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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?

Argonaut
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2 Answers2

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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.

1muflon1
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  • Thanks this was pleasant and thorough explanation. One quick question though: You wrote that you can not get the $\beta$ of ARMA from ARIMA directly. Is there an indirect method? – Argonaut Oct 03 '20 at 00:56
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    @Argonaut there is no way to get exactly $\beta$ with the same interpretation but it is possible to run ARDL model (if you dont care about the MA part) where using the 'bounds tests' for cointegration you can include also levels of the variable. The interpretation of $\beta$ will be bit different there. – 1muflon1 Oct 03 '20 at 09:08
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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

$$X_t = 1.01 X_{t-1} + \epsilon_t$$

example

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.

Sextus Empiricus
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  • Hi: Both of those answers were great I just want to emphasize that differencing and taking the log are two separate approaches so trying to relate them is not the way to go. differencing is often used to possibly eliminate a unit root (which is a specific type of non-stationarity ). A log transformation would be more appropriate if the histogram of the response looked skewed in some way ( which is another specific type on non-stationarity ). There's no relation between the two approaches aside from the fact that another possible way to obtain stationarity is to take the difference of the logs. – mlofton Oct 06 '20 at 02:46