner)
There we see the following pairs of Ranks
(1,4) (2,5) (3,3) (4,1) (5,2), we can evaluate two out of these five pairs of observations in 10 ways
Out of these pairs, look at each of the ranks in the pair, concordant pairs are those where the x value is more than the y value in both the set of two numbers in the pair (or x value is less than the y value), these are (1,4)(2,5) and (4,1)(5,2) which makes it two
Discordant pairs are those wherein either the x is greater than y in one set and y is greater than x in the other set or vice versa. these are (1,4)(4,1); (1,4)(5,2); (2,5)(4,1); (2,5)(5,2); which makes it four
whenever a set in a pair has x and y as equal they are neither, these are (1,4)(3,3); (2,5)(3,3); (3,3)(4,1); (3,3)(5,2);
In plain English, concordant pairs are those that have the sizes of their x and y values in concordance (in agreement) with each other, discordant pairs are those that have the sizes of their x and y values to be in disagreement with each other ( Definition mine)
I was thinking about this visually. If (X,Y) is the first pair represented by the intersection, then concordant pairs fall into the green regions and discordant fall into the red regions, relatively speaking:
Hi @Kavita.bhangdia No, the arrangement of the pairs is irrelevant. Take QuantMan's example above (which is Meissner's example); with 5 pairs there 5*4/2 = 10 comparisons and the order does not matter. It might be a little easier to count them if the X are ordered, but it won't matter on the result; e.g.,
(1,4) <- 1 C and 3 D; i.e., comparing to all other four with greater Xs, so any lesser Y is discordant
(2,5) <- 0 C and 3 D; i.e., comparing to three with greater Xs, but all have lesser Ys so all three discordant
(3,3) <- 0 C and 2 D; i.e., comparing to two with greater Xs but both have lesser Ys so both discordant
(4,1) <- 1 C and 0 D; i.e., comparing only to (5,5) which has both X and Y greater, so concordant. Total concordant = 2 and total discordant = 8, such that Kendall's tau = (2-8)/10 = -0.60. I hope that helps!