Secular trend
The secular trend is the smooth, long-term direction of a time series. It may be upward (rising sales), downward (falling death rate) or constant. It is measured to describe long-term growth and to forecast. There are four main methods.
1. Freehand (graphic) method
Plot the data and draw a smooth line by eye through the middle of the points. It is quick but subjective: different people draw different lines.
2. Method of semi-averages
Divide the data into two equal halves (if the number of years is odd, drop the middle year). Find the average of each half and plot it against the middle year of that half. Join the two points with a straight line. Example: values 20, 22, 24 | 30, 32, 34 give semi-averages 22 and 32. It is objective but assumes a straight-line trend and is affected by extreme values.
3. Method of moving averages
Replace each value by the average of a fixed number of consecutive values centred on it. For data 10, 14, 18, 16, 20, the 3-year moving averages are (10 + 14 + 18)/3 = 14, (14 + 18 + 16)/3 = 16 and (18 + 16 + 20)/3 = 18, placed against the 2nd, 3rd and 4th years. Moving averages smooth out short-term fluctuations. For an even period (such as 4 years), the averages fall between years and must be centred by averaging pairs. The trend values for the first and last few years are lost.
4. Method of least squares
Fit the straight line Y = a + bX that makes the sum of squared deviations of actual from trend values least. The normal equations are ΣY = na + bΣX and ΣXY = aΣX + bΣX². If the years are coded so that ΣX = 0 (for an odd number of years, take the middle year as 0), these become a = ΣY/n and b = ΣXY/ΣX².
Example: X = −2, −1, 0, 1, 2 and Y = 10, 12, 15, 17, 21. ΣY = 75, ΣXY = 27, ΣX² = 10, so a = 15, b = 2.7 and Y = 15 + 2.7X. The forecast for X = 3 is 23.1.
Key points
- Freehand: simplest but most subjective.
- Semi-averages: two averages, one straight line.
- Moving averages: smooth the series; even periods need centring.
- Least squares: best-fitting line, used for forecasting.
Common mistakes
- Forgetting to centre an even-period moving average.
- Not coding years so that ΣX = 0, which makes the arithmetic much harder.