# Copula- AR simulation

I am estimating different copulas for bond factors that i also fit AR(1) models on.

Now i would like to test and compare durations and VaRs with my model vs empiric.

But how can i simulate AR(1) series with my copula properties? I can simulate both independently but i am unsure how to proceed to do both simulatneously

I hope my question isn't too specific.

Thanks!

As you know, simulating AR(1) is to simulate the distributed error path.

Assume the bivariate errors distributed $\sim F(x),\sim F(y)$ with copula $C(u,v)$ to model their dependence.

Then the bivariate joint error distribution is given by Sklar's theorem:

$$F(x,y)=C(F(x),F(y))$$

You can simulate from this distribution using Conditional Sampling:

To obtain a realization of a bivariate Copula $C(u,v)$, one draws the first variable $u$ as random number $\sim U(0,1)$. The second variable $v$ is generated from another independent random number $z$ plugged into the inverse Copula $C^{-1}(z\,|u=u)$ under the first generated (conditional) random number $u$:

1. Draw $\bar{u},\bar{z}\sim U(0,1)$
2. Set $\bar{v} = C_{\bar{u}}^{-1}(\bar{z})$ (quasi-inverse Copula under $\bar{u}$, or conditional $C^{-1}(t,u\,|u=\bar{u})$ )

From this you get $$(x=F(\bar{u})^{-1},y=F(\bar{v})^{-1})$$ as your two simulated errors for the AR(1) process.