I'm reading Hayashi's "Econometrics", and in Chapter 2, page 101 he discusses the following white noise process:
I understand the calculations for the expected mean and variance, but I can't understand why the covariance would be 0 for $i\neq j$.
Since $E(z_i)=0$, the expression for the covariance would be reduced to $E(\cos(iw)\cdotp \cos(jw))$. Would someone mind showing me why this goes to zero? Assuming for the moment that $cos(iw)$ is positive, then $\cos((i+1)w)$ would be negative and so on. But why would there be 0 covariance for any two $i$ and $j$, $i\neq j$? I know this is a simple question, but can't seem to wrap my head around it.