I'm stuck with the definition of the Covariance in the classical test theory (CTT) framework:
In CTT it is assumed that for two items:
$X = \tau_x + \epsilon_x$
$E(X) = \tau_x$
$Y = \tau_y + \epsilon_y$
$E(Y) = \tau_y$
From these assumptions it follows that:
$Cov(X,Y) = E[(\tau_x+\epsilon_x)*(\tau_y+\epsilon_y)] - E(\tau_x+\epsilon_x)*E(\tau_y+\epsilon_y)$
$= E(\tau_x\tau_y) - E(\tau_x)E(\tau_y) $
$= Cov(\tau_x,\tau_y)$
I tried to come to the same conclusion using a slightly different approach:
$Cov(X,Y) = E((X-E(X))(Y-E(Y))$
$=E((\tau_x+\epsilon_x-\tau_x)*(\tau_y+\epsilon_y-\tau_y))$
$=E(\epsilon_x*\epsilon_y)$
As $E(\epsilon_x) = 0$ the following terms should be equivalent:
$E(\epsilon_x*\epsilon_y) = E[(\epsilon_x - E(\epsilon_x))*(\epsilon_y - E(\epsilon_y))] = Cov(\epsilon_x, \epsilon_y)$
As the conclusion drawn from my approach contradicts with the first conclusion I assume that there is a flaw in my math. Unfortunately I seem to be unable to identify my mistake. Can someone point me to it?