# Practical interpretation for $u_t = \log(x_t) - \log(x_{t-1})$

I have a time series $x_t$. If I use the transformation $u_t = log(x_t) - log(x_{t-1})$, my new time series $u_t$ has properties of white noise (random). I wonder whether there is any practical interpretation for $u_t$?

If a continuous-time process $x_t$ is geometric brownian motion it would have this property, or the discrete-time equivalent (geometric random walk).

A difference in logs is is (for $u_t$ small at least) effectively a percentage change.

See also the connection to the force of mortality (what actuaries used to call the hazard function, or rather they seem to be using it less these days) and the force of interest, which are 'instantaneous' equivalents of your annualized (or more generally, periodized) discrete measure.

• Why only for $u_t$ small? – Peter Flom - Reinstate Monica Jan 29 '13 at 13:18
• Hi @PeterFlom, for $u_t$ small because of the approximation of $ln(1 + u) = u$. This is what I think. – Sam Jan 29 '13 at 16:43
• @peterFlom The actual percentage change would be $(x_t-x_{t-1})/x_{t-1}$ (well, times 100% I guess); this is not the same as $log(x_t/x_{t-1})$ when the percentage change is large. – Glen_b -Reinstate Monica Jan 29 '13 at 22:24

If $u_{t}$ is near 0, then after multiplication by 100 it could be interpreted as percentage change of $x$ minus 100% from period $t-1$ to $t$ , that is beacause we could approximate $log(x_{t}/x_{t-1})$ by $x_{t}/x_{t-1}-1$ "very near" the point $x=1$, when $x$ is far away from 1 this approximation doesn't hold. Put functions $y=log(x)$ and $y=x-1$ on one plot.

• Thanks @Qbik, this is what I also guess. Thanks for your help. – Sam Jan 29 '13 at 16:45