# A Tau Measure of Serial Dependendence, from Victor Niederhoffer

February 7, 2013 |

It is interesting and useful to measure the tendency to continuation or reversal in a series. It's particularly useful for markets because many traders like to go with or against in a period. And some measure of whether this works or not, and how it's changing provides a rudder.

The usual methods of measuring it rely on the serial correlation coefficient, but this tends to be disrupted by extreme or missing observations and doesn't have stable properties for many non-normal distributions. Non-parametric measures that rely on ranks, runs, or moves above and below the median, or curve fitting for consecutive observations have often been used. Cowles started the whole subject for stock prices by looking at sequences and reversals in consecutive prices.

A measure that I have been working with that is relatively new and has many advantages is to consider the concordances and discordances in a series. This method is based on work done by Kendall in rank correlations with his statistic, Kendall's Tau. A key article in this area that provides an excellent foundation is Ferguson, Genet and Hallin, "Kendall's tau for serial dependence" and "Bandt Ordinal Time series analysis".

The method of concordances and discordances starts with looking at 3 consecutive observations in a series. Let's call them p1, p2, and p3. If p2 > 1 and P3 > p2, that's a positive concordance. If p2< 1 and P3 < p2, that's a negative concordance. All the other rises followed by declines, or declines followed by rises are discordances. (Note that there are 6 permutations of the 3 numbers and only 2 yield concordances.)

To make it more tangible consider the levels in stocks from Friday 1/4/2013 to  Friday 1/11/2013

day        Date                  Level       change    rank of change

```Fri           1/4/2013         1458
```
```Mon          1/7/2013       1456       -2          2
```
```Tues         1/8/2013        1452       -4         1
```
`Wed          1/9/2013       1456        4         4                                      `
`Thurs        1/10/2013      1467        11        5                                     `
`Fri            1/11/2013      1467      0         3`

To measure the momentum in the series of changes, one must compute all the consecutive one day discordances, + the number of consecutive 2 day discordances + the number of 3 day discordances. It is best to focus on the ranks. If the consecutive pairs of ranks reverse there is a discordance. If they are in the same direction, there is a concordance.

Comparing Mon to Tue and Tue to Wed, one notes a discordance.

Comparing Tues to Wed, and Wed to Thur, there is a concordance.

Comparing Wed to Thurs, and Thurs to Friday, there is a discordance.

Mon and Wed and Tue and Thur are in concord.

Tues and Thur and Wed and Friday are in discord as Thur rank is higher than Tues and Friday's rank is lower than Wed.

There is one 3 day comparison. Mon and Thur, and Tuesday to Friday are in concordance. Thus, there were 3 discordances and 3 concordances. It turns out that the expected number of discordances for a time series is ( n-2) ( 3n-1 ) / 12. since n is 5 , the expected number of discordances is 3.5. An exact calculation is possible and shows that 3 or less discordances has a prob of 20%.

How can this measure be used? First, it provides a nice estimate of the degree of correlation between the consecutive values of a series. The question then arises, how can one predict subsequent momentum based on past momentum. It turns out that that there is a tendency in the series that we have looked at , for periods with high concordances to be followed by periods with high discordances, i.e. momentum changes from period to period. This would have to be quantified with the period one is interested in, a week, a month etc.

I will report further work on this in future. I would like to thank Doc Castaldo and Mike Chuprin for their kind assistance on this project.

## Fabrice Rouah writes:

Very good point. Non-parametric methods are definitely preferable for financial time series that rarely meet the normality or linearity assumptions required of many parametric methods. Another example of parametric methods are t-tests and ANOVA. To compare returns between different groups one is better off using their non-parametric counterparts, namely the tests of Wilcoxon, Mann-Whitney, Kruskal-Wallis and many others.