# How to judge whether to model a time series additively or multiplicatively?

I don't know how to to identify whether my time series is additive or multiplicative using decompose() command in R. It is a monthly time series. • For those unfamiliar with this command: in an exploratory spirit, it smooths a time series into a "trend"; extracts a seasonal component "by averaging, for each time unit, over all periods"; and finally computes the residuals. Upon request it will fit an additive model or a multiplicative model. Thus, presumably, the present question concerns how to judge whether to model a time series additively or multiplicatively based on the two possible decompositions offered by decompose. – whuber Oct 9 '17 at 17:40
• @whuber yes this is what i'm looking for : how to judge whether to model a time series additively or multiplicatively. – Karima Touati Oct 9 '17 at 17:44
• I think this question is about how to interpret the output, not about the R commands, so I am voting to leave it open. – Peter Flom Oct 10 '17 at 11:17
• Try exploring both and not only one. – Nuclear03020704 Aug 10 '20 at 11:37

## 1 Answer

One method would be to use the test provided in JDemetra+, which is explained in the manual and the reference therein (though it's unclear from their list of references which one they are referencing):

The test for a log-level specification used by TRAMO is based on the maximum likelihood estimation of the parameter $\lambda$ in the Box-Cox transformation, which is a power transformation such that the transformed values of the time series $y$ are a monotonic function of the observations, i.e. $y^\alpha = \begin{cases} \frac{(y^\alpha_i - 1)}{\lambda}, & \lambda \neq 0 \\ \log y^\alpha_i, & \lambda = 0 \end{cases}$. The program first fits two Airline models (i.e. ARIMA (0,1,1)(0,1,1) with a mean) to the time series: one in logs ($\lambda = 0$), other without logs ($\lambda = 1$). The test compares the sum of squares of the model without logs with the sum of squares multiplied by the square of the geometric mean of the (regularly and seasonally) differenced series in the case of the model in logs. Logs are taken in the case this last function is the minimum. GÓMEZ, V., and MARAVALL, A. (2010).