Log Returns for Trend-Stationary Series with Changing Levels

Summary

This note asks whether log differences are an appropriate transformation for a trend-stationary series modeled as a linear trend plus random noise. It contrasts examples where the series has a relatively large baseline and modest trend and noise with cases where the baseline is smaller and the noise is larger. The concern is that taking the logarithm of consecutive price ratios may produce a more or less suitable transformed series depending on those scales.

The document does not propose or demonstrate an econometric transformation, nor does it define a statistical criterion for judging the examples as good or bad. It presents the setup as a question about how to transform trend-stationary data. A reader should therefore treat it as a prompt about transformation choice, not as evidence that log returns are generally valid or invalid under the stated conditions. The examples alone do not establish stationarity or identify a preferred method.

Key ideas

• The note considers log differences as a transformation of a trend-stationary series.
• Its model combines a linear deterministic trend with random noise.
• It suggests that the baseline level and noise scale may affect the behavior of log differences.
• The examples are illustrative and do not establish a statistical test of transformation quality.
• No alternative transformation or definitive recommendation is provided.

Tags

# Log transformation of TS-stationary time series


# 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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.