Copulas

[ibrahim-1987]

hi david

1. what is meant by : ".... correlation does not work for trended return series that are not cointegrated" ?

2. can y plz, explain to me what is the difference between:
a. variate
b. bivariate
c. multivariate.
distributions?

 

[David Harper CFA FRM]

1. I am not sure exactly what Dowd means by that. I am not aware (I did not believe) that a cointegrated series cannot be tested for correlation. His full paragraph is below. However, please note: 1. His general point is that correlation is not so useful when distributions are non-elliptical (non-normal), among its several problems; for exam purposes, the most salient is that correlation is merely linear, while in practice, dependence is often non-linear; 2. Cointegrated time series is beyond our scope (http://en.wikipedia.org/wiki/Cointegration) but we can think of cointegration as a long run relationship between two time series; versus correlation, which measures the same-period co-movement. For example, maybe google and amazon's stocks tend to react similarly on a daily basis. This would manifest as positive daily return correlations. But, over long time periods, maybe they do not drift in unison or similarly, there is nothing that keeps them tracking together (like mean reversion but reverting to each other!] ... this lack of "mutual reversion" (I just made up that term) would manifest as non-cointegrated. Both are views of a relationship, so I do not know what Dowd means exactly by the last part of:

Correlation bad with Non-elliptical distributions: "Unfortunately, the correlation measure runs into more serious problems once we go outside elliptical distributions. One problem is that correlation is not even defined unless variances are finite, and this is obviously a problem if we are dealing with a heavy-tailed distribution that has an infinite variance (e.g., a Levy distribution with α< 2), or if we are dealing with trended return series that are not cointegrated." - Dowd

2. It's just the number of random variables that constitute the distribution. It's maybe easier to start with discrete. Classic case of a univariate distribution is a single die (uniform distribution); e.g.,
P [die = 1] = 1/6

If we roll two die, we can refer to a bivariate distribution; e.g.,
P [first die = 1, second die = 1] = (1/6)^2
... it is bivariate b/c we query the probability of an outcome in two variables
... note this is a bit different than P[sum of two die] = 1/6th. The bivariate (multivariate) specifically asks about the simultaneous outcome of two (n) random variables.

We have an example of multivariate with; e.g.,
P[first die = X, second die = Y, third die = Z] = (1/6)^3

Just as we can visualize the bivariate with a matrix, we can visualize a three-variable multivariate (discrete) distribution in three-dimensional space; eg.,
P[GDP > 3%, interest rates < 2%, unemployment < 9%) = ?

We need this terminology, i think, to comprehend the copula (in Dowd), which joins a multivariate function to several univariates. That is, the multivariate distribution is hard to figure, but the coupula gives a function to produce the multivariate with each of the marginal univarates; e.g., P[GDP] is a univariate, P[interest rates] is a univariate ... the copula gives us a function to take these univariate marginals as inputs and produce the non-trivial multivariate.

I hope that helps, David

 

[ibrahim-1987]

daivd, i find my self speechless in front of y, y r the best, AMAZING

 

[Aleksander Hansen]

1) Think of it this way: why do we measure correlation using returns instead of prices?
More generally, if any two non-stationary time series are cointegrated we can capture the [otherwise blurred/spurious] trend.
Why do we care? Because it makes all the stuff we usually would like to do with time series (serial correlation, metrics,moments) possible [well, it's always possible, but it makes us more comfortable that we are not just goofing around]