Let $X = (X_t : t ∈ Z)$ be a $MA(1)$-Process. Define the time series $Y = (Y_t : t ∈ Z)$ with $Y_t = \mathbb 1_{(X_t > 0)}$. What is the Autocorrelation function of $Y_t$?
Anyone has any idea how to even start solving this?
There was a hint to show $P(U > 0,V > 0) = 1/ 4 + 1 /2π\cdot \arcsin(\rho)$ where $U$ annd $V$ are both normally distributed with mean $0$, same Variance $σ^2$ und correlation $\rho$. I have shown this equality but how does it help me further?
Let $t,s\in\mathbb Z$. Then the correlation of $Y_t$ with $Y_s$ is $$\rho_{t,s}=\frac{\operatorname{Cov}(Y_t,Y_s)}{\sqrt{\operatorname{Var}(Y_t)\operatorname{Var}(Y_s)}} = \frac{\mathbb E[Y_tY_s]-\mathbb E[Y_t]\mathbb E[Y_s]}{\sqrt{\operatorname{Var}(Y_t)\operatorname{Var}(Y_s)}}. $$ Now $$\mathbb E[Y_t]=\mathbb E\left[\mathbf 1_{(X_t>0)}\right] = \mathbb P(X_t>0), $$ $$\mathbb E[Y_tY_s] = \mathbb E\left[\mathbf 1_{(X_t>0)}\mathbf 1_{(X_s>0)}\right] =\mathbb P(X_t>0,X_s>0), $$ and $$\operatorname{Var}(Y_t)=\mathbb E[Y_t^2]- \mathbb E[Y_t]^2 = \mathbb P(X_t>0)(1-\mathbb P(X_t>0)). $$ So $$\rho_{t,s} = \frac{\mathbb P(X_t>0,X_s>0)-\mathbb P(X_t>0)\mathbb P(X_s>0)}{\mathbb P(X_t>0)(1-\mathbb P(X_t>0))\mathbb P(X_s>0)(1-\mathbb P(X_s>0))} .$$
Using the result from your hint it seems that you should be able to compute this quantity.