Stationarity, ADF/KPSS, Autocorrelation and Heteroscedastiy i have a time series which is not stationary due to ADF/KPSS test, but is is in its first difference. So ADF and KPSS tell me it is starionary so it has a constant mean/variance/autocorrelation. But i still need to check for autocorrelation and homoscedastie and i dont get why, since ADF/KPSS told me for example the variance is constant, so why would i check with other tests for homoscedastie?
First, it is actually heteroskedasticity or heteroscedasticity and homoskedasticity or homoscedasticity (depending on whether you prefer to be faithful to the Greek root of the word or prefer the latinized English spelling).
Unit root test only tell you if the dependent variable has constant variance. That is for a dependent variable $y_t$ that is non-stationary (among other things) it will hold that:
$$\operatorname {Var}(y_{t})=\sum _{{j=1}}^{t}\sigma ^{2}=t\sigma ^{2}.$$
However, note this is variance of $y_t$. When you test for presence of presence of heteroskedasticity in subsequent model that uses $y_t$, for example:
$$y_t = a + b x_t + e_t$$
you are testing for constant variance of $e_t$ not $y_t$, since one of the Guass-Markov assumptions behind OLS is that errors should be homoskedastic (although this can be adjusted for by reestimating the errors with different method as it only affects efficiency of OLS estimator).
When it comes to autocorrelation, the same issue as above applies. In addition, unit root tests won't even tell you if $y_t$ is autocorrelated or not. In fact, most unit root tests will be biased in the presence of autocorrelation (e.g. Augmented Dickey-Fuller test) so that is something that you should test separately before you even perform your unit root test.