Qualitative and quantitative approaches
Forecasting methods fall into two families. Qualitative methods rely on judgment, and they are used when there is no history, such as launching a new product. Quantitative methods use data, and they split into time-series methods (which use past values of the variable itself) and causal methods (which relate it to other variables, such as regression).
| Approach | Examples | Best when |
|---|---|---|
| Qualitative | Expert judgment, Delphi method, market surveys, sales force estimates | No past data, or a major change makes the past a poor guide |
| Time series | Naive, moving average, exponential smoothing, trend, seasonal indices | There is a history of the series and patterns are stable |
| Causal | Regression using price, advertising, income or other drivers | You understand what drives the variable |
Time-series data usually combine four components: trend (long-term direction), seasonality (regular repeating patterns, such as holiday peaks), cycles (longer swings tied to the economy) and random noise. Good forecasting separates the first three from the last.
The worked dataset
Monthly demand for a product over eight months (hypothetical units):
| Month (t) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Demand | 100 | 108 | 104 | 112 | 118 | 115 | 124 | 128 |
The series rises with some noise. We will forecast it three ways, then compare accuracy.
Method 1: moving average
A moving average forecasts the next period as the average of the last n periods. It smooths noise. A larger n smooths more but reacts more slowly to change.
| Month | Actual | 3-month moving average forecast | Error (actual minus forecast) |
|---|---|---|---|
| 4 | 112 | (100 + 108 + 104) / 3 = 104.00 | 8.00 |
| 5 | 118 | (108 + 104 + 112) / 3 = 108.00 | 10.00 |
| 6 | 115 | (104 + 112 + 118) / 3 = 111.33 | 3.67 |
| 7 | 124 | (112 + 118 + 115) / 3 = 115.00 | 9.00 |
| 8 | 128 | (118 + 115 + 124) / 3 = 119.00 | 9.00 |
| 9 (forecast) | (115 + 124 + 128) / 3 = 122.33 |
Every error is positive, which means the forecast lags the rising series. This is the main weakness of moving averages when there is a trend. A weighted moving average gives more weight to recent periods, for example weights of 0.5, 0.3 and 0.2 on the latest three months.
Method 2: exponential smoothing
Exponential smoothing updates the previous forecast by a fraction of its error: new forecast = old forecast + alpha x (actual - old forecast). Alpha is between 0 and 1. A high alpha reacts quickly. A low alpha smooths strongly. Start the series by setting the first forecast equal to the first actual.
With alpha = 0.3:
| Month | Actual | Forecast | Error | Working for next forecast |
|---|---|---|---|---|
| 1 | 100 | |||
| 2 | 108 | 100.00 | 8.00 | 100 + 0.3 x 8 = 102.40 |
| 3 | 104 | 102.40 | 1.60 | 102.40 + 0.3 x 1.60 = 102.88 |
| 4 | 112 | 102.88 | 9.12 | 102.88 + 0.3 x 9.12 = 105.62 |
| 5 | 118 | 105.62 | 12.38 | 105.62 + 0.3 x 12.38 = 109.33 |
| 6 | 115 | 109.33 | 5.67 | 109.33 + 0.3 x 5.67 = 111.03 |
| 7 | 124 | 111.03 | 12.97 | 111.03 + 0.3 x 12.97 = 114.92 |
| 8 | 128 | 114.92 | 13.08 | 114.92 + 0.3 x 13.08 = 118.85 |
The forecast for month 9 is 118.85. Like the moving average, simple exponential smoothing lags a trending series, because it has no trend component. Holt's method adds a trend term, and Holt-Winters adds seasonality. In Excel, the Analysis ToolPak's Exponential Smoothing asks for a damping factor, which equals 1 minus alpha, so 0.7 for alpha of 0.3.
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Get an instant quoteMethod 3: linear trend
When a series moves steadily up or down, fit a straight line against time using regression, with t as the x variable. This is the same least-squares method used in our guide to regression analysis.
