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I want to test whether a time series $\{x_1,x_2,\dots,x_t\}$ is stationary (weak stochastic stationarity/WSS). I first want to show that the time series has no deterministic trend and then use an ADF (augmented Dickey-Fuller) test to show that it is not a unit root process. From eyeballing, I know that the time series has an at most linear deterministic trend, but there may be several periodic signals of different frequencies present.

I would like to use the following approach to test whether my data is stationary:

Step 1A: I make an OLS fit of my time series with the model $x_t = a+bx_t+e_t$, where $e_t$ is the error term.

Step 1B: I show that there are no endogenous variables by showing that there are $e_t$ has no serial correlation. (I can use a Durbin-Watson, Breusch-Godfrey, Ljung-Box, or similar test for this.) This shows that I do not need to include periodic deterministic trends in my model.

Step 1C: I use a t-test to show that b=0. This shows that there is no linear deterministic trend either.

Step 2: I use an ADF test to show that the time series is not a realisation of a unit-root process.

Is this a valid approach to test my data for stationarity? Are there approaches that would be easier, more robust, or more common?

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