Question

4. Time-series data contain both trend and seasonal variations. Use an example of quarterly data to explain how you would measure the trend and how you would measure the seasonal variation

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Answer 1

Time series data can contain numerous unique variations that increase the complexity in forecasting and the resulting model error in predicting Y. This complexity is due to the types of variation that exist in the time series data. Four common types of variation can be present in time series data: trend, seasonal variation, cyclical variation and random variation.

The main use of time series analysis is to predict trends. A trend is a consistently upward or downward movement in the data values over the time period. A trend is described as general movement of data over time. A trend can be positive or negative depending on the data in question. Once calculated, trends can be used as a means of analyzing historic data so that forecasts can be made for the future. A classic example of trend analysis would be in the retail industry, where sales of goods could be analyzed over time and the underlying trend could be calculated. In turn, the underlying trend can then be used to make forecasts for sales in the future. Once a forecast has been made then the business can be managed accordingly, for example inventory management, decisions about whether to continue selling certain products and whether the business needs any further investment to support growth.

Although the trend line is linear, the actual data fluctuates around this line. In some months the sales are above the trend line, in others the sales are below it. The trend line cannot go beyond November based on the data given because we would require the actual data for November, December and January (added together and divided by three). The gap between the actual data and the trend line is known as the seasonal variation.

A seasonal variation is where a pattern in the movements repeats itself at regular intervals.Seasonal variation can be described as the difference between the trend of data and the actual figures for the period in question. A seasonal variation can be a numerical value (additive) or a percentage (multiplicative). The term ‘seasonal’ is applied to a time period, not necessarily a traditional season (summer, autumn etc.). For example sales may be a lot higher for a store around Christmas, but lower in January.

Seasonal Variation = Actual Data or Forecast Data – Trend

With time-series analysis we need to calculate both the seasonal variation and the trend.

Seasonal variation

A Seasonal Variation (SV) is a regularly repeating pattern over a fixed number of months. If you look at our time-series you might notice that sales rise consistently from month 1 to month 3, and then similarly from month 4 to month 6. There appears to be a SV repeating over a three month period, where sales get higher each month for three months. We could expect this pattern to repeat in the future, so sales are likely to rise from month 7 to month 9.

Trend

A Trend (T) is a long-term movement in a consistent direction. Trends can be hard to spot because of the confusing impact of the SV. The easiest way to spot the Trend is to look at the months that hold the same position in each set of three period patterns. For example, month 1 is the first month in the pattern, as is month 4. The sales in month 4 are higher than in month 1.

Identifying the trend

To identify the T, we need to smooth out the impact of the SV. We do this by calculating what are known as ‘three-period moving averages’. This involves averaging the sales for three months at a time and then ’moving’ down to the next three months.

Month	Sales (the time-series)	Three-period moving average	You can see that compared to the original time-series, the three-period moving average figures show a much more consistent increase; in fact it is increasing by 20 each month. We would expect this trend to continue in the future. (Notice that you can’t work out figures for the first or last month).
1	70
2	80	300/3 = 100
3	150	360/3 = 120
4	130	420/3 = 140
5	140	480/3 = 160
6	210

Identifying the seasonal variation

Now that we know the trend we can identify the specific impact of the SV. We do this by comparing the time-series to the trend, to see whether it is above or below what we would expect. In month 2, the time series of 80 is 20 below the trend giving a SV of -20. In month 3, 150 is 30 above the trend giving a SV of +30.

Month	Sales (the time-series)	Three-period moving average (the trend)	Seasonal variation
1	70
2	80	100	-20
3	150	120	+30
4	130	140	-10
5	140	160	-20
6	210