Source: APEC 3002 Lab Session 3, University of Minnesota Twin Cities
Tags: regression analysis, demand function, OLS, price elasticity, advertising expenditure, personal selling expenditure, R-squared, p-value, t-statistic, managerial economics
Difficulty: Intermediate Prerequisites: Basic statistics (mean, variance, hypothesis testing), introductory microeconomics (demand curves, elasticity concepts), familiarity with Excel regression tools.
This lab sits at the intersection of econometrics and managerial decision-making. You are estimating a demand function for an IT company using ordinary least squares (OLS) regression, then interpreting the output to understand how price, advertising, and personal selling expenditures each influence quantity demanded. The broader point is that firms do not guess at demand; they measure it with data, and the regression coefficients tell managers which levers matter and by how much. You should already be comfortable reading a basic demand curve and know what "statistically significant" means in a hypothesis-testing context.
An IT company's monthly demand was regressed on price, advertising spend, and personal selling spend. The model fits well (R-squared of 0.97), all three explanatory variables are statistically significant at the 95% level, and the estimated price elasticities are inelastic across a wide price range. In plain terms: price, advertising, and personal selling all measurably affect how many units the firm sells, but demand is not very sensitive to price changes in this market.
Demand function (estimated)
A mathematical equation that expresses quantity demanded as a function of its determinants (here: price, advertising, and personal selling expenditure), with coefficients estimated from observed data using regression. Think of it as the firm's best statistical guess at how each variable drives sales.
Ordinary least squares (OLS) regression
The standard method for fitting a linear equation to data by minimising the sum of squared differences between observed and predicted values. In simple terms, OLS draws the "best fit" line through your data points.
R-squared (coefficient of determination)
The proportion of variation in the dependent variable (Q) that is explained by the independent variables in the model. An R-squared of 0.97 means 97% of the variation in quantity demanded is accounted for by price, advertising, and personal selling expenditure. Think of it as a score for how well the model fits the data, where 1.0 is perfect.
Adjusted R-squared
A modified version of R-squared that penalises the addition of variables that do not improve the model. It is always less than or equal to R-squared. Useful when comparing models with different numbers of independent variables.
Degrees of freedom
The number of observations minus the number of estimated parameters (including the intercept). Here: 12 observations minus 4 parameters = 8 degrees of freedom. In simple terms, it is how much independent information is left over after estimating the model.
P-value
The probability of observing a test statistic as extreme as (or more extreme than) the one calculated, assuming the null hypothesis is true. A p-value below 0.05 means you reject the null hypothesis at the 95% confidence level. Think of it as the chance that the result happened by luck.
Statistical significance (at the 95% confidence level)
A result is statistically significant when its p-value is less than 0.05, meaning there is less than a 5% chance the observed relationship is due to random noise. In simple terms, you can be fairly confident the effect is real.
Coefficient (regression coefficient)
The estimated change in the dependent variable for a one-unit change in the independent variable, holding all other variables constant. For example, a price coefficient of -0.296 means that for every one-dollar increase in price, quantity demanded falls by about 0.296 units.
Price elasticity of demand
The percentage change in quantity demanded divided by the percentage change in price. A value between 0 and -1 (in absolute terms, between 0 and 1) indicates inelastic demand, meaning quantity is not very responsive to price changes. Think of it as "how much do buyers flinch when the price moves?"
Inelastic demand
Demand is inelastic when the absolute value of price elasticity is less than 1. A price increase leads to a proportionally smaller decrease in quantity demanded, so total revenue rises when the firm raises its price. In simple terms, buyers keep buying even when the price goes up.
Residual
The difference between the observed value and the value predicted by the regression model for a given observation. Small residuals indicate the model fits individual data points well.
The dataset contains 12 monthly observations for an IT company, with four variables:
Q (quantity demanded): ranges from 1,000 to 3,000 units, mean of about 2,021
P (price in USD): ranges from $2,600 to $3,800, mean of $3,175
AD (advertising expenditure in USD): ranges from $10,400 to $29,800, mean of about $19,967
PSE (personal selling expenditure in USD): ranges from $26,000 to $45,000, mean of about $35,667
The OLS regression produces:
Q = -117.51 - 0.296P + 0.036AD + 0.066PSE
Reading this equation:
The intercept (-117.51) has no practical interpretation on its own; it is simply the baseline the model starts from when all independent variables are zero.
Each coefficient tells you how Q changes when that variable increases by one unit, holding the others constant.
Price (P): coefficient = -0.296
For every $1 increase in price, quantity demanded falls by about 0.296 units.
The negative sign is consistent with the law of demand.
P-value = 0.019 (below 0.05), so this effect is statistically significant at the 95% level.
