A four-year moving average has to be centred because:

an even period places each average between two time points

moving averages must add to 400

four is larger than the cycle length

Time Series Analysis

A time series is a variable recorded at successive points of time — monthly prices, quarterly output, annual population. Its movement is the combined effect of a long-run trend, a regular seasonal pattern, slower cycles and random shocks. This chapter covers the four components and the additive and multiplicative models, the methods of determining trend (free-hand, semi-averages, moving averages and least squares), and the methods of measuring seasonal variation.

The four components

Components of a time series
Component	What it is	Example
Secular trend (T)	the smooth long-run direction	population growth over decades
Seasonal variation (S)	a pattern repeating within a year	sales rising every festive season
Cyclical variation (C)	swings lasting more than a year, of irregular length	business cycle: boom, recession, depression, recovery
Irregular variation (I)	unpredictable, random shocks	a flood, a strike, a pandemic

Two models combine them. The additive model Y = T + S + C + I treats each component as an amount added on, in the units of Y. The multiplicative model Y = T × S × C × I treats S, C and I as ratios (indices) applied to the trend, and is the usual choice because seasonal swings tend to grow with the level of the series.

⚠️ Exam trap

Seasonal means a cycle within one year (months, quarters), whatever its cause — weather, festivals or fiscal year-ends. A swing of five to seven years is cyclical, not seasonal.

Measuring trend

Free-hand curve: draw a smooth line through the plotted series by eye — quick, but subjective.
Semi-averages: split the series into two halves, average each, plot the averages at the middle of each half and join them; the slope is the difference of the averages divided by the gap between their mid-points.
Moving averages: replace each value by the average of the m values centred on it. An odd period sits on a time point; an even period falls between two points and must be centred by averaging consecutive moving averages in pairs. A moving average of the right period removes a regular cycle of that length.
Least squares: fit y = a + bx by minimising Σ(y − a − bx)². With x coded so that Σx = 0, simply a = Σy/n and b = Σxy/Σx².

Worked least-squares trend, five years coded −2 to 2
Year code x	y	xy	x²
-2	10	-20	4
-1	12	-12	1
0	15	0	0
1	16	16	1
2	17	34	4
Total	70	18	10

a = 70/5 = 14 and b = 18/10 = 1.8, so the trend is y = 14 + 1.8x (origin at the middle year, x in years). The estimate two years after the last year (x = 4) is 14 + 7.2 = 21.2. For the series 10, 12, 15, 16, 17, 20 the three-year moving averages are 12.33, 14.33, 16, 17.67.

Measuring seasonal variation

Four methods
Method	How	Note
Simple averages	average each month or quarter over the years; express as % of the overall average	assumes no trend
Ratio to trend	divide each value by its trend value, then average the ratios season by season	removes trend; cycles remain
Ratio to moving average	divide each value by a centred 12-month or 4-quarter moving average	most widely used; removes trend and cycle
Link relatives	each season as % of the previous season, chained and corrected for trend	due to Karl Pearson

Seasonal indices are finally adjusted so that they average 100: quarterly indices must add to 400 and monthly indices to 1,200. If four raw quarterly indices add to 404, multiply each by 400/404. Dividing the original data by the seasonal index (× 100) gives deseasonalised data.

Exam traps and a quick revision

T, S, C, I; multiplicative Y = T × S × C × I is the usual model.
Seasonal = within a year; cyclical = longer than a year.
Least squares with Σx = 0: a = Σy/n, b = Σxy/Σx².
Even-period moving averages must be centred.
Quarterly seasonal indices add to 400, monthly to 1,200.

Key takeaways

A time series combines trend, seasonal, cyclical and irregular components.
The multiplicative model Y = T × S × C × I is standard; the additive model is Y = T + S + C + I.
Trend by free-hand, semi-averages, moving averages or least squares.
Ratio to moving average is the standard method for seasonal indices.
Seasonal indices are adjusted to average 100: sum 400 for quarters, 1,200 for months.

Practice questions (10)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

Fluctuations in a time series that repeat regularly within a year are called:
1. secular trend
2. irregular variations
3. seasonal variations
4. cyclical variations
Show answer
Answer: C — seasonal variations
A pattern with a period of one year or less is seasonal. Swings lasting several years are cyclical.
The prosperity–recession–depression–recovery pattern of business activity is an example of:
1. irregular variation
2. seasonal variation
3. cyclical variation
4. secular trend
Show answer
Answer: C — cyclical variation
The business cycle has phases spread over several years and of irregular length, which is the definition of cyclical variation.
The multiplicative model of a time series is:
1. Y = (T + S) × (C + I)
2. Y = T × S × C × I
3. Y = T + S + C + I
4. Y = T × S + C × I
Show answer
Answer: B — Y = T × S × C × I
In the multiplicative model the trend is in the units of Y and S, C, I are ratios applied to it. The additive model sums all four.
For five years coded x = −2, −1, 0, 1, 2, Σy = 70, Σxy = 18 and Σx² = 10. The least-squares trend line is:
1. y = 14 + 18x
2. y = 14 + 1.8x
3. y = 70 + 1.8x
4. y = 1.8 + 14x
Show answer
Answer: B — y = 14 + 1.8x
With Σx = 0, a = Σy/n = 14 and b = Σxy/Σx² = 1.8. Using Σy itself as a gives the wrong 70.
Using the trend y = 14 + 1.8x (x = 0 at the middle year of five, x in years), the trend value two years after the last of the five years is:
1. 19.4
2. 23
3. 21.2
4. 17.6
Show answer
Answer: C — 21.2
The last year is x = 2, so two years later x = 4: y = 14 + 1.8 × 4 = 21.2. Taking x = 2 gives 17.6, the trend of the last year itself.
The first three-year moving average of the series 4, 6, 8, 10, 12 is:
1. 6
2. 18
3. 5
4. 8
Show answer
Answer: A — 6
(4 + 6 + 8)/3 = 6, placed against the second year, the middle of the three. The next ones are 8 and 10; 18 is the moving total before dividing by 3.
A four-year moving average has to be centred because:
1. an even period places each average between two time points
2. it removes the trend
3. moving averages must add to 400
4. four is larger than the cycle length
Show answer
Answer: A — an even period places each average between two time points
The average of years 1 to 4 belongs at the midpoint between years 2 and 3. Averaging consecutive pairs of such averages brings them back onto actual years.
Four quarterly seasonal indices computed from data add up to 404. Before use, each should be multiplied by:
1. 100/404
2. 404/400
3. 400/404
4. 4/404
Show answer
Answer: C — 400/404
Quarterly indices must average 100, so their total must be 400. Multiplying each by 400/404 rescales the total to exactly 400.
The method of link relatives for measuring seasonal variation was suggested by:
1. Irving Fisher
2. Karl Pearson
3. A. L. Bowley
4. R. A. Fisher
Show answer
Answer: B — Karl Pearson
Karl Pearson's link-relative method expresses each season as a percentage of the one before and chains them. Irving Fisher is known for the ideal index number.
A six-year series is split into halves of three years each. The first half averages 20 and the second 26. By the method of semi-averages, the annual increase in trend is:
1. 6
2. 1
3. 2
4. 3
Show answer
Answer: C — 2
The two averages sit at the middles of the halves, which are 3 years apart. Annual increase = (26 − 20)/3 = 2. 6 is the total rise, not the yearly one.