I have another question about main econometric time series transformation. I usually see the $log$ transformation of prices: $$p_{new}\left(t\right) = \ln\left(\frac{p_t}{p_{t-1}}\right), t \in [2\dotsc N].$$
Let's our series be a trend stationary time series like: $$p\left(t\right) = kt + b + \xi(t)$$, where $k,b$ are numbers, $t \in [1...N]$, $\xi(t)$ - the random variable like $\xi(t)- N\left(\mu, \sigma\right)$.
For big $b$ and small $k$ and also small $\sigma$ we have "good" transformed series, but if $b$ small and $\sigma$ big, so, we have "bad" transformed series.
"Good"($k = 2, b = 100, \sigma = 3, t \in \left[0...100\right]$).
"Bad"($k = 2, b = 10, \sigma = 10$).
So, what's the correct method for TS-series transformation (econometric-style transformation)?