95% confidence interval: roughly -0.53 to -0.06.
Advertising expenditure (AD): coefficient = 0.036
For every additional $1 spent on advertising, quantity demanded rises by about 0.036 units.
The positive sign makes economic sense: more advertising should increase demand.
P-value = 0.033 (below 0.05), statistically significant at 95%.
95% confidence interval: roughly 0.004 to 0.068.
Personal selling expenditure (PSE): coefficient = 0.066
For every additional $1 spent on personal selling, quantity demanded rises by about 0.066 units.
P-value = 0.002 (below 0.05), statistically significant at 95%.
PSE has the strongest statistical significance of the three variables (lowest p-value).
95% confidence interval: roughly 0.033 to 0.099.
R-squared = 0.970: The model explains about 97% of the variation in quantity demanded. This is a very strong fit.
Adjusted R-squared = 0.958: Even after penalising for the number of variables, the model still explains about 96% of the variation.
Standard error = 123.9: The typical prediction error is about 124 units.
F-statistic = 85.4, p-value = 2.04 x 10⁻⁶: The overall model is highly significant. The probability that all three coefficients are simultaneously zero is essentially nil.
Degrees of freedom: 3 (regression) and 8 (residual), from 12 observations and 4 estimated parameters.
The lab computes point price elasticities using the regression coefficient for price and specific price/quantity combinations. The formula is:
Elasticity = (dQ/dP) x (P / Q)
Where dQ/dP is the price coefficient from the regression (-0.296), P is the price at the point being evaluated, and Q is the estimated quantity demanded at that point.
The lab evaluated elasticity at six combinations of price ($1,000, $2,500, $4,000) and advertising ($15,000 and $30,000):
Price | AD | Estimated Q | Price Elasticity |
|---|---|---|---|
$1,000 | $15,000 | 2,940 | (baseline) |
$2,500 | $15,000 | 2,496 | -0.049 |
$4,000 | $15,000 | 1,598 | -0.079 |
$1,000 | $30,000 | 3,027 | -0.010 |
$2,500 | $30,000 | 2,582 | -0.025 |
$4,000 | $30,000 | 2,137 | -0.040 |
All elasticities are between 0 and -1 in absolute value. This means demand is inelastic at every price-advertising combination tested.
When demand is inelastic, a 1% increase in price leads to less than a 1% decrease in quantity demanded.
Inelastic demand implies that raising the price will increase total revenue, because the revenue gained from higher prices exceeds the revenue lost from fewer units sold.
As price rises, elasticity increases in absolute value. At higher prices, demand becomes somewhat more responsive to price changes, but it remains firmly inelastic throughout the range tested.
Higher advertising spending slightly reduces the absolute elasticity. With more advertising support, buyers are even less price-sensitive. This makes intuitive sense: advertising can build brand loyalty or perceived value, making consumers less likely to switch away when prices rise.
A manager at this IT company would conclude that there is pricing power in this market. Because demand is inelastic, modest price increases would raise revenue. The data also suggests that personal selling expenditure has a larger per-dollar effect on demand than advertising does (coefficient of 0.066 vs 0.036), which could inform budget allocation between the two.
Estimated demand function:
Q = -117.51 - 0.296P + 0.036AD + 0.066PSE
R-squared:
R² = 1 - (SS_residual / SS_total) = 1 - (122,854 / 4,057,292) = 0.970
Degrees of freedom:
df = n - k = 12 - 4 = 8
Where n = number of observations, k = number of estimated parameters (intercept + 3 coefficients).
Point price elasticity of demand:
E_p = (dQ/dP) x (P / Q) = -0.296 x (P / Q)
F-statistic (overall model significance):
F = (SS_regression / k_regression) / (SS_residual / df_residual) = (3,934,438 / 3) / (122,854 / 8) = 85.4
t-statistic for individual coefficients:
t = (estimated coefficient) / (standard error of coefficient)
For price: t = -0.296 / 0.102 = -2.91 For AD: t = 0.036 / 0.014 = 2.58 For PSE: t = 0.066 / 0.014 = 4.61
This is how firms set prices in practice. A company like Dell or HP might estimate a demand function from historical sales data to decide whether a $100 price cut would generate enough extra units sold to be worth it. The regression output answers that directly: plug in the new price and compare predicted quantity.
Advertising and personal selling budgets are also allocated this way. If personal selling has a larger coefficient per dollar than advertising, the marketing team can argue for shifting budget toward the sales force.
Students often confuse R-squared with the significance of individual variables. A high R-squared means the model as a whole fits well, but an individual variable could still be insignificant (high p-value). Here, all three happen to be significant, but that will not always be the case.
