Simulating stock returns as normally distributed to run through a copula

I am trying to model stock index returns for 4 different indices as normally distributed, to then run these distributions through a copula to obtain their correlation structure. I keep encountering the error ,

Error using copulafit (line 75)

U must contain values strictly between 0 and 1.

I know this is because the normal cdf of each set of returns varies between [0,1] , but when I model the returns as having a kernel distribution, the code runs fine, so how do I get a normal distribution cdf estimate for each set of returns that is between (0,1), so that the returns are able to be inputted to copulafit.m ?

Here's the code for a normal distribution estimate, that doesn't work.

% returns_N(:,1234) = returns for each of the 4 indices 

u= normcdf(returns_N(:,1),mean(returns_N(:,1)),std(returns_N(:,1)));
v= normcdf(returns_N(:,2),mean(returns_N(:,2)),std(returns_N(:,2)));
w= normcdf(returns_N(:,3),mean(returns_N(:,3)),std(returns_N(:,3)));
x= normcdf(returns_N(:,4),mean(returns_N(:,4)),std(returns_N(:,4)));

correlation_gaussian_normal = copulafit('Gaussian',[u v w x]);

%Error using copulafit (line 75)
%U must contain values strictly between 0 and 1.

Heres the code for a kernel distribution estimate, that works fine

u = ksdensity(returns_N(:,1),returns_N(:,1),'function','cdf');
v = ksdensity(returns_N(:,2),returns_N(:,2),'function','cdf');
w = ksdensity(returns_N(:,3),returns_N(:,3),'function','cdf');
x = ksdensity(returns_N(:,4),returns_N(:,4),'function','cdf');

correlation_gaussian_kernel = copulafit('Gaussian',[u v w x]);

[Jeff Miller]

Change normcdf values of 0 to eps and of 1 to 1-eps?

[Phillip McCart]

Did you ever find the answer to this?

I'm having a the same problem but I'm using ecdf instead of normcdf

[fx,~]=ecdf(x);
[fy,~]=ecdf(y);
u = [x,y];

[rhohat,nuhat,nuci]=copulafit(u)

Error using copulafit (line 75)
U must contain values strictly between 0 and 1.

I've verified that the values of fx and fy lie between 0 and 1 or are 0 and 1. I've tried deleting the zeroes and the beginning of fx, fy and same goes for the 1s and the end.

I'm so confused! U doesn't contain any values greater than 1 or less than zero!!

[Kausthub]

Hi Phillip, shouldn't it be u = [fx,fy] instead of u = [x,y] since you have verified the values of fx and fy lie between (0,1)?

Answer 1

[Kausthub]

Hi Pete and Phillip,

I understand that you are facing issues regarding the "normcdf" function. The range of "normcdf" and "ecdf" is [0,1] which is inclusive of both 0 and 1, whereas "copulafit" requires values strictly between (0,1) i.e., exclusive of both 0 and 1. Since, there is possiblity of "normcdf" and "ecdf" returning 0 or 1, the "copulafit" function errors when it recieves an input containing either of 0 or 1. For example:

x = [-2,-1,0,1,11];
mu = 2;
sigma = 1;
u = normcdf(x,mu,sigma)

u = 1×5

0.0000 0.0013 0.0228 0.1587 1.0000

correlation_gaussian_kernel = copulafit('Gaussian', [u]);

Error using copulafit
U must contain values strictly between 0 and 1.

To solve to issue, as rightly suggested by Jeff you could try to subtract or add a small value (epsilon) to make 0 to 0+ and 1 to 1-.

Hope this helps!