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These are results from ADF test as implemented in the aTSA package in R:

> adf.test(x)
Augmented Dickey-Fuller Test 
alternative: stationary 
 
Type 1: no drift no trend 
     lag   ADF p.value
[1,]   0 -2.14  0.0339
[2,]   1 -2.53  0.0137
[3,]   2 -2.91  0.0100
[4,]   3 -2.84  0.0100
Type 2: with drift no trend 
     lag   ADF p.value
[1,]   0 -2.12  0.2822
[2,]   1 -2.44  0.1592
[3,]   2 -2.81  0.0669
[4,]   3 -2.78  0.0728
Type 3: with drift and trend 
     lag   ADF p.value
[1,]   0 -2.02   0.559
[2,]   1 -1.59   0.736
[3,]   2 -1.69   0.695
[4,]   3 -1.86   0.624
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Note: in fact, p.value = 0.01 means p.value <= 0.01 

Might be a little bit basic, but I don't completely understand the implications of the difference between the three types. For example, does "Type 3" case mean that if I account for a deterministic trend (i.e. $\alpha t$) in and a "departure" point (i.e. constant "c") in my series (i.e. x), then all what is left in x excluding those previous things is stationary?

And also, what are the implications of that? For example if I get from the test that "Type 1" and "Type 2" are NOT stationary, but "Type 3" IS stationary, it means that in order to get a stationarity from x I need to remove a deterministic trend and an intercept?

Would appreciate some help on that. Thanks!

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