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Are 30 or 40 observations enough or we need more?

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    $\begingroup$ It depends on what you mean by meaningful. The bigger the sample size, the more power it will have. $\endgroup$ – Henry Oct 16 '20 at 14:27
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If by 'meaningful' you mean 'discriminatory', so that the test tells correctly when the process is stationary and when it is not - you need to look both at size, and power of the test (or probabilities of errors of the first and the second type).

The size of the test is chosen in advance. It is simply one minus significance level. The test shows false positive with assumed probability. You can decrease false positive rate, but it is trade-off for the power of the test.

The power of the test, which is probability, that null hypothesis is correctly rejected, is the other side. As @Henry correctly mentioned, the bigger sample size, the higher probability of correct rejection. However, such power depends on many things, not only on the sample size. It depends also on coefficient values, satisfaction of test assumptions, the test itself, data-generating process, and other factors.

Without calculating the power in a given situation by Monte Carlo simulation, you have no possibility to answer your question. However there is huge branch of literature which already analysed this problem. Take for example: "Testing for unit roots using the augmented Dickey-Fuller test: Some issues relating to the size, power and the lag structure of the test" You can try to get some rules of thumb from there, however the problem is quite complicated with no straightforward answer.

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  • $\begingroup$ The only way to calculate the power is via a Monte Carlo simulation? $\endgroup$ – adrCoder Oct 16 '20 at 14:37
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    $\begingroup$ As far as I know, it is hard to do it different way, so the Monte Carlo simulations are definitively a mainstream in power analysis. $\endgroup$ – cure Oct 16 '20 at 14:48

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