Students sometimes think a negative intercept means something is wrong with the model. It does not. The intercept of -117.51 simply anchors the equation. It would only have a real-world interpretation if it were plausible for all independent variables to equal zero, which is not the case here (a price of $0 with $0 in advertising and $0 in selling effort is not a realistic scenario).
Students frequently report price elasticity as a positive number. By convention, price elasticity of demand is negative (price and quantity move in opposite directions). The lab report discusses elasticities as "bigger than zero" in absolute terms, but be careful with the sign when writing your own answers.
Students sometimes believe inelastic demand means demand does not change at all. Inelastic means demand changes proportionally less than price, not that it is unresponsive. The quantity still falls when price rises; it just falls by a smaller percentage.
⚠️ You will almost certainly be asked to interpret regression coefficients. Know how to read the sign, the magnitude, and the p-value together.
⚠️ Be able to compute price elasticity from a regression coefficient and a price-quantity pair. The formula E_p = (dQ/dP) x (P/Q) is a staple exam question.
⚠️ Know the difference between R-squared and adjusted R-squared, and when each is more appropriate.
⚠️ Understand what the F-statistic tests (joint significance of all variables) versus what individual t-tests show (significance of each variable separately).
⚠️ Be ready to explain what "inelastic demand" implies for a firm's revenue when it raises or lowers its price.
True or false: An R-squared of 0.97 means that 97% of the variation in price is explained by quantity demanded.
Answer: False. R-squared measures how much variation in the dependent variable (Q) is explained by the independent variables, not the other way around.
Fill in the blank: A p-value of 0.019 for the price coefficient means the coefficient is statistically significant at the ____% confidence level.
Answer: 95% (because 0.019 < 0.05).
True or false: If demand is inelastic, a firm should lower its price to increase total revenue.
Answer: False. With inelastic demand, raising the price increases total revenue.
Fill in the blank: The degrees of freedom in this regression are ____, calculated as ____ observations minus ____ parameters.
Answer: 8, calculated as 12 observations minus 4 parameters.
True or false: The intercept of -117.51 means the company would have negative demand if all other variables were zero.
Answer: Technically the equation predicts that, but this is not meaningful because all independent variables equalling zero is not a realistic scenario. The intercept is a mathematical anchor, not a practical prediction.
Q: Given the estimated demand function Q = -117.51 - 0.296P + 0.036AD + 0.066PSE, what is the predicted quantity demanded when P = $3,000, AD = $20,000, and PSE = $35,000?
A: Q = -117.51 - 0.296(3,000) + 0.036(20,000) + 0.066(35,000) = -117.51 - 888 + 720 + 2,310 = 2,024.49, so approximately 2,024 units.
Q: The coefficient on PSE is 0.066. Interpret this in plain language.
A: Holding price and advertising expenditure constant, every additional $1 spent on personal selling is associated with an increase of 0.066 units in quantity demanded. Equivalently, an additional $1,000 in personal selling expenditure is associated with about 66 more units sold.
Q: The p-value for the advertising coefficient is 0.033. At a significance level of 0.01, would you conclude that advertising expenditure has a statistically significant effect on quantity demanded?
A: No. The p-value of 0.033 is greater than 0.01, so you would fail to reject the null hypothesis at the 99% confidence level. However, it is significant at the 95% level (0.033 < 0.05).
Q: Calculate the price elasticity of demand when P = $3,000, AD = $20,000, and PSE = $35,000 (use the predicted Q from the first question).
A: E_p = -0.296 x (3,000 / 2,024.49) = -0.296 x 1.482 = -0.439. Demand is inelastic at this point (absolute value less than 1).
Q: If the R-squared is 0.970 and the adjusted R-squared is 0.958, what does the gap between them suggest?
A: The small gap suggests that the independent variables are genuinely contributing to the model's explanatory power. A large gap would indicate that some variables might be in the model without adding real predictive value. Here, the penalty for three explanatory variables is modest, confirming the model is not overfitted.
Q: The F-statistic for this regression is 85.4 with a p-value of 2.04 x 10⁻⁶. What does this tell you?
A: The F-test evaluates whether all the independent variables jointly have a significant effect on Q. A p-value this small means you can reject the null hypothesis that all coefficients are simultaneously zero. The model as a whole is highly significant.
This lab connects directly to pricing strategy and revenue management: once you know demand is inelastic, the profit-maximising pricing decision follows. It also links to the theory of the firm, where managers allocate budgets across marketing channels (advertising vs personal selling) based on marginal returns, which is exactly what the regression coefficients measure.
The econometrics here (OLS, hypothesis testing, confidence intervals) is the same toolkit used throughout applied economics, from labour economics to public policy evaluation. If you are comfortable interpreting this regression, the same skills transfer to any empirical study you encounter in the course.
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