Let (r_{t}) be the log return of an asset at time (t). Assume that (left{r_{t} ight}) is

Let \(r_{t}\) be the log return of an asset at time \(t\). Assume that \(\left\{r_{t}\right\}\) is a Gaussian white noise series with mean 0.02 and variance 0.04 . Suppose that the probability of a trade at each time point is \(50 \%\) and is independent of \(r_{t}\). Denote the observed return by \(r_{t}^{0}\). Is \(r_{t}^{0}\) serially correlated? If yes, calculate the first three lags of autocorrelations of \(r_{t}^{0}\).