CMA Final · Strategic Cost Management

Business Forecasting Models - Time Series and Regression Analysis: formula sheet

Full chapter guide

Key formulas

Forecast error: Error = Actual value − Forecast value; Positive error means you under-forecast. Use it to judge accuracy after the period ends.
Mean Absolute Deviation (MAD): MAD = Σ|Actual − Forecast| ÷ n; Measures average size of error ignoring sign. A lower MAD means a better method on the same data.
Simple moving average forecast: Forecast for next period = Sum of last n periods ÷ n; A quantitative time series method. Larger n gives a smoother but slower-reacting forecast.
Classification rule: Forecasting methods = Qualitative (judgement) + Quantitative (time series + causal); Use this structure to organise any theory answer.
Additive model: Y = T + S + C + I; Components are in the units of Y. Seasonal indices (variations) sum to zero over a full cycle.
Multiplicative model: Y = T × S × C × I; S, C and I are ratios. Seasonal indices average 100% (or sum to 400 for quarters, 1200 for months).
Removing seasonality (additive): Deseasonalised value = Y − S; Subtract the seasonal variation, which can be negative.
Removing seasonality (multiplicative): Deseasonalised value = Y ÷ S (S as a ratio) = Y ÷ Seasonal index × 100; Divide by the index and multiply by 100 if the index is in percentage.
Isolating cyclical and irregular (multiplicative): Y ÷ (T × S) = C × I; Divide actual by trend and seasonal to leave cyclical and irregular together.
Isolating cyclical and irregular (additive): Y − T − S = C + I; Subtract trend and seasonal from actual.
Forecast using trend and seasonal: Additive: Forecast = T + S. Multiplicative: Forecast = T × S; Assumes cyclical and irregular effects are ignored or taken as zero (additive) or 1 (multiplicative).
Straight-line trend: Y = a + bX; a is the trend value at X = 0. b is the change in Y for a one-unit change in X.
Normal equations (least squares): ΣY = na + bΣX ; ΣXY = aΣX + bΣX²; Use these when X is not coded from the middle. Solve the two equations for a and b.
Least squares with deviations from the middle (ΣX = 0): a = ΣY ÷ n ; b = ΣXY ÷ ΣX²; Take X = 0 at the middle period. For odd n, X runs ..., -2, -1, 0, 1, 2, .... This is the fastest form.
Coding for an even number of years: X = 2 × (year − midpoint of the two middle years), where the midpoint is the half-year point between the two middle years; Find the midpoint by averaging the two middle years. For 2020 to 2025, the middle years are 2022 and 2023, so the midpoint is 2022.5 and X = 2 × (year − 2022.5). X values become ..., -3, -1, 1, 3, ... in half-year units. Then b is the change per half year, and the yearly change is 2b.
Semi-average slope: b = (Average of second half − Average of first half) ÷ (Years between the two mid-points); Each average is placed at the middle of its own half. If n is odd, leave out the middle value.
Moving average of odd period m: MA = (sum of m consecutive values) ÷ m, placed against the middle period; For a 3-year average, the first value is placed against year 2.
Moving average of even period: Take the m-period totals, then average each pair of consecutive totals to centre it on a year; A 4-year moving average falls between years, so you centre it with a second 2-term average.
Simple average seasonal index: Seasonal index = (Average of the period across years ÷ Grand average of all periods) × 100; Use when there is no clear trend. The grand average is the average of the period averages.
Ratio-to-trend: Seasonal ratio = (Actual Y ÷ Trend value T) × 100; Fit the trend first (for example by least squares). Then average the ratios for each period across years.
Centred moving average (quarterly data): Centred MA = (Sum of two consecutive 4-quarter totals) ÷ 8; The first value falls at the third quarter of the first year. You lose two periods at each end. For monthly data use 12-month totals and divide the sum of two consecutive totals by 24.
Ratio-to-moving-average: Seasonal ratio = (Actual Y ÷ Centred MA) × 100; Average the ratios for each quarter (or month) across years. A median is also accepted if the question asks for it.
Adjustment to total 400 (or 1200): Adjusted index = Average ratio × (400 ÷ Sum of average ratios); Use 1200 for monthly data. Skip this step if the averages already sum to 400 (or 1200).
Deseasonalised value (multiplicative model): Deseasonalised value = (Actual Y ÷ Seasonal index) × 100; Model Y = T × S × C × I. An index above 100 lowers the actual value; an index below 100 raises it.
