Time series models: Addition and Multiplication model

Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people’s wages by reference to their respective education levels, where the individuals’ data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values.

Additive Model:

1. Data is represented in terms of addition of seasonality, trend, cyclical and residual components
2. Used where change is measured in absolute quantity
3. Data is modeled as-is

Additive model is used when the variance of the time series doesn’t change over different values of the time series.

On the other hand, if the variance is higher when the time series is higher then it often means we should use a multiplicative models.

Returni=pricei−pricei−1=trendi−trendi−1+seasonali−seasonali−1+errori−errori−1returni=pricei−pricei−1=trendi−trendi−1+seasonali−seasonali−1+errori−errori−1

If error’s increments have normal iid distributions then returni has also a normal distribution with constant variance over time.

Multiplicative model:

1. Data is represented in terms of multiplication of seasonality, trend, cyclical and residual components
2. Used where change is measured in percent (%) change
3. Data is modeled just as additive but after taking logarithm (with base as natural or base 10)

If log of the time series is an additive model then the original time series is a multiplicative model, because:

log(pricei)=log(trendi⋅seasonali⋅errori)=log(trendi)+log(seasonali)+log(errori)log(pricei)=log(trendi⋅seasonali⋅errori)=log(trendi)+log(seasonali)+log(errori)

So the return of logarithms:

log(pricei)−log(pricei−1)=log(pricei/pricei−1)