I am curious that if the ADF test indicates the time series data has no unit root, can we conclude that the time series is stationary (time-invariant mean, variance and covariance)?
Here is a small example. First we generate a white noise sequence with different variances in the first half and second half. Then I pass the white noise through a stable first order filter to get the output.
e = [randn(500,1);2*randn(500,1)]; % white noise F = tf([1 0],[1 -0.5],1); % stable filter with root 0.5 y = lsim(F,e); % output h = adftest(y); % adftest, unit root check
Since the second half shows larger variance than the first output, we may say this time series sequence is not variance stationary. But if we check the ADF test, we will get h = 1, which means that there is no unit root (stationary).
Is there anything wrong with this process and my understanding? Many thanks!