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Linear trend regression and forecast error

Short answer

Least squares trend: y = 37.6 + 4.114x (thousand units). Year7 forecast 66.4, year8 forecast 70.5. MAD = 0.72, MSE = 0.63, MAPE = 1.4%. Causal methods use regression on related variables; time-series uses only historic values of the variable itself.

Annual demand (in thousand units) for a product over six years is: Year 1: 42, Year 2: 45, Year 3: 51, Year 4: 53, Year 5: 59, Year 6: 62. (a) Fit a straight-line trend y=a+bxy = a + bx by the least squares method. (b) Forecast the demand for years 7 and 8. (c) Compute MAD, MSE and MAPE of the fitted line, and comment on the fit. Also name the causal quantitative techniques and the difference from time-series methods.

(a) Least squares line

Take xx = year number, n=6n = 6.

xxyyxyxyx2x^2
142421
245904
3511539
45321216
55929525
66237236
21312116491
b=n∑xy−∑x∑yn∑x2−(∑x)2=6(1164)−21(312)6(91)−441=6984−6552105=4.114b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} = \frac{6(1164) - 21(312)}{6(91) - 441} = \frac{6984 - 6552}{105} = 4.114 a=yˉ−bxˉ=52−4.114(3.5)=37.6a = \bar{y} - b\bar{x} = 52 - 4.114(3.5) = 37.6

Trend line: y=37.6+4.114xy = 37.6 + 4.114x (demand rises by about 4.1 thousand units per year).

(b) Forecasts

  • Year 7: 37.6+4.114(7)=66.437.6 + 4.114(7) = 66.4 thousand units
  • Year 8: 37.6+4.114(8)=70.537.6 + 4.114(8) = 70.5 thousand units

(c) Forecast error of the fit

xxActualFittedError
14241.710.29
24545.83-0.83
35149.941.06
45354.06-1.06
55958.170.83
66262.29-0.29
  • MAD =4.34/6=0.72= 4.34/6 = 0.72 thousand units
  • MSE =3.77/6=0.63= 3.77/6 = 0.63
  • MAPE =1.4%= 1.4\%

The errors are small and change sign, so the straight line fits the data well (correlation coefficient r≈0.994r \approx 0.994). Forecasting two years ahead is reasonable, but the line should be revised as new data arrive.

Causal methods

Causal (associative) techniques forecast a variable from other variables that influence it, for example demand of cement from construction activity, using simple or multiple regression and correlation, or econometric models. Time-series methods (moving average, exponential smoothing, trend) use only the past values of the variable itself.

Answer: y=37.6+4.114xy = 37.6 + 4.114x; forecasts 66.4 (year 7) and 70.5 (year 8) thousand units; MAD = 0.72, MSE = 0.63, MAPE = 1.4%.

✓ Answer checked by Claude Opus 5.5 (high effort)

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