Lines of Best Fit and Seasonal Variations

A moving average column shows the trend at each date. To forecast, you need the trend as a straight line that can be extended into the future. Then you need the seasonal pattern, so you can put it back. This topic completes the forecasting method started in the last topic.

It shows two ways of getting a trend line (from the centred moving averages and by least squares regression), how they compare, how to measure seasonal variation under the additive and the proportional (multiplicative) models, and how to combine trend and season to produce a forecast.

It is a favourite exam topic, often worth 10 to 20 marks, and the arithmetic is long. The key is a neat layout and careful checking, so every step is shown here on one data set with the totals checked.

The data used in this topic

Quarterly sales (₦000) for three years, with the quarters numbered 1 to 12:

Four-quarter totals (Q1-Q4, Q2-Q5 and so on): 195, 200, 205, 212, 217, 223, 229, 233, 238. Centred moving averages are (sum of two adjacent totals) ÷ 8:

Line of best fit: high-low method on the moving averages

The trend is a general movement. To forecast, it must be a straight line. One way is to use the centred moving averages as points. The first value is the starting point of the equation, and the slope comes from the first and last values:

In the example, the first CMA is 49.375 at Q3 and the last is 58.875 at Q10, which is 7 periods later. Slope = (58.875 − 49.375) ÷ 7 = 9.5 ÷ 7 = 1.357. The trend line is:

y = 49.375 + 1.357x, where x = 0 at Q3.

Take care with the starting point. The equation starts at Q3 of year 1, so x must be counted from there. Q4 of year 1 is x = 1, Q4 of year 2 is x = 5, and Q4 of year 4 (quarter 16) is 13 quarters after Q3, so x = 13. The trend for Q4 of year 4 is 49.375 + 1.357 × 13 = 49.375 + 17.64 = 67.02.

Line of best fit: least squares regression

Linear regression, used earlier for cost estimation, can also fit a line to a time series. The periods are numbered sequentially, starting from 1 (or 0), to give the values of x. There is no loss of data, because all 12 values are used.

b = (12 × 4,440 − 78 × 650) ÷ (12 × 650 − 78²) = (53,280 − 50,700) ÷ (7,800 − 6,084) = 2,580 ÷ 1,716 = 1.5035.

a = 650 ÷ 12 − 1.5035 × (78 ÷ 12) = 54.167 − 9.773 = 44.394.

The line is y = 44.394 + 1.5035x. Here a is the value at x = 0, one quarter before the first observation. The forecast for quarter 16 is 44.394 + 1.5035 × 16 = 44.394 + 24.056 = 68.45.

Comparing the two lines

The two lines give slightly different forecasts. When comparing, make sure each x is measured from the right starting point. Either line can then be used to find the seasonal variations. The next parts use the centred moving averages as the trend.

Seasonal variation

A trend line alone cannot forecast a quarter, because the actual figure is always above or below the trend. A seasonal variation is measured from past data as the difference between the actual value for a period and the corresponding trend value. It is then used to adjust a forecast of the trend. Two models estimate it.

The additive model

Seasonal variations above the trend are positive and those below it are negative. The additive model assumes they add to zero over a full cycle.

1. For each period, calculate actual − centred moving average.
2. Group the results by season (Q1, Q2, Q3, Q4).
3. Average the variations for each season.
4. If the four averages do not add to zero, spread the difference evenly over the seasons (add or subtract the same amount to each).
5. The adjusted figures are the seasonal variations.

The averages add to −8.9375 + 3.75 + 11.3125 − 6.0625 = +0.0625. They should add to zero, so subtract 0.0625 ÷ 4 = 0.015625 from each:

Forecast for Q4 of year 4 (quarter 16), additive: trend 67.02 + seasonal variation (−6.08) = 60.94, that is ₦60,940.

Forecast for Q1 of year 4 (quarter 13, x = 10): trend = 49.375 + 1.357 × 10 = 62.95. Adjusted: 62.95 − 8.95 = 54.00.

The proportional (multiplicative) model

Here the actual value in each season is expressed as a proportion of the trend. The proportions for the four quarters should add to 4 (they average 1).

