Univariate time series
A univariate time series records the values of one variable over time, such as the monthly production of a single factory. (A multivariate series records several variables together.) Analysing it means studying how this one variable has behaved in the past in order to understand it and forecast its future.
Why analyse a time series?
- To understand past behaviour and the causes of change.
- To forecast future values for planning (production, staffing, budgets).
- To compare the actual performance with expected performance.
- To compare two or more series, for example sales in two regions.
Steps in analysis
1. Plot the data. A time-series graph (also called a historigram) has time on the x-axis and the variable on the y-axis, with points joined by straight lines. The graph shows trend and patterns at a glance. 2. Identify the components present: trend, seasonal, cyclical and irregular. 3. Choose a model. Use the additive model Y = T + S + C + I if components act independently and seasonal swings are of similar size each year; use the multiplicative model Y = T × S × C × I if swings grow with the level of the series. 4. Measure the trend (next topic) and remove it. In the additive model, Y − T = S + C + I; in the multiplicative model, Y/T = S × C × I. 5. Measure seasonal variation using seasonal indices. In a multiplicative model with quarterly data, the four seasonal indices average 100, so they add to 400. 6. Deseasonalise the data to see the underlying movement: in the multiplicative model, deseasonalised value = Y ÷ (seasonal index/100). If Y = 660 and the seasonal index is 120, the deseasonalised value is 660/1.2 = 550.
Key points
- Univariate = one variable observed over time.
- Time is always on the horizontal axis of a time-series graph.
- Additive: subtract components; multiplicative: divide by them.
- Quarterly seasonal indices add to 400; monthly indices add to 1200.
Exam tips
- When asked for S + C + I in an additive model, subtract T from Y.
- A seasonal index above 100 means that season is above the average level.