What type of model is used when seasonal fluctuations increase over time with a growing dependent variable due to a long-term linear trend?

When seasonal fluctuations increase over time in conjunction with a growing dependent variable influenced by a long-term linear trend, a multiplicative model is the appropriate choice. This type of model operates on the principle that the seasonal component alters the trend component multiplicatively.

In a multiplicative framework, the components of the model are combined such that the effect of seasonality increases in proportion to the size of the trend. For instance, if the overall level of a variable rises, the seasonal effects that are traditionally fixed will also expand, leading to larger deviations from the trend during seasonal spikes.

This contrasts with additive models, where seasonal fluctuations are added to the trend, maintaining a consistent influence regardless of the trend's level. Linear models do not account for seasonality or trends that change in magnitude over time, and exponential models capture growth rates that increase multiplicatively but do not inherently incorporate seasonal fluctuations or trends in the same manner as a multiplicative model. Thus, given the context of increasing seasonal fluctuations tied to a long-term trend, the multiplicative model fits best.