Multiple regression extends simple regression by including more than one independent variable. In this lab, a log-log demand function is estimated with both price (P) and advertising (A) as predictors of sales (S). The result is a power-function demand model where each coefficient is a partial elasticity, telling you the percentage change in sales for a 1% change in that variable, holding the other constant.
Difficulty: Intermediate | Prerequisites: simple regression (Part C of this lab), natural log transformations, concept of ceteris paribus (holding other variables constant).
In real markets, demand depends on more than just price. Advertising spend, income, competitor pricing, and other variables all shift how much consumers buy. Multiple regression lets you estimate a demand function that accounts for several of these drivers at once, isolating the effect of each one while controlling for the others.
This section of the lab adds advertising expenditure (A) alongside price (P) as predictors of sales (S), using a log-log specification. If you are comfortable with the simple log-log regression from Part C, this extends the same logic to two independent variables.
Multiple regression
A regression with two or more independent variables. The model estimates the partial effect of each variable while holding the others constant. Think of it as the statistical version of ceteris paribus.
Partial elasticity
In a multiple log-log model, each slope coefficient is the elasticity of the dependent variable with respect to that particular independent variable, holding the others fixed. In simple terms, it answers: "If I change just this one thing by 1%, what happens to sales?"
Advertising elasticity of demand
The percentage change in quantity demanded (or sales) for a 1% change in advertising expenditure, all else equal. In this lab's model, it is the coefficient on ln(A).
Price elasticity of demand (in a multiple regression context)
The percentage change in sales for a 1% change in price, holding advertising constant. In this lab's model, it is the coefficient on ln(P).
Omitted variable bias
The distortion in an estimated coefficient that occurs when a relevant variable is left out of the model. If advertising affects both sales and is correlated with price, leaving it out would bias the price coefficient.
Multicollinearity
When two or more independent variables are highly correlated with each other, making it difficult to isolate each one's individual effect. Standard errors inflate, coefficients become unstable.
Statistical significance vs. practical significance
A coefficient can be statistically significant (p-value < 0.05) but economically small, or statistically insignificant but economically large in a small sample. Both dimensions matter for interpretation.
Dataset: 10 observations with three variables: Sales (S), Advertising (A), and Price (P).
S ranges from 453.60 to 926.40
A ranges from 585.00 to 1,395.00
P ranges from 85.50 to 128.25
Model specification: ln(S) = a + b₁ ln(A) + b₂ ln(P). This is equivalent to the power function S = e^a × A^b₁ × P^b₂.
Estimated equation: ln(S) = 2.310 + 0.766 ln(A) + (−0.224) ln(P)
In exponential form: S = e^2.31 × A^0.766 × P^(−0.224), or approximately S = 10.07 × A^0.766 × P^(−0.224).
Interpretation of coefficients:
Advertising elasticity = 0.766: a 1% increase in advertising is associated with a 0.766% increase in sales, holding price constant.
Price elasticity = −0.224: a 1% increase in price is associated with a 0.224% decrease in sales, holding advertising constant. Demand is inelastic with respect to price.
R² = 0.965: The model explains 96.5% of the variation in ln(S). This is a strong fit.
Adjusted R² = 0.955.
Degrees of freedom = 7 (10 observations minus 3 estimated parameters: intercept, b₁, b₂).
Significance F = 7.62 × 10⁻⁶: The overall model is highly significant.
Coefficient on ln(A) is significant (p-value = 7.33 × 10⁻⁶, t-stat = 11.749). Advertising has a strong, well-measured effect on sales.
Coefficient on ln(P) is not significant at the 5% level (p-value = 0.197, t-stat = −1.425). The 95% confidence interval for the price coefficient spans from −0.597 to 0.148, which includes zero. This means we cannot rule out that price has no effect on sales in this sample.
Intercept is marginally significant (p-value = 0.005), but the intercept's statistical significance matters less than the slopes' in most demand analysis.
What this means practically: Advertising appears to be the dominant driver of sales in this dataset. The price effect, while negative as expected, is not precisely estimated enough to be confident it differs from zero. This could reflect the small sample size (only 10 observations) or a genuinely weak price effect in this market.
Comparing elasticities: |0.766| > |−0.224|, so sales are more responsive to changes in advertising than to changes in price. Advertising is the more "elastic" lever here.
\text{Multiple log-log model: } \ln(S) = a + b_1 \ln(A) + b_2 \ln(P)\text{Power form: } S = e^{a} \cdot A^{b_1} \cdot P^{b_2}df = n - k = 10 - 3 = 7Where n = 10 observations and k = 3 estimated parameters (intercept + two slopes).