1. For each period, calculate actual ÷ centred moving average.
2. Group by season and average.
3. If the total is not 4, spread the difference evenly.

The average proportions total 4.0068. The excess is 0.0068, so subtract 0.0068 ÷ 4 = 0.0017 from each: Q1 0.8350; Q2 1.0648; Q3 1.2150; Q4 0.8852 (total 4.0000).

Forecast for Q4 of year 4, proportional: trend 67.02 × 0.8852 = 59.33, that is ₦59,330. This differs a little from the additive forecast of 60.94, because the two models treat the size of the seasonal effect differently.

An odd cycle: daily variations

The same method works for daily data. Using the five-day shop data from the previous topic (days 1 to 15), the moving averages give the trend y = 70 + 3x. The variations from the moving average (actual − moving average) for each weekday are:

The total is zero, so no adjustment is needed. To forecast day 21, a Monday: trend = 70 + 3 × 21 = 133. Additive forecast = 133 − 8 = 125 units. For day 20, a Friday: trend 130 + 0 = 130.

Under the proportional model, the Monday proportions are 80 ÷ 88 = 0.9091 and 95 ÷ 103 = 0.9223, average 0.9157. Tuesday 1.0000, Wednesday (85 ÷ 79, 100 ÷ 94, 115 ÷ 109) averages 1.0649, Thursday (84 ÷ 82, 99 ÷ 97) averages 1.0225, Friday 1.0000. The total is 5.0031, which should be 5 for five days of the week, so subtract 0.0031 ÷ 5 = 0.0006 from each. Monday becomes 0.9151, and the forecast for day 21 is 133 × 0.9151 = 121.7 units.

Forecasting with the regression trend

The regression line from earlier can also supply the trend. In the quarterly example, for quarter 16 the regression trend is 68.45. With the additive variation of −6.08 for Q4 the forecast is 68.45 − 6.08 = 62.37 (₦62,370). Note that seasonal variations were estimated from the centred moving averages.

In an exam, if the question says to use the regression line for the trend, use its value for the forecast but keep the seasonal variations from the moving average analysis unless told otherwise. The difference between 60.94 and 62.37 shows the effect of the choice of trend line.

The ideas in more depth

Why the adjustment is needed

The seasonal pattern is assumed to be only a re-shuffling of sales within the year. Over a full year, the quarters above trend must exactly cancel those below. Because averaging a small number of observations leaves some rounding and noise, the raw averages seldom sum to zero, so the small difference is spread equally. It does not change the shape of the pattern.

Interpreting the figures

An additive seasonal variation of +11.30 for Q3 means that Q3 sales are on average ₦11,300 above the trend. A proportional figure of 1.215 means Q3 sales are 21.5% above trend. Q1 is the weak quarter: ₦8,950 below trend (or 16.5% below).

Choosing the model

Compare the variations in different years. In this data the Q3 additive variations are +10.6 (year 1) and +12.0 (year 2), growing slightly as the trend rises. The proportions are 1.215 and 1.218, almost identical. The proportional model therefore fits a little better. When the seasonal variations in the data are fairly steady in naira, use the additive model. When they grow with the trend, use the proportional model.

A forecasting checklist

1. Identify the cycle length and calculate (centred) moving averages.
2. Fit a trend line and write its equation, with x measured from the right starting point.
3. Calculate the seasonal variations for each model asked for and adjust them.
4. Calculate the trend value for the future period. Add the variation (additive) or multiply by the proportion (proportional).
5. State the assumptions: the trend continues, the seasonal pattern repeats, and random and cyclical effects are ignored.

How reliable is the forecast?

Forecasts for periods far beyond the data (extrapolation) are less reliable. A forecast one quarter ahead is more trustworthy than one four years ahead. The forecast depends on the assumption that growth continues at the same rate, and on the small amount of data used to estimate the seasonal variation (only two or three observations for each quarter here).