For the dataset: mean of t is 4.5 and mean of demand is 113.625. The sum of (t minus mean t) squared is 42, and the sum of products is 157.5. Slope = 157.5 / 42 = 3.75. Intercept = 113.625 - 3.75 x 4.5 = 96.75.
Trend forecast = 96.75 + 3.75 x t. For month 9 that is 96.75 + 33.75 = 130.5, and for month 10, 134.25. Each month, demand is expected to rise by about 3.75 units. In Excel use =FORECAST.LINEAR(9, demand_range, t_range) or =TREND().
Notice how different the month 9 forecasts are: 122.3 (moving average), 118.9 (smoothing) and 130.5 (trend). The first two lag a rising series. The trend model extends the pattern, which is appropriate only if the rise continues.
Measuring accuracy
Never present a forecast without saying how accurate the method has been. Compare methods on the same periods using error measures.
| Measure | Formula | What it tells you |
|---|---|---|
| Bias (mean error) | Average of errors (actual minus forecast) | Whether forecasts are systematically too low or too high |
| MAD or MAE | Average of absolute errors | Typical size of error in units |
| MSE | Average of squared errors | Penalizes large errors heavily |
| RMSE | Square root of MSE | Typical error in units, weighted toward large misses |
| MAPE | Average of absolute error divided by actual | Typical error as a percentage, easy to compare across items |
Accuracy of the 3-month moving average (months 4 to 8)
- Bias: (8 + 10 + 3.67 + 9 + 9) / 5 = 7.93, so the forecast is on average about 8 units too low.
- MAD: also 7.93, since every error is positive.
- MSE: (64 + 100 + 13.44 + 81 + 81) / 5 = 67.89, so RMSE is 8.24.
- MAPE: the average of 7.14, 8.47, 3.19, 7.26 and 7.03 percent is 6.62 percent.
A bias equal to the MAD shows the forecasts are always on one side, which suggests the method does not suit a trending series. A good method has bias near zero and a small MAD or MAPE.
Seasonal indices
When demand follows a yearly pattern, a seasonal index tells you how much above or below average each period usually is. Suppose quarterly sales over two years are:
| Q1 | Q2 | Q3 | Q4 | Year total | |
|---|---|---|---|---|---|
| Year 1 | 80 | 100 | 120 | 100 | 400 |
| Year 2 | 88 | 110 | 132 | 110 | 440 |
| Average by quarter | 84 | 105 | 126 | 105 | |
| Seasonal index (average / 105) | 0.80 | 1.00 | 1.20 | 1.00 |
The overall average per quarter is (400 + 440) / 8 = 105. Q3 runs 20 percent above average, and Q1 runs 20 percent below. To forecast next year, estimate the annual total and distribute it with the indices. If next year is expected to reach 484 (10 percent growth), the average quarter is 121, so Q1 = 121 x 0.80 = 96.8, Q2 = 121.0, Q3 = 145.2 and Q4 = 121.0, which add to 484.
Choosing and defending a method
| Pattern in the data | Suitable method | Why |
|---|---|---|
| Stable level, random noise | Moving average or simple exponential smoothing | Smooths noise |
| Steady trend | Linear trend or Holt's method | Captures direction of change |
| Regular seasonal pattern | Seasonal indices or Holt-Winters | Captures repeating peaks and troughs |
| A known driver such as price or advertising | Regression | Uses causes rather than only history |
| No history | Judgment, analogies, surveys | There is nothing to extrapolate |
- Plot the data first The chart shows trend, seasonality and outliers before you choose a method.
- Compare at least two methods And report the accuracy of each on the same periods.
- Hold out some data Test on periods you did not use to build the forecast.
- State assumptions For example that the trend continues, or that no promotion changes demand.
- Give a range, not only a point Forecasts are uncertain, so mention scenarios or intervals.
- Update regularly Re-forecast as new data arrive and track the errors.
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