# Are log difference time series models better than growth rates?

Often I see authors estimate a "log difference" model, e.g.

$\log (y_t)-\log(y_{t-1}) = \log(y_t/y_{t-1}) = \alpha + \beta x_t$

I agree this is appropriate to relate $x_t$ to a percentage change in $y_t$ while $\log (y_t)$ is $I(1)$.

But the log difference is an approximation, and it seems one could just as well estimate a model without the log transformation, e.g.

$y_t/y_{t-1} -1 = (y_t - y_{t-1}) / y_{t-1}=\alpha+\beta x_t$

Moreover the growth rate would precisely describe the percent change, while the log difference would only approximate the percent change.

However, I've found the log difference approach is used much more often. In fact, using the growth rate $y_t/y_{t-1}$ seems just as appropriate to address stationarity as taking the first difference. In fact, I have found that forecasting becomes biased (sometimes called the retransformation problem in the literature) when transforming the log variable back to the level data.

What are the benefits to using the log difference compared to the growth rate? Are there any inherent problems with the growth rate transformation? I'm guessing I am missing something, otherwise it would seem obvious to use that approach more often.

• Thank you for your comments. I agree the symmetry and bounding is a significant advantage. It seems the bounding would help control heteroskedasticity and the symmetry would help hold the mean constant. – A. Smith Jul 2 '16 at 15:46
• The log-difference is not an approximation. It is a continuously compounded or exponential growth rate, as opposed to a period-over-period rate. They are different things. Laypersons understand the second one better, but the first one has cleaner mathematical properties (e.g. average growth is just the mean of the growth rates, growth rate of product is the sum of the rates, etc). The bit about forecasting is either unnecessary transformation leading to explosive forecasts, or median-unbiased but not mean-unbiased, which is fine. It has nothing to do with continuous vs. period rates. – Chris Haug Jul 22 '17 at 15:58

## 2 Answers

One major advantage of log-differences is symmetry: if you have a log difference of $0.1$ today and one of $-0.1$ tomorrow, you are back from where you started. In contrast, 10% growth today and 10% decline tomorrow will not bring you back to the initial value.

• The symmetry / bounding is the main advantage I see. Going from 100 to 10 is a log10 difference of -1, but -90%. Going from 100 to 1000 is also a log difference of 1, but 900%. A linear model is going to pay inordinate attention to that 900% observation. – zbicyclist Jul 2 '16 at 14:40

Many macroeconomic indicators are tied to population growth, which is exponential, and thus have an exponential trend themselves. So the process before modelling with ARIMA, VAR or other linear methods is usually:

• Take logs to get a series with a linear trend
• Then difference to get a stationary series