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Multiple Regression and Forecast Evaluation with MAD, MSE and MAPE

Updated 11 October 2026 · Fact-checked

Multiple regression predicts one dependent variable, such as overhead cost, from two or more independent variables, using Y = a + b1X1 + b2X2. You find a, b1 and b2 from the normal equations, then forecast. You judge the forecast with errors: MAD, MSE and MAPE. Lower values mean a better forecast.

Understand Multiple Regression and Forecast Evaluation

Simple regression uses one driver to explain a cost or demand figure. Real costs rarely move with one driver. Overhead may depend on both machine hours and number of batches. Sales may depend on price and advertising spend. Multiple regression puts two or more drivers in one equation.

With two independent variables the equation is Y = a + b1X1 + b2X2. Y is the dependent variable you want to forecast. X1 and X2 are the independent variables. a is the value of Y when both X's are zero. b1 is the change in Y for one unit change in X1, holding X2 constant. b2 is read the same way for X2.

You find a, b1 and b2 by the least squares method. It picks the values that make the sum of squared gaps between actual and fitted Y as small as possible. In the exam you solve three normal equations, or use the deviation form when the data are given in a way that makes it easy.

A model is only useful if its forecasts are close to what happens. Forecast evaluation measures this. The forecast error is actual minus forecast. MAD is the average size of errors. MSE squares the errors, so large misses are punished more. MAPE shows the error as a percentage of actual, so you can compare across products of different scale. R² tells you how much of the variation in Y the model explains.

In decisions, regression gives a cost estimate for budgeting, a demand estimate for pricing and capacity, and a view on which cost drivers matter. Use it only within the range of data observed. Forecasting far outside that range is risky.

Key rules to remember

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.

How to solve Multiple Regression and Forecast Evaluation questions

Use this order for any question on multiple regression or forecast evaluation.

1Identify Y and the independent variables. Write the model as Y = a + b1X1 + b2X2.
2Check what is given. If the sums (ΣY, ΣX1, ΣX1² and so on) are given, use them directly. If only raw data are given, compute the means and sums first.
3Pick the method. Use the three normal equations when sums are given. Use the deviation form when the question gives or easily yields the deviations.
4Solve for b1 and b2 first, then find a from the means.
5Write the final equation and state what each coefficient means in rupees or units.
6Substitute the given values of X1 and X2 to get the forecast. Check they lie within the range of the data.
7If asked to evaluate, compute each error as Actual − Forecast, then MAD, MSE and MAPE as required. Comment on accuracy and bias.
8Close with a recommendation: use the model, add a driver, or collect more data.

Quickest way: Deviation form with a check

When to use it: Use when raw data for five or six observations are given and you must find the equation quickly.

Find the three means and write the deviations x1, x2 and y in a small table.
Compute Σx1², Σx2², Σx1x2, Σx1y and Σx2y. Do this once and carefully.
Find the denominator D = Σx1²·Σx2² − (Σx1x2)². Then b1 and b2 from the formulas.
Compute a = Ȳ − b1X̄1 − b2X̄2.
Check by putting one data row into the equation. The result should be close to the actual Y.
For errors, list the errors in one column, then take sums of |e|, e² and |e|÷Actual.

Common mistakes in Multiple Regression and Forecast Evaluation

Using deviations instead of original means when finding a
Students work in deviation form all through and carry on to the intercept.
Fix: Always compute a = Ȳ − b1X̄1 − b2X̄2 using the means of the original data.
Mixing up the cross products in the b1 and b2 formulas
The two formulas look alike and the numerators swap terms.
Fix: For b1 start with Σx1y·Σx2². For b2 start with Σx2y·Σx1². Subtract the other variable's term times Σx1x2. Then test one data row.
Reading b1 as the effect of X1 alone
Students carry over the simple regression habit.
Fix: Say that b1 is the change in Y per unit of X1 with X2 held constant.
Taking the error as Forecast − Actual and then mixing signs in bias
Textbooks differ in sign convention.
Fix: Use Actual − Forecast and state it. A negative mean error then means the forecast is too high on average. MAD, MSE and MAPE are not affected by the sign.
Dividing by the forecast in MAPE
Students recall 'percentage error' without the base.
Fix: Divide each absolute error by the actual value, then average and multiply by 100.
Forecasting outside the data range and presenting it as reliable
Students plug in any value asked.
Fix: Calculate it, but add a line that the relationship is tested only within the observed range.

Worked examples

Example 1

A plant records monthly overhead cost Y (₹ in thousands) against machine hours X1 (in hundreds) and number of set-ups X2 for five months. (X1, X2, Y): (1, 2, 17), (2, 1, 18), (3, 4, 27), (4, 3, 28), (5, 5, 35). Fit Y = a + b1X1 + b2X2 and forecast overhead for 600 machine hours and 4 set-ups.

