088. Time-Series Models

A time-series model can be expressed as some combination of these four components. The model is simply a mathematical statement of the relationship amon g the four components. Two types of models are commonly associated with time series:

1) the additive model;

2) the multiplicative model.

The Additive model is expressed as

YT = TT + ST + CT + IT,

Where YT is the Value of time series for time period t, and the right-hand side values are the Trend, the Seasonal variation, the Cyclical variation, and the Random or Irregular variation, respectively, for the same time period. In the additive model, all values are expressed in original units, and S, C, and I are deviations around T.

Example. If we were to develop a time-series model for sales in dollars for a local retail store, we might find that T = $500, S = $100, C = -$25, and I = -$10. Sales would be

Y = $500 + $100 - $25 - $10 = $565.

Notice that the positive value for S indicates that existing seasonal influences have had a positive impact on sales. The negative cyclical value suggests that the business cycle is currently in a downswing. There was apparently some random event that had a negative impact on sales.

The additive у model suffers from the somewhat unrealistic assumption that the components are independent of each other. This is seldom the case in the real world. In most instances, movements in one component will have an impact on other components, thereby negating the assumption of independence. Or, perharps even more commonly, we often find that certain forces at work in the economy simultaneously affect two or more components. Again, the assumption of independence is violated.

As a result, the multiplicative model is often preferred. It assumes that the components interact with each other and do not move independently. The multiplicative model is expressed as

YT = TT × ST × CT × IT .

In the multiplicative model, only T is expressed in the original units, and S, C, and I are stated in terms of percentages.

Example. Values for bad debts at a commercial bank might be recorded as T = $ 10 million, S = 1.7, C = 0.91 and I = 0.87. Bad debts could then be computed as

Y = (10)(1.7)(0.91)(0.87) = $13.46 million.

Since seasonal fluctuations occur within time periods of less than one year, they would not be reflected in annual data. A Time series for annual data would be expressed as

YT = TT × CT × IT.

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