Difficulty: Intermediate | Prerequisites: basic statistics (mean, variance, standard deviation), introductory econometrics or familiarity with simple linear regression.
This lab sits in the applied econometrics portion of APEC 3002 (Managerial Economics). You are moving from simple linear regression (one explanatory variable) to multiple regression (two or more). The practical question is whether advertising spending and price jointly explain variation in sales, and which of those levers matters more. You should already be comfortable with scatter plots, the idea of a "line of best fit," and the basics of hypothesis testing (null vs. alternative hypotheses, confidence intervals). If those feel shaky, revisit your introductory statistics notes first.
A multiple regression model was fitted to 10 observations, using advertising expenditure (A) and price (P) as predictors of sales (S). The model explains about 97% of the variation in sales (R² = 0.969). T-tests show advertising is a statistically significant driver of sales at the 95% level, while price is not statistically significant in this dataset.
Multiple regression
A statistical method that models the relationship between one dependent variable and two or more independent (explanatory) variables. Think of it as drawing a "best-fit surface" instead of a best-fit line.
R-squared (R²)
The proportion of variation in the dependent variable that is explained by the independent variables in the model. Ranges from 0 to 1. In simple terms, an R² of 0.969 means 96.9% of the movement in sales is accounted for by advertising and price together.
Adjusted R-squared
A modified version of R² that penalises the addition of variables that do not improve the model. It only increases if a new variable improves the fit more than would be expected by chance. Useful when comparing models with different numbers of predictors.
Degrees of freedom (df)
The number of independent observations minus the number of estimated parameters. With 10 observations and 3 coefficients (intercept, A, P), df = 10 - 3 = 7. Think of it as how much "room" is left in your data to estimate error after fitting the model.
t-statistic (t-stat)
The ratio of an estimated coefficient to its standard error. It measures how many standard errors the coefficient sits away from zero. A larger absolute t-stat means the variable is more likely to be a real (non-random) predictor.
Critical t-value
The threshold value from the t-distribution at a chosen confidence level and degrees of freedom. If the absolute calculated t-stat exceeds this threshold, the coefficient is statistically significant. For 7 df at 95% confidence, the critical value is 2.365.
Null hypothesis (H₀)
The default assumption in hypothesis testing, typically that a coefficient equals zero (no effect). In simple terms, "this variable does nothing."
Alternative hypothesis (H₁)
The claim you are testing for, typically that the coefficient is not equal to zero (the variable does have an effect).
P-value
The probability of observing a t-stat as extreme as the one calculated, assuming the null hypothesis is true. A small p-value (typically < 0.05) gives reason to reject H₀.
ANOVA (Analysis of Variance)
A framework that partitions total variation in the dependent variable into variation explained by the regression and unexplained (residual) variation. The F-statistic from ANOVA tests whether the overall regression model is statistically significant.
F-statistic
The ratio of the mean square of the regression to the mean square of the residuals. A large F (here 109.87, p ≈ 0.0000) indicates the model as a whole is significant.
Residual
The difference between an observed value and the value predicted by the regression model. Think of it as "what the model got wrong" for each observation.
Standard error of the coefficient
A measure of the precision of a coefficient estimate. Smaller standard errors mean the estimate is more tightly pinned down.
The estimated equation is: S = 284.81 + 0.5497A - 1.2716P
S = sales (dependent variable)
A = advertising expenditure (independent variable)
P = price (independent variable)
284.81 = the intercept (predicted sales when A = 0 and P = 0)
The coefficient on A (0.5497) means that, holding price constant, each additional dollar of advertising is associated with roughly 0.55 units of additional sales.
The coefficient on P (-1.2716) means that, holding advertising constant, each additional dollar of price is associated with roughly 1.27 fewer units of sales. The negative sign is consistent with the law of demand.
Multiple R: 0.9844, indicating a very strong linear relationship between the predictors and sales.
R² = 0.9691: about 96.9% of the variation in sales is explained by advertising and price jointly.
Adjusted R² = 0.9603: after penalising for the number of predictors, the model still explains about 96% of variation.
Standard error = 30.10: the typical prediction error is around 30 units of sales.
Observations = 10, df (residual) = 7: a small sample, which matters when choosing critical values.
Total SS (sum of squares) = 205,466.26, split into regression SS (199,123.08) and residual SS (6,343.18).
The F-statistic = 109.87 with a significance F of approximately 0.0000. This tells you the model as a whole is highly significant: the combination of A and P explains a statistically meaningful amount of variation in S.
The null hypothesis for each coefficient: H₀: βᵢ = 0 (no effect).
The alternative hypothesis: H₁: βᵢ ≠ 0 (variable has an effect).
Critical t-value at 95% confidence with 7 df = 2.365.
Results:
Intercept: t-stat = 3.623, p = 0.0085. Exceeds critical value, so reject H₀. The intercept is statistically significant.
A (advertising): t-stat = 13.224, p = 0.0000. Far exceeds critical value, so reject H₀. Advertising has a strong, statistically significant effect on sales.
P (price): t-stat = -1.440, p = 0.1930. Does not exceed the critical value in absolute terms, so fail to reject H₀. Price is not statistically significant in this model at the 95% level.
Failing to reject H₀ for price does not prove price has zero effect. It means the data do not provide enough evidence, at this sample size and confidence level, to conclude that price has a significant impact on sales.
With only 10 observations, power is limited. A larger dataset might yield a different result.
A: 95% CI is [0.4514, 0.6480]. The interval does not include zero, confirming significance.
P: 95% CI is [-3.3597, 0.8164]. The interval spans zero, consistent with the failure to reject H₀.
