Seasonal exponential smoothing without trend

Question

Why multiplicative property exists only for the exponential smoothing with seasonality and trend (Winter's additive and Winter's multiplicative models) and not for the exponential smoothing with only seasonality (without trend)? As I have seen in forecasting software, the exponential smoothing model with only seasonality only exists in an additive form. Why not exist in multiplicative form also? Is there a technical problem with this form (i.e. it does not exist either in the literature) or is it a deficiency of the software (i.e. it has not be implemented although it could be implemented)?

Answer 1

You can have an exponential smoothing model that involves multiplicative seasonality but no trend. For example, in R:

> library(forecast)
> x <- ts(rnorm(100,10,1),f=4)
> fit <- ets(x,"MNM")
> fit
ETS(M,N,M) 

Call:
 ets(y = x, model = "MNM") 

  Smoothing parameters:
    alpha = 1e-04 
    gamma = 0.0449 

  Initial states:
    l = 10.2173 
    s=0.9706 1.0156 1.0105 1.0033

  sigma:  0.0895

     AIC     AICc      BIC 
455.0367 455.9399 470.6677