Time Series Analysis

YouTube video

1 Introduction

Managers often wish to look at the trend of costs or sales over time as a basis for forecasting the future. It is unlikely in practice that past results will follow a smooth pattern for various reasons.

Of particular interest to us in this chapter are seasonal variations which we can attempt to identify.

2 Definitions

Time series: a set of observations taken at equal intervals of time e.g. monthly

Variations in observations:

Trend: the underlying pattern of a time series when the short term fluctuations have been smoothed out.
Cyclical Variations: the wave-like appearance of a number of time series graph when taken over a number of years. Generally this corresponds to the influence of booms and slumps in the industry.
Seasonal variations: the regular rise and fall over shorter periods of time. For example, umbrella sales are likely to be higher than average every winter and lower than average every summer.
Random (residual) variations: these are other, unpredictable variations.

3 Moving averages

In order to estimate the trend and the seasonal variations, we use the method of moving averages.

Set out below are the sales per quarter (in 000’s of units) of a company over the last 3 years.

	*Quarter*
	1	2	3	4
2000	80	87	82	90
2001	90	95	93	102
2002	105	112	103	116

Identify the trend and calculate the average seasonal variation.

		Actual sales	4 ¼ average	TREND (centered average)	Seasonal variation
2000	1	80

	2	87
			84.75
	3	82		86.00	–4.00
			87.25
	4	90		88.25	+1.75
			89.25
2001	1	90		90.63	–0.63
			92.00
	2	95		93.50	+1.50
			95.00
	3	93		96.88	–3.88
			98.75
	4	102		100.88	+1.12
			103.00
2002	1	105		104.25	+0.75
			105.50
	2	112		107.25	+4.75
			109.00
	3	103

	4	116

	1	2	3	4
2000			–4.00	+1.75
2001	–0.63	+1.50	–3.88	+1.12
2002	+0.75	+4.75
	+0.12	+6.25	–7.88	+2.87
average	+0.06	+3.13	–3.94	+1.44

4 The multiplicative model

In the previous example we calculated the seasonal variations in terms of units.

However, if the trend is increasing it would perhaps be more sensible to accept an increasing seasonal variation.

The multiplicative model deals with this by measuring the actual seasonal variation as a percentage of trend.

Using the data from example 1 together with the trend already calculated, calculate the average seasonal variation using the multiplicative model.

		Actual sales	TREND	Seasonal variation
2000	1	80

	2	87

	3	82	86.00	95.3%

	4	90	88.25	102.0%

2001	1	90	90.63	99.3%

	2	95	93.50	101.6%

	3	93	96.88	96.0%

	4	102	100.88	101.1%

2002	1	105	104.25	100.7%

	2	112	107.25	104.4%

	3	103

	4	116

	1	2	3	4
2000			95.3	102.0
2001	99.3	101.6	96.0	101.1
2002	100.7	104.4

average	100%	103%	95.7%	101.6%

Answer the questions one at a time. Your progress is saved so you can leave and come back.

Open chapter practice