Criteria for existence of iid common factors

Question

Let $x$ be a $n$-dimensional random vector with $Cov(x)=I_n$. Under which conditions do exist a $l$-dimensional $(l\leq n)$ random vector $z$, a matrix $\Gamma\in\mathbb{R}^{n\times l}$ and a $n$-dimensional random vector $\epsilon$ such that the following conditions are satisfied:

1) $x=\Gamma z+\epsilon$

2) The elements of $z$ are iid, the elements of $\epsilon$ are iid and the elements of $z$ and $\epsilon$ are independent.

Answer 1

If 1) and 2) hold, then $$\text{Cov}(\Gamma z + \epsilon) = \text{Cov}(\Gamma z) + \text{Cov}(\epsilon) = \Gamma \text{Cov}(z) \Gamma^\top + \text{Cov}(\epsilon)$$ by independence of $z$ and $\epsilon$. Since the elements of $z$ are i.i.d. and the elements of $\epsilon$ are i.i.d., both covariance matrices $\text{Cov}(z)$ and $\text{Cov}(\epsilon)$ are multiplies of the identity, specifically $\text{Var}(z_1) I$ and $\text{Var}(\epsilon_1) I$. So, $$\text{Cov}(\Gamma z + \epsilon) = \text{Var}(z_1) \Gamma \Gamma^\top + \text{Var}(\epsilon_1) I.$$

If this equals $I$, then one of the following must happen.

• $\Gamma$ has orthonormal rows (so that $\Gamma \Gamma^\top = I$) and $\text{Var}(z_1) + \text{Var}(\epsilon_1) = 1$, or
• $\text{Var}(z_1) = 0$ (i.e. $z_1$ is a constant) and $\text{Var}(\epsilon_1)=1$.

$\Gamma\Gamma^\top = I$ is only possible if $l \ge n$, so given your restriction $l \le n$ we must have $l=n$ in the first situation.

On top of all these constraints on $z$, $\Gamma$, $\epsilon$, I still don't think this is enough to guarantee existence. (For instance, I have not checked that $E[x] = E[\Gamma z + \epsilon]$.) Maybe I am not approaching the problem in the right way...