# Analysis of Time Series

The objective of the time series analysis is to identify the magnitude and direction of trends, to estimate the effect of seasonal and cyclical variations and to estimate the size of the residual component. This implies the decomposition of a time series into its several components. Two lines of approach are usually adopted in analyzing a given time series:

(i) The additive model

(ii) The multiplicative model

Thus, if we denote the time series by $Y$, the secular trend by $T$, the seasonal or short term periodic movements by $S$, the long term cyclical movements by $C$ and the irregular or residual component by $R$, then the additive model can be described as

$Y = T + S + C + R$

while the multiplicative model can be described as

$Y = T \times S \times C \times R$

The additive model is generally used when the time series is spread over a short time span or where the rate of growth or decline in the trend is small. The multiplicative model, which is used more often than the additive model, is generally used whenever the time span of the series is large or the rate of growth or decline is

$Y - T = S + C + R$

or

$\frac{Y}{T} = S \times C \times R$

Similarly, a de-trended, de-seasonalized series may be obtained as

$Y - T - S = C + R$

or

$\frac{Y}{{T \times S}} = C \times R$

It is not always necessary for the time series to include all four types of variations; rather, one or more of these components might be missing altogether. For example, when using annual data the seasonal component may be ignored, while in a time series of a short span having monthly or quarterly observations, the cyclical component may be ignored.