Intercept: 95% CI is [98.92, 470.70].
Residuals are the gaps between observed and predicted sales. If the model fits well, residuals should scatter randomly around zero with no visible pattern.
The residual plots for A and P both show points scattered around zero without an obvious curve or fan shape, which suggests the linear specification is reasonable.
S = \beta_0 + \beta_1 A + \beta_2 P + \varepsilonWhere β₀ is the intercept, β₁ is the coefficient on advertising, β₂ is the coefficient on price, and ε is the error term.
t = \frac{\hat{\beta}_i}{SE(\hat{\beta}_i)}The t-statistic for each coefficient: estimated coefficient divided by its standard error.
R^2 = 1 - \frac{SS_{\text{residual}}}{SS_{\text{total}}} = 1 - \frac{6343.18}{205466.26} = 0.9691df_{\text{residual}} = n - k = 10 - 3 = 7Where n = number of observations, k = number of estimated coefficients (including the intercept).
F = \frac{MS_{\text{regression}}}{MS_{\text{residual}}} = \frac{99561.54}{906.17} = 109.87Firms routinely run regressions like this one to decide how to allocate their marketing budget. If the data show that advertising significantly drives sales but price does not (at current levels), a manager might prioritise increasing the ad spend rather than cutting prices. Advertising effectiveness studies of this kind underpin media buying decisions, campaign ROI calculations, and demand forecasting across industries from consumer goods to software.
Students often think "fail to reject H₀" means the variable has no effect at all. It does not. It means the evidence in this sample is too weak to rule out zero effect at the chosen significance level. With more data, the result could change.
Students sometimes confuse R² with the significance of individual coefficients. A high R² tells you the model as a whole fits well; it does not guarantee that every individual predictor is significant. Here, R² is 0.969, yet the price coefficient is not significant.
Comparing raw coefficient sizes across variables (e.g. saying "price matters more because its coefficient is -1.27 versus 0.55 for advertising") is misleading when the variables are measured in different units or scales. The units of A and P differ, so the magnitudes are not directly comparable without standardisation.
Students sometimes think a negative coefficient means the variable is "bad" or "wrong." A negative coefficient on price simply reflects the expected inverse relationship between price and quantity demanded.
⚠️ Know how to calculate degrees of freedom: n minus the number of estimated coefficients (including the intercept).
⚠️ Be able to compare a calculated t-stat against the critical t-value and state whether you reject or fail to reject H₀. This is a near-certainty on any econometrics exam.
⚠️ Understand that the F-test and the individual t-tests answer different questions. The F-test asks whether the model as a whole is significant; the t-tests ask whether each individual coefficient is significant.
⚠️ Be able to interpret a confidence interval that includes zero (variable not significant) versus one that does not (variable significant).
⚠️ Know the difference between R² and adjusted R². Expect a question asking when adjusted R² is preferable (when comparing models with different numbers of predictors).
True or False: An R² of 0.969 means that 96.9% of variation in sales is explained by the regression model. (True.)
Fill in the blank: With 10 observations and 3 estimated coefficients, the residual degrees of freedom equal ____. (7.)
True or False: If a coefficient's p-value is 0.19, we reject the null hypothesis at the 5% significance level. (False. 0.19 > 0.05, so we fail to reject.)
Fill in the blank: The critical t-value at 95% confidence with 7 df is ____. (2.365.)
True or False: A high R² guarantees that every individual predictor in the model is statistically significant. (False.)
Q: A regression of sales on advertising and price yields the equation S = 284.81 + 0.5497A - 1.2716P. Interpret the coefficient on A.
A: Holding price constant, each additional dollar spent on advertising is associated with an increase of approximately 0.55 units in sales.
Q: The t-statistic for the price coefficient is -1.44 with a critical value of 2.365 (two-tailed, 7 df, 95% confidence). What do you conclude?
A: Because |-1.44| < 2.365, we fail to reject the null hypothesis that the price coefficient equals zero. At the 95% confidence level, the data do not provide sufficient evidence that price significantly affects sales in this model.
Q: The overall F-statistic is 109.87 with a significance F of approximately 0.0000. What does this tell you?
A: The model as a whole is highly statistically significant. The combination of advertising and price explains a significant portion of the variation in sales; the probability of getting an F this large by chance alone is essentially zero.
Q: Explain why a variable can be statistically insignificant even when R² is high.
A: R² reflects the explanatory power of all predictors combined. An individual variable may add little beyond what the other variables already explain, or its effect may be imprecisely estimated (large standard error relative to the coefficient), making its t-stat too small to clear the critical value.
Q: The 95% confidence interval for the price coefficient is [-3.36, 0.82]. Why does this interval support the conclusion from the t-test?
A: The interval spans zero, meaning zero is a plausible value for the true price coefficient. This is consistent with failing to reject H₀: β₂ = 0.
This material connects directly to demand estimation, which you will encounter when computing price elasticity from the same regression output (see the companion notes on Demand Analysis and Price Elasticity). Understanding how to test whether a coefficient is significant is also foundational for later topics in econometrics, including omitted variable bias, multicollinearity diagnostics, and model specification testing. The ANOVA decomposition introduced here reappears in experimental design and in comparing nested regression models.
Multiple regression, OLS, ordinary least squares, regression analysis, t-test, t-statistic, hypothesis testing, null hypothesis, alternative hypothesis, R-squared, adjusted R-squared, coefficient of determination, ANOVA, F-test, F-statistic, degrees of freedom, residuals, standard error, confidence interval, p-value, significance level, advertising and sales, APEC 3002, managerial economics, applied microeconomics, regression output interpretation, Excel regression, fail to reject