# Log transformation of TS-stationary time series?

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)?

• you might want to look at stats.stackexchange.com/questions/18844/… and for a counter example autobox.com/pdfs/vegas_ibf_09a.pdf Commented May 7, 2019 at 14:56
• What happens when you difference (without logging) your model? Check the algebra. Also your model is overparameterized--you don't need $b$ and $\mu$. You don't even need either if you only analyze the differenced series. Commented Sep 8 at 18:06

1. I don't think you wrote your price process correctly, or else it is a bad model of prices. $$p(t) = kt + b + \xi(t)$$ where $$\xi(t)\sim N(\mu,\sigma)$$ does not have a stochastic trend you need to difference out to make the series stationary.

Presumably $$\xi(t)$$ is supposed to be something like a Weiner process integrated from 0 to t. It is also strange to let $$\mu$$ be anything other than 0 while also including a deterministic drift term $$b$$.

1. The important point about your transformation is not so much the logarithm, but that $$\log(p_{t}/\log(p_{t-1}) = \log(p_t) - \log(p_{t-1})$$. Prices are integrated of order 1, so you first difference to remove that non-stationarity.

2. The $$kt$$ term is called a deterministic trend. The simplest way to remove a deterministic trend of this sort is to run a regression of the time series on $$t$$ to estimate $$k$$ and then subtract out $$\hat{k}t$$. Of course, if you also have a stochastic trend you need to deal with that first.

• Yes, I know about TS- and DS-trends. So, If I have TS-trend in my price series, It's not a good idea to take logarithms, because you'll have your transformed data going from big to small variations when $t$ gonna grow. And same financial instruments have TS-trend behavior (by unit root tests). Process $p(t) = kt + b + \xi(t)$ I wrote as an example, but It's a correct example for same tradeable instruments in the short window. So, my question is about correctness of $log(\frac{p_{t}}{p_{t-1}})$ usage everywhere in econometric models. Commented May 7, 2019 at 15:06
• And also If I'll use I(1) to TS-stationary series, I'll have MA part overdifferencing and non-invertable ARMA model, for example. Commented May 7, 2019 at 15:14
• @Dmitry: The log transformation is used for modelling percent changes ( of the the dependent variable ) rather than changes in its level. The log DIFFERENCE is used to model returns by differencing the log price because ( as Matt P said ), $log(p_t) - log(p_{t-1} =$ the return of the stock over t-1 to t and returns tend to be stationary compared to prices. So, one needs to be careful when talking about using the log transormation versus taking the log difference. Econometric papers probably take the log but they wouldn't take the log difference unless they were modelling asset returns. Commented Jun 1, 2021 at 23:59
• Also, I wouldn't even try ( or think about ) modelling prices by using a time trend. It's not going to be constant enough no matter how short the time frame. Go straight to returns and forget about prices. Well, that's just my suggestion of course. :). Commented Jun 2, 2021 at 0:02
• I don't know if I explained above clearly enough. The point is that, once you log difference a price series, you've almost for sure have a zero-mean series so there so a time trend should not be included. I hope this clarifies my previous comment. Commented Sep 9 at 0:41