Worked example: additive and proportional forecasts

1. Additive variations.
   Q3: 70 − 54.75 = +15.25
   Q4: 50 − 56.75 = −6.75
   Q5 (Q1): 48 − 58.75 = −10.75
   Q6 (Q2): 63 − 60.75 = +2.25
   Q7 (Q3): 78 − 62.75 = +15.25
   Q8 (Q4): 58 − 64.75 = −6.75
   Q9 (Q1): 56 − 66.75 = −10.75
   Q10 (Q2): 71 − 68.75 = +2.25.
2. Averages. Q1 −10.75; Q2 +2.25; Q3 +15.25; Q4 −6.75.
   Total = −10.75 + 2.25 + 15.25 − 6.75 = 0.
   No adjustment is needed.
3. Trend for quarter 16.
   x = 13: 54.75 + 2 × 13 = 80.75.
4. Additive forecast.
   80.75 − 6.75 = 74.00 (₦74,000).
5. Proportional variations.
   Q4: 50 ÷ 56.75 = 0.8811 and 58 ÷ 64.75 = 0.8958, average 0.8884.
   Similarly Q1: 0.8279; Q2: 1.0349; Q3: 1.2608.
   Total = 4.0121.
   Subtract 0.0121 ÷ 4 = 0.0030 from each.
   Q4 = 0.8854.
6. Proportional forecast.
   80.75 × 0.8854 = 71.50 (₦71,500).
7. Comment. The models differ by about ₦2,500. The additive model suggests that the Q4 shortfall stays at ₦6,750 as sales grow. The proportional model scales the shortfall up with the trend (11.5% of 80.75 is 9.3).

Seasonally adjusted data

Background: the syllabus uses seasonal variations only for forecasting, but the same figures are used to remove the seasonal pattern from past data. A seasonally adjusted (or deseasonalised) figure shows what the value would have been without the seasonal effect, so that the underlying movement can be seen. Under the additive model, subtract the seasonal variation; under the proportional model, divide by the seasonal proportion.

The raw data jumps up and down (Q3 is always high and Q1 always low), but the adjusted column rises steadily, from about ₦47,000 to ₦61,000. It confirms that the underlying trend is upward at roughly ₦1,300 a quarter, and that most of the movement between quarters is seasonal.

Management can use the adjusted figures to judge performance: a Q1 sale of 49 looks weak against Q3’s 71, but adjusted it is 57.95, a strong quarter.

Forecasting a whole year

Budgets need forecasts for every period of the next year. Using the centred moving average trend line (y = 49.375 + 1.357x, with x = 0 at Q3 of year 1), the four quarters of year 4 are quarters 13 to 16, so x = 10, 11, 12 and 13.

The additive seasonal variations add to zero, so the additive forecast for the year equals the sum of the trend values (₦259,930). The proportional total is slightly different (₦260,140) because the proportions average 1 but are applied to different trend values. Year 3 sales were ₦238,000, so the forecast shows growth of about 9.2%. A budget accountant would compare this with market knowledge before adopting it for the sales budget.

An examination layout for a seasonal forecast

A clear layout earns marks and avoids slips. Use this order.

1. Number the periods 1 to n beside the data.
2. Moving totals and averages. Show the total of each cycle, the average, and (for an even cycle) the centred average, each against its period.
3. Trend line. State the first and last centred values, the number of periods between them, the slope and the equation, saying where x = 0.
4. Variations. A table of actual − trend (or actual ÷ trend) by year and season, with averages.
5. Adjustment. Show the total, the error and the adjustment per season, then the final variations.
6. Forecast. Work out the x value for the forecast period, the trend value, then the adjusted forecast.
7. Comment. State the assumptions (trend continues, season repeats) and one limitation.

Using the forecast in a budget

The forecast quarterly sales feed the sales budget. If the average selling price is ₦500 per unit and the forecast figures are in thousands of units, quarterly sales revenue can be calculated, then production, purchases and the cash budget follow as in the budgeting topics.

Management may then adjust the forecast for known factors that the time series cannot see, such as a planned price rise, a new competitor or a promotion. The time series forecast is the starting point, and management judgement completes it.

How good is a seasonal forecast?