Seasonal forecast: Forecast = Trend forecast × (Seasonal index ÷ 100); Project the trend first, then apply the index of the target period.
Additive model: Deseasonalised value = Actual Y − Seasonal variation S; Used when the question gives seasonal variations in absolute units. The variations should sum to zero over the year.
Regression line of Y on X: Y − Ȳ = b_yx (X − X̄); Use it to estimate Y for a given X.
Regression line of X on Y: X − X̄ = b_xy (Y − Ȳ); Use it to estimate X for a given Y.
Regression coefficient using r and standard deviations: b_yx = r × σy ÷ σx ; b_xy = r × σx ÷ σy; The standard deviation of the dependent variable is in the numerator.
Regression coefficient from raw data: b_yx = [nΣXY − ΣX ΣY] ÷ [nΣX² − (ΣX)²] ; b_xy = [nΣXY − ΣX ΣY] ÷ [nΣY² − (ΣY)²]; Use when actual values are given. Take care with the denominator for each line.
Regression coefficient from deviations: b_yx = Σxy ÷ Σx² ; b_xy = Σxy ÷ Σy² (x = X − X̄, y = Y − Ȳ); Useful when deviations from actual means are small or already given.
Assumed mean method: b_yx = [nΣdxdy − Σdx Σdy] ÷ [nΣdx² − (Σdx)²]; dx = X − A, dy = Y − B. The coefficient is unchanged by shifting the origin, not by changing scale.
Correlation from the coefficients: r² = b_yx × b_xy , so r = ±√(b_yx × b_xy); r takes the common sign of both coefficients.
Properties of regression coefficients: Arithmetic mean of b_yx and b_xy ≥ r ; b_yx and b_xy have the same sign ; b_yx × b_xy ≤ 1; The AM statement holds when r is positive, comparing with its absolute value in general. Both coefficients cannot be greater than 1 in absolute terms.
Slope and intercept form: Y = a + bX, where b = b_yx and a = Ȳ − b X̄; The intercept a is the value of Y when X = 0.
Karl Pearson correlation coefficient: r = [nΣXY − ΣXΣY] ÷ √{[nΣX² − (ΣX)²] × [nΣY² − (ΣY)²]}; Use when you have raw data. The value always lies between -1 and +1.
Covariance form of r: r = Cov(X, Y) ÷ (σx × σy), where Cov(X, Y) = ΣXY ÷ n − X̄ × Ȳ; Use when means and standard deviations are given or easy to find. Use the same divisor (n) throughout.
Coefficient of determination: r² = Explained variation ÷ Total variation = 1 − (Unexplained variation ÷ Total variation); Express as a percentage when interpreting. It lies between 0 and 1.
Link between r and regression coefficients: r = ±√(byx × bxy); byx = r × σy ÷ σx; bxy = r × σx ÷ σy; Take the sign of the regression coefficients. Both have the same sign as r. Their product, r², cannot exceed 1.
Standard error of estimate (definition): Se = √[Σ(Y − Ŷ)² ÷ (n − 2)]; Divisor n − 2 is used for a simple linear regression fitted from sample data. Some questions divide by n; follow the question's instruction.
Standard error of estimate (computational): Se = √[(ΣY² − aΣY − bΣXY) ÷ (n − 2)]; Use for Y = a + bX when the sums are given and residuals are not worked out.
Se from r: Unexplained variation = Σ(Y − Ȳ)² × (1 − r²); Se = √(Unexplained variation ÷ (n − 2)); Quick route when r and the total variation in Y are given. If using σy with divisor n, then Se = σy × √(1 − r²).
Approximate forecast range: Ŷ ± 1 Se ≈ 68% range; Ŷ ± 2 Se ≈ 95% range; A rough rule that assumes residuals are roughly normal and the sample is reasonably large. Use it only when the question asks for it.
Multiple regression equation (two variables): Y = a + b1X1 + b2X2; Y is the dependent variable. b1 and b2 are the effects of X1 and X2, each with the other held constant.
Normal equation 1: ΣY = na + b1ΣX1 + b2ΣX2; n is the number of observations.
Normal equation 2: ΣX1Y = aΣX1 + b1ΣX1² + b2ΣX1X2; Multiply the model by X1 and sum.
Normal equation 3: ΣX2Y = aΣX2 + b1ΣX1X2 + b2ΣX2²; Multiply the model by X2 and sum.
Slope b1 (deviation form): b1 = (Σx1y·Σx2² − Σx2y·Σx1x2) ÷ (Σx1²·Σx2² − (Σx1x2)²); Here x1 = X1 − mean of X1, x2 = X2 − mean of X2, y = Y − mean of Y.
Slope b2 (deviation form): b2 = (Σx2y·Σx1² − Σx1y·Σx1x2) ÷ (Σx1²·Σx2² − (Σx1x2)²); Same denominator as b1.
Intercept: a = Ȳ − b1X̄1 − b2X̄2; Use the means of the original data, not the deviations.
Forecast error: e = Actual − Forecast; Positive means the forecast was too low.
MAD: MAD = Σ|e| ÷ n; Average absolute error, in the units of Y.
MSE: MSE = Σe² ÷ n; In squared units. Large errors weigh heavily.
MAPE: MAPE = (Σ(|e| ÷ Actual) ÷ n) × 100; Needs non-zero actual values.
Mean forecast error (bias): MFE = Σe ÷ n; Far from zero means the forecast is consistently too high or too low.
Coefficient of determination: R² = 1 − SSE ÷ SST; SSE is the sum of squared errors. SST is the total sum of squares of Y around its mean.
Adjusted R²: Adjusted R² = 1 − (1 − R²)(n − 1) ÷ (n − k − 1); k is the number of independent variables. It penalises useless variables.