Show the solution

Means: X̄1 = 15 ÷ 5 = 3. X̄2 = 15 ÷ 5 = 3. Ȳ = 125 ÷ 5 = 25.
Deviations x1: −2, −1, 0, 1, 2. x2: −1, −2, 1, 0, 2. y: −8, −7, 2, 3, 10.
Σx1² = 4 + 1 + 0 + 1 + 4 = 10. Σx2² = 1 + 4 + 1 + 0 + 4 = 10.
Σx1x2 = 2 + 2 + 0 + 0 + 4 = 8.
Σx1y = 16 + 7 + 0 + 3 + 20 = 46. Σx2y = 8 + 14 + 2 + 0 + 20 = 44.
D = 10 × 10 − 8² = 100 − 64 = 36.
b1 = (46 × 10 − 44 × 8) ÷ 36 = (460 − 352) ÷ 36 = 108 ÷ 36 = 3.
b2 = (44 × 10 − 46 × 8) ÷ 36 = (440 − 368) ÷ 36 = 72 ÷ 36 = 2.
a = 25 − 3 × 3 − 2 × 3 = 10.
Equation: Y = 10 + 3X1 + 2X2. Check row 1: 10 + 3 + 4 = 17, which matches.
Forecast: 600 hours means X1 = 6. Y = 10 + 3 × 6 + 2 × 4 = 10 + 18 + 8 = 36.

Answer: Y = 10 + 3X1 + 2X2. Forecast overhead is ₹36 thousand, which is ₹36,000. Each extra 100 machine hours adds ₹3,000 with set-ups held constant. Each extra set-up adds ₹2,000 with machine hours held constant. The data fit exactly, which is rare in real data. X1 = 6 is just outside the observed range of 1 to 5, so treat the forecast with caution.

Example 2

Actual demand for four months (units) is 100, 120, 110 and 130. The regression model forecast was 105, 115, 120 and 125. Compute the mean forecast error, MAD, MSE and MAPE, and comment.

Show the solution

Errors (Actual − Forecast): 100 − 105 = −5. 120 − 115 = 5. 110 − 120 = −10. 130 − 125 = 5.
Mean forecast error = (−5 + 5 − 10 + 5) ÷ 4 = −5 ÷ 4 = −1.25 units.
Absolute errors: 5, 5, 10, 5. Sum = 25. MAD = 25 ÷ 4 = 6.25 units.
Squared errors: 25, 25, 100, 25. Sum = 175. MSE = 175 ÷ 4 = 43.75.
Absolute percentage errors: 5 ÷ 100 = 5.00%. 5 ÷ 120 = 4.17%. 10 ÷ 110 = 9.09%. 5 ÷ 130 = 3.85%.
Sum = 22.10%. MAPE = 22.10 ÷ 4 = 5.53% (approx).

Answer: Mean forecast error = −1.25 units, MAD = 6.25 units, MSE = 43.75, MAPE ≈ 5.53%. On average the forecast misses by about 5.5% of actual demand. The small negative mean error shows a slight tendency to over-forecast, but it is small compared with MAD, so there is no strong bias. The month 3 error of 10 units drives most of the MSE. Check what happened in that month before accepting the model.

Exam tips

In the MCQ section, expect questions on the meaning of a coefficient, the effect of adding a variable on R², and simple MAD or MAPE computations. Practise these until they take under two minutes.
In written questions, show the equation, the coefficients with units and the forecast separately. Marks go for each stage even if one figure slips.
When sums are given in the question, use the normal equations or the deviation formulas. Do not rebuild the sums from raw data.
Always add a decision comment: whether the forecast is reliable, which driver matters more and what you recommend for budgeting or pricing.
State the sign convention for errors at the start of any forecast evaluation answer.

Practice questions from Business Forecasting Models - Time Series and Regression Analysis

Multiple Regression and Forecast Evaluation in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Multiple Regression and Forecast Evaluation: frequently asked questions

What is the multiple regression equation with two independent variables?

It is Y = a + b1X1 + b2X2. Y is the variable you forecast. X1 and X2 are the drivers. Each b shows the change in Y for one unit change in its own driver, with the other driver held constant.

How do I find a, b1 and b2 in the exam?

Use the three normal equations or the deviation form. Find b1 and b2 first, then a = Ȳ − b1X̄1 − b2X̄2. Check the answer by putting one data row back into the equation.

What is the difference between MAD, MSE and MAPE?

MAD is the average absolute error in the units of the data. MSE is the average squared error and punishes large misses. MAPE is the average absolute error as a percentage of actual, so it works across items of different scale.

Does a higher R² always mean a better model?

No. R² never falls when you add a variable, even a useless one. Adjusted R² corrects for the number of variables. You should also check that the signs and sizes of the coefficients make business sense.