# Log transformation of TS-stationary time series

I usually see the $$log$$ transformation of prices: $$p_{new}\left(t\right) = ln\left(\frac{p_t}{p_{t-1}}\right), t \in [2...N]$$.

Let's our series be a trend stationary time series like: $$p\left(t\right) = kt + b + \xi(t)$$, where $$k,b$$ are numbers, $$t \in [1...N]$$, $$\xi(t)$$ is the random variable like $$\xi(t) \sim N\left(\mu, \sigma\right)$$.

For large $$b$$ and small $$k, \sigma$$ 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)?

Thank you.

• it's not entirely clear what you're trying to do and/or what your objective is. typically we transform one series or set of values to another because it makes them easier to work with or allows us to apply a certain set of techniques to the result. you wouldn't need to do that if you're simply simulating a price time-series with some set of parameters. it's also not clear what framework you're using to establish one TS as 'good' and the other 'bad' please consider re-working your question to make it clearer. – Chris May 7 at 17:18
• Okay, I'll try it today. Thnxs for response. – Dmitriy May 8 at 12:03