Quick revision

Time series components: trend (T), seasonal (S), cyclical (C), irregular (I).
Additive model: Y = T + S + C + I. Multiplicative model: Y = T × S × C × I.
Seasonal variation repeats within a year; cyclical variation spans longer periods.
Moving average smooths data; use an even period with centring.
Least squares trend: Y = a + bX, with X coded so that ΣX = 0 where possible.
Regression normal equations: ΣY = na + bΣX and ΣXY = aΣX + bΣX².
Slope b = [nΣXY − ΣXΣY] ÷ [nΣX² − (ΣX)²]; a = Ȳ − bX̄.
Correlation r lies between −1 and +1; r² is the share of variation in Y explained by X.
Standard error of estimate measures the spread of actual values around the regression line; smaller means a better fit.
Deseasonalised value = actual value ÷ seasonal index (multiplicative model).
Forecasts are most reliable close to the data range; extrapolating far beyond it is risky.
Regression shows association, not proof of cause.

Common mistakes

Treating a forecast as a certain prediction or a target. Fix: Write that a forecast is an estimate of the likely outcome, while a budget or plan is what management decides to achieve.
Classifying regression as qualitative, or Delphi as quantitative. Fix: Ask whether the method runs on numbers and a model. If it runs on opinion, it is qualitative.
Confusing seasonal and cyclical variation. Fix: Seasonal repeats within one year at a fixed time. Cyclical lasts more than a year and has no fixed length.
Subtracting the seasonal index in a multiplicative model. Fix: In the multiplicative model, divide by the index (as a ratio). Subtract only in the additive model.
Putting the moving average against the first or last period instead of the middle period. Fix: Always place the average at the centre of the group. A 3-year average of years 1 to 3 belongs against year 2.
Not centring a 4-year (even-period) moving average. Fix: Average each pair of consecutive 4-year totals (or averages) so that the result sits on an actual year.
Taking an ordinary 4-quarter moving average and not centring it Fix: Add two consecutive 4-quarter totals and divide by 8. Place the result against the third quarter of the first window.
Placing the moving average against the wrong quarter Fix: The first centred moving average belongs to Q3 of Year 1. The ratio table therefore starts at Q3 and ends two quarters before the last observation.
Using the wrong standard deviation ratio, such as b_yx = r σx ÷ σy. Fix: Put the dependent variable's standard deviation on top. For Y on X, Y is dependent, so b_yx = r σy ÷ σx.
Using the line of Y on X to estimate X from a given Y. Fix: Use the line of X on Y to estimate X. Rearranging Y on X gives a different, wrong answer unless r = ±1.

Exam tips

Structure theory answers as meaning, purpose, steps, types, limitations. Use short bullet points.
In case questions, justify the method by data availability and horizon. This earns marks that a bare list does not.
For MCQs, learn the classification: Delphi, expert opinion and surveys are qualitative; moving average, trend and regression are quantitative.
Practise forecast error and MAD. They are quick, safe numerical marks.
Link forecasting to its use in budgeting, pricing and capacity decisions, since this paper is decision-oriented.
Write the model equation at the start of every numerical answer. It earns method marks even if arithmetic slips.
Read whether seasonal figures are in units or percentages before choosing a model, unless the question names it.
In theory questions, give one business example for each component. It shows application, not recall.