0
$\begingroup$

I'd like to find the correlation matrix of $Y_1$ and $Y_2$, for $Y_1=X_1+X_2$ and $Y_2=X_1-X_2$, where $X_1$ and $X_2$ are random variable with correlation matrix $\Omega_X=\begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix} $, where $-1\leq \rho \leq 1$.

Here is my what I did:

I think that since the main diagonal entries are equal to 1, $\Omega_X=\Sigma_X$. Then, I apply the Cholesky decomposition: $\Sigma_X=U^t\,U$

$\begin{pmatrix} X_1\\ X_2 \end{pmatrix} = U^t \, \begin{pmatrix} Z_1\\ Z_2 \end{pmatrix}$, $Z_1$ and $Z_2$ are independent standard normal variables.

$\Rightarrow \begin{pmatrix} X_1\\ X_2 \end{pmatrix} = \begin{pmatrix} 1 & 0\\ \rho & \sqrt{1-\rho^2} \end{pmatrix} \, \begin{pmatrix} Z_1\\ Z_2 \end{pmatrix} $

$\Rightarrow \begin{cases} X_1 = Z_1\\ X_2 = \rho\,Z_1 + \sqrt{1-\rho^2}Z_2 \end{cases} $

$ \Rightarrow \begin{cases} Y_1 = (1+\rho)\,Z_1 + \sqrt{1-\rho^2}\,Z_2\\ Y_2 = (1-\rho)\,Z_1 - \sqrt{1-\rho^2}\,Z_2 \end{cases} $

I deduce: $Y_1 \sim N(0,2\rho+2)$ and $Y_2 \sim N(0,2-2\rho)$

But I am not able to get corr($Y_1$,$Y_2$).

Any hint about what I'm doing wrong? Thank you

$\endgroup$

1 Answer 1

Reset to default
0
$\begingroup$

I don't see anything about normal variables here, so Cholesky is out.

Also I don't see any assumptions about the variance of $X_1$ and $X_2.$ I will call them $\sigma_1^2$ and $\sigma_2^2$ so that the covariance matrix of $X_1$ and $X_2$ is $\begin{pmatrix}\sigma_1^2 & \rho\sigma_1\sigma_2\\\rho\sigma_1\sigma_2 & \sigma_2^2\end{pmatrix}.$ The final answer depends on these variances (for instance, if they're equal then the correlation of $Y_1$ and $Y_2$ is zero).

First get variance of $Y_1$ and $Y_2.$ $$ Var(Y_1) = Var(X_1) + Var(X_2) + 2Cov(X_1,X_2) = \sigma_1^2 + \sigma_2^2 + 2\rho\sigma_1\sigma_2$$ and similarly $$ Var(Y_2) = \sigma_1^2+\sigma^2_2-2\rho\sigma_1\sigma_2.$$

Then you can get the covariance of $Y_1$ and $Y_2$ by $$ Cov(Y_1,Y_2) = E(Y_1Y_2)-E(Y_1)E(Y_2) = E(X_1^2-X_2^2)-(E(X_1)^2-E(X_2)^2) \\= Var(X_1)-Var(X_2) = \sigma_1^2-\sigma_2^2.$$ Then the correlation of $Y_1$ and $Y_2$ is $$ Cor(Y_1,Y_2) = \frac{Cov(Y_1,Y_2)}{\sqrt{Var(Y_1)Var(Y_2)}}$$

$\endgroup$
2
  • $\begingroup$ I tried to use Cholesky, to work with independent standard normal variable. From the matrix given, $Cov(Y_1,Y_2)=1-1=0$, so $Corr(Y_1,Y_2)=0$. Is it wrong? $\endgroup$ Commented Mar 12, 2017 at 14:35
  • $\begingroup$ You are not wrong about cov equalling zero if the variance of $X_1$ equals the variance of $X_2.$ You cannot use cholesky to work with standard normal random variables unless $X_1$ and $X_2$ are bivariate normal. I see neither assumption in your problem statement. $\endgroup$ Commented Mar 12, 2017 at 22:17

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.