Higher Order Conditional Expectation

Question

This may be a very silly question, but does $E[y_t| I_{t-1}] \neq 0$, where $E[y_t|I_{t-1}]$ is the expectation of $y_t$ conditional on the information available at time $t-1$, implies $E[y^2_t| I_{t-1}] \neq 0$?

Answer 1

If $E(y_t^{2}|I_{t-1})=0$ then (taking expectation) $Ey_t^{2}=0$ which implies that $y_t=0$ almost surely. This contradicts the hypothesis that $E(y_t|I_{t-1})\neq 0$.