• Data. Three years give only two or three observations for each season. A longer record gives steadier seasonal averages.
• Stability. The method assumes that the seasonal pattern and the rate of growth stay the same. A new product, a recession or a change in customer habits breaks this.
• Horizon. Forecasts one or two periods ahead are more reliable than those several years ahead.
• Random variation. The method ignores it. Large irregular movements in the data suggest that the forecast has wider limits than the single figure implies.
• Model choice. If the actual variation of a season grows with the trend, the proportional model forecasts better than the additive one.

A second complete example with clean numbers

Quarterly sales (₦000): Year 1: 20, 28, 34, 24. Year 2: 24, 32, 38, 28. Year 3: 28, 36, 42, 32. Four-quarter totals: 106, 110, 114, 118, 122, 126, 130, 134, 138. Centred moving averages for Q3 to Q10 are (sum of two adjacent totals) ÷ 8: 27, 28, 29, 30, 31, 32, 33 and 34.

• Trend:
  first centred value 27 (Q3, x = 0), last 34 (Q10, x = 7).
  Slope = (34 − 27) ÷ 7 = 1.
  Trend: y = 27 + x.
• Additive variations:
  Q3: 34 − 27 = +7
  Q4: 24 − 28 = −4
  Q5 (Q1): 24 − 29 = −5
  Q6 (Q2): 32 − 30 = +2
  Q7 (Q3): 38 − 31 = +7
  Q8 (Q4): 28 − 32 = −4
  Q9 (Q1): 28 − 33 = −5
  Q10 (Q2): 36 − 34 = +2.
  Averages: Q1 −5, Q2 +2, Q3 +7, Q4 −4.
  Total = 0, so no adjustment.
• Forecasts, year 4:
  Q1 is quarter 13 (x = 10): trend 37, forecast 37 − 5 = 32.
  Q3 is quarter 15 (x = 12): trend 39, forecast 39 + 7 = 46.
• Proportional:
  Q3: 34 ÷ 27 = 1.2593 and 38 ÷ 31 = 1.2258, average 1.2425.
  Q4: 24 ÷ 28 = 0.8571 and 28 ÷ 32 = 0.8750, average 0.8661.
  Q1: 24 ÷ 29 = 0.8276 and 28 ÷ 33 = 0.8485, average 0.8380.
  Q2: 32 ÷ 30 = 1.0667 and 36 ÷ 34 = 1.0588, average 1.0627.
  Total = 4.0093, so subtract 0.0023 from each: Q1 0.8357, Q2 1.0604, Q3 1.2402, Q4 0.8638.
  Forecast Q1 year 4 = 37 × 0.8357 = 30.92
  Q3 = 39 × 1.2402 = 48.37.

The additive and proportional forecasts differ (32 against 30.92 for Q1; 46 against 48.37 for Q3) because the sales swings grow a little with the trend in this data, so the proportional model fits slightly better.

Formula summary for this topic

Common mistakes in the exam

• Measuring x from the wrong starting point in the CMA line equation. x = 0 is the first CMA period.
• Using actual minus trend for the proportional model, or actual divided by trend for the additive model.
• Forgetting to adjust the seasonal variations to total zero (additive) or the number of seasons (proportional).
• Adjusting by dividing the error by the wrong number. Divide by the number of seasons.
• Adding the seasonal variation when the model requires multiplying.
• Using the trend value for the wrong quarter, for example the quarter number instead of the offset from the starting point.
• Rounding too early. Keep three or four decimals until the end.
• Comparing the regression line with the CMA line without aligning the x values.

Quick revision

• CMA trend line: slope = (last − first CMA) ÷ periods between; start at first CMA with x = 0.
• Regression trend: number periods from 1; b = (n∑xy − ∑x∑y) ÷ (n∑x² − (∑x)²); a = ∑y/n − b∑x/n.
• Additive: variation = actual − trend; averages must total 0; forecast = trend + variation.
• Proportional: proportion = actual ÷ trend; averages must total the number of seasons; forecast = trend × proportion.
• Spread any adjustment equally across seasons.

Practice questions

These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer.

Section A style: multiple choice

Section B style: written questions

A further written question

Model answers

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