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

Question

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.

Answer 1

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).

Answer 2

Similar to Fato39's answer, you could try both decompositions and check which one yields the smallest additive residuals with respect to the original series, i.e., minimized the mean squared residual.

Especially if your end goal is to forecast your series, you could use a holdout sample, like the last 12 months. Fit both models - additive and multiplicative - to the beginning of the series, forecast them out, and check which one yields the lowest MSE.