Following the (Vasicek 2002) paper, on page 2, it is specified that if the variables $X^i$ are jointly standard normal with equal pairwise correlations $\rho$ they can be represented as:
$X^i = Y\sqrt{\rho} + Z^i\sqrt{1 – \rho}$
where $Y$ and $Z^1…Z^n$ are mutually independent standard normal variables. The paper states that
"This is a property of the equicorrelated normal distribution."
How can I derive this property?
I tried finding the correlation of two variables $X^i$ and $X^j$ constructed in that way, but I don't get $\rho$ as the result.
Best Answer
Which property is being called out as a property enjoyed by jointly normal standard normal random variables? That they have a representation as $$X_i = \sqrt \rho\, Y + \sqrt{1-\rho}\,Z_i \tag{1}$$ where $Y$ and the $Z_i$ are independent standard normal random variables? The joint normality of the $X_i$ follows from the fact that they are obtained as linear combinations of jointly normal random variables. That the $X_i$ have expected value $0$ follows straightforwardly from the linearity of expectation applied to $(1)$ while bilinearity of covariance and independence gives \begin{align} \operatorname{cov}(X_i,X_j) &= \operatorname{cov}\left(\sqrt \rho\, Y + \sqrt{1-\rho}\,Z_i,\sqrt \rho\, Y + \sqrt{1-\rho}\,Z_j\right)\\ &= \rho\operatorname{cov}(Y,Y) + 0 + 0 + (1-\rho)\operatorname{cov}(Z_i,Z_j)\\ &= \rho + \begin{cases}1-\rho, & i=j,\\ 0, &i \neq j, \end{cases}\\ &= \begin{cases}1, & i=j,\\ \rho, &i \neq j, \end{cases} \end{align} showing that the $X_i$ have unit variance and correlation coefficient as desired.
Note that all this works only when $\rho \geq 0$. The common correlation $\rho$ satisfies $-\frac{1}{n-1}\leq \rho \leq 1$ but the construction above does not work for $-\frac{1}{n-1}\leq \rho < 0$. For a construction of unit random variables with common correlation coefficient $-\frac{1}{n-1}$, see this answer of mine.