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.

• Ahhhh okay now i get this! Thanks a lot! So if all Gauss-Markov-Assumption are satisfied i can assume my ADF-test gives valid results, if not i need to adjust for autocorrelation for exmaple, right? Jan 28 '21 at 15:11
• @Guest1 the Gauss-Markov assumptions are separate assumptions for OLS regression ADF has its own assumptions that are related because at the core of ADF model there is plain regression, but they wont be exactly same. For example, in standard OLS autocorrelation does not bias results only affects efficiency, but in ADF test autocorrelation does bias result so if you don't get rid of it there is no way how to fix the estimates (with standard OLS you can just replace ordinary errors with Newey-West errors or other autocorrelation consistent errors).
– 1muflon1
Jan 28 '21 at 15:18
• so its basically that the t statistic isnt valid anymore and the adf test wrong then? Jan 28 '21 at 16:32
• @Guest1 not just t statistics the estimated coefficient that then ADF tests is invalid. At that point it does not matter what the t statistics is because the whole test is invalid
– 1muflon1
Jan 28 '21 at 16:33
• i think because of a wrong standard error the t and F statistics are wrong and thats why the ADF is not valid, atleast read that a lot Jan 28 '21 at 16:41