The coefficient b₁ is the advertising elasticity of sales, and b₂ is the price elasticity of sales, each measured ceteris paribus.
This is how companies decide where to allocate marketing budgets. If the advertising elasticity (0.766) is much larger than the price elasticity (0.224), a firm might get more sales growth from increasing advertising spend than from cutting prices.
Consumer goods companies, tech firms, and media businesses routinely estimate these multi-variable demand models to guide pricing and promotional strategy.
"A high R² means every coefficient is reliable." The model can explain most of the variation overall (R² = 0.965) while individual coefficients remain imprecisely estimated. In this lab, the price coefficient is not statistically significant despite the high R². A small sample and possible multicollinearity can cause this.
"An insignificant coefficient means the variable has no effect." It means we cannot confidently distinguish the effect from zero with this sample. With more data, the same true effect might become significant.
"The larger coefficient is the more important variable." In a log-log model, both coefficients are elasticities (percentage terms), so they are comparable. But "larger elasticity" does not automatically mean "more important" in a business sense, because the cost of a 1% change in advertising may differ greatly from the cost of a 1% change in price.
"Adding more variables always improves the model." It raises R² mechanically, but Adjusted R² can fall if the added variable does not carry enough explanatory power to justify the lost degree of freedom.
⚠️ Know how to write the power-function demand equation from the regression output: S = e^a × A^b₁ × P^b₂.
⚠️ Be ready to identify which coefficients are statistically significant and which are not, and to explain what that means for the demand function.
⚠️ Understand that in a log-log multiple regression, each slope is a partial elasticity (ceteris paribus).
⚠️ Degrees of freedom in multiple regression = n − k, where k includes the intercept. With 10 observations and 3 parameters, df = 7.
⚠️ Be prepared to compare advertising elasticity to price elasticity and discuss which lever is more effective for increasing sales.
⚠️ A high overall R² does not guarantee that every individual coefficient is significant. This is a common exam trap.
True or false: In a multiple log-log model, each slope coefficient is the partial elasticity of the dependent variable with respect to that independent variable. (True)
Fill in the blank: With 10 observations and 3 estimated parameters, degrees of freedom = ___. (7)
True or false: If the coefficient on ln(P) has a p-value of 0.197, it is statistically significant at the 5% level. (False)
Fill in the blank: In the model ln(S) = 2.31 + 0.766 ln(A) − 0.224 ln(P), a 1% increase in advertising is associated with a ___% increase in sales. (0.766%)
True or false: An R² of 0.965 guarantees that all individual coefficients are statistically significant. (False)
Q: Write the estimated demand function from the regression output in both log and power-function form.
A: Log form: ln(S) = 2.310 + 0.766 ln(A) − 0.224 ln(P). Power form: S = e^2.31 × A^0.766 × P^(−0.224), or approximately S = 10.07 × A^0.766 × P^(−0.224).
Q: The coefficient on ln(P) has a p-value of 0.197. What does this mean, and should you drop price from the model?
A: It means we cannot reject the null hypothesis that the price coefficient is zero at the 5% significance level. However, dropping it is not automatic. Economic theory says price should matter, and the insignificance may be due to the small sample (n = 10). You would typically keep it and note the imprecision.
Q: Which variable, advertising or price, has a larger effect on sales? How do you know?
A: Advertising. The advertising elasticity (0.766) is larger in absolute value than the price elasticity (0.224), meaning a 1% increase in advertising has a bigger percentage impact on sales than a 1% increase in price.
Q: The R² is 0.965 but the price coefficient is insignificant. How is that possible?
A: R² measures the model's overall explanatory power, which is driven primarily by advertising in this case. The price variable adds relatively little additional explanation, and with only 10 observations there is not enough statistical power to distinguish its small effect from zero.
Q: If a firm in this market wants to boost sales, should it lower prices or increase advertising? Justify using the regression results.
A: The regression suggests increasing advertising is more effective. The advertising elasticity (0.766) is larger and statistically significant, while the price elasticity (0.224) is smaller and not statistically distinguishable from zero. A 1% increase in advertising spend is associated with roughly three times the sales response of a 1% price cut.
This connects directly to simple regression (Part C) because the log-log specification and interpretation of elasticities are the same, just extended to more variables.
It also connects to pricing strategy and revenue optimisation: once you know the price and advertising elasticities, you can model how changes in each lever affect total revenue and profit.
The concept of omitted variable bias matters here: if you had only run the simple regression of S on P (ignoring advertising), your price coefficient might have been biased because advertising was left out.
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