Use log ARMA instead of ARIMA? And what about ARIMA coefficients? So bascially in times series analysis, if the data is not stationary instead of the arma model one should use arima. But couldn't you just log the data to eliminate stationarity and use an arma model instead?
Also the ARIMA data is shifted/integrated. But what effect does that have on the coefficients. For example in an AR or ARMA model a coefficient of +0.5 means that $x_t=\beta_0+ x_{t-1}*0.5$. So adding 50% of the last period.
But how would this 0.5 coefficient be interpreted in an ARIMA model. Does the coefficient apply to the differences and if so how can I get the coeffiecents for the absolute values?
 A: Generally speaking if you have some variable $X$ just taking a log of variable $\ln (X)$ will not solve the unit-root problem.
A general way of solving the unit-root problem is to take first differences of the data $x_t-x_{t-1}$. This is where ARIMA comes to play since $ARIMA(p,d,q)$ will  beside modelling the autocorrelation of order $p$ and moving average $q$ also differences data where the $d$ will be equal to the order of integration of the series as to make the data stationary.
Once you estimate ARIMA the coefficients will not have the same interpretation anymore. For example, suppose we are using log of real GDP $\ln (Y_t) = y_t$ in a simple $ARMA(1,0)$ we would have:
$$y_t = \alpha + \beta y_{t-1} + \epsilon_t$$
and the $\beta$ would tell us how log of present GDP depends on the past GDP.
If we would use ARIMA $(1,1,0)$ the model would look like:
$$\Delta y_t = a + b\Delta y_{t-1}+ \epsilon_t$$
where $\Delta y_t = y_t-y_{t-1}$. Now the $b$ would tell us how the present growth of GDP depends on its past. While this is not exactly the same as what the ARMA model tells us it still gives us indirect information about how output behaves based on its past. However, you can't get $\beta$ of ARMA from ARIMA directly.
A: 
But couldn't you just log the data to eliminate stationarity and use an arma model instead?

Yes, in some situations you can use a log-transform to make a series a stationary time series that can be well modelled with an ARMA model (See also the question When to log transform a time series before fitting an ARIMA model).
So that is when the model is multiplicative, and then the logarithm makes sense. But for a model generated by linear additions, I think, it does not make sense.
For instance when you have something like
$$X_t = 1.01 X_{t-1} + \epsilon_t$$

then the curve may look like having an exponential trend. But just because it looks like an exponential curve taking the logarithm is not automatically turning it into a pretty ARMA model.  I think ( I am not sure) it might be better, after all, to fit such an explosive model with an ARMA model anyway.
