This summary covers the execution of computer exercises related to simple linear regression models using Stata, as detailed in the introductory econometrics textbook (seventh edition).

Simple Linear Regression Example 1: Participation Rate vs. Match Rate
📌 The estimated equation for participation rate ($\text{P rate}$) based on match rate ($\text{M rate}$) was $\hat{P} = 83.08 + 5.86 \times \text{M rate}$, with a sample size ($n$) of 1334.
📉 The $\text{R}^2$ value was very low at 0.075, indicating that only 7.5% of the variation in participation rate is explained by the match rate.
⚠️ Prediction for $\text{M rate} = 3.5$ resulted in $\hat{P} = 103.6\%$, which is an unreasonable prediction as participation rate cannot exceed 100%.

Simple Linear Regression Example 2: Log Salary vs. CEO Tenure
📊 The sample means showed an average salary of \$864,000 and average CEO tenure of 10.95 years.
📈 Regression of $\text{log(salary)}$ on $\text{CEO tenure}$ yielded $\widehat{\text{log(salary)}} = 6.51 + 0.0097 \times \text{CEO tenure}$.
🔄 The slope coefficient of 0.0097 implies that one additional year as CEO increases salary by 0.97% on average.

Simple Linear Regression Example 3: Sleep Time vs. Total Work Time
😴 The estimated intercept (sleep time if working 0 hours) was 3586 minutes per week (approx. 8.5 hours/day).
🤏 An increase of 2 hours (120 minutes) in work time was associated with an estimated sleep reduction of 2.6 minutes per night ($120 \times 0.15 / 7$).

Simple Linear Regression Example 4: Wage vs. IQ Score
💵 Regressing wage on IQ resulted in an estimated wage increase of \$8.30 for a one-point increase in IQ.
📉 The $\text{R}^2$ was 0.096, suggesting that IQ explains less than 10% of the wage variation.
📊 Using $\text{log(wage)}$ on IQ, a 15-point IQ increase led to an approximate 13.2% wage increase.

Simple Linear Regression Example 5: Constant Elasticity Model (Log-Log)
🔬 A log-log model was used to estimate elasticity: $\text{log(R\&D expenditure)}$ on $\text{log(sales)}$.
🧮 The estimated elasticity of R\&D expenditure with respect to sales was 1.08, meaning a 1% increase in sales raises R\&D spending by 1.08%.

Simple Linear Regression Example 6: Diminishing Marginal Effect (Log-Level Model)
📚 For a log-level model ($\text{mass}$ vs. $\text{log(expenditure)}$), the slope coefficient ($\beta_1$) represents the percentage point change in the outcome for a 100% change in the explanatory variable.
📈 A 10% spending increase was linked to a 1.1 percentage point increase in the mass pass rate, for an average spending of \$4,300/student.

Simple Linear Regression Example 7: Charitable Gift vs. Mailings
🎁 The average gift amount was 7.44 Dutch guilders, with 60% of people ($n=2561$) giving no gift.
💌 The regression $\text{gift}$ on $\text{mailings}$ yielded a slope of 2.65, implying one more mailing increases the gift by 2.65 guilders on average.
🔭 Predicted gift values ranged significantly less than actual values, and the model could not predict zero gift.

Simulation Exercise: Verifying OLS Properties (Exercise 8)
🎲 A sample of $n=500$ was simulated with true parameters $\beta_0=1$ and $\beta_1=2$, using uniform $X \sim U(0, 10)$ and normal errors $\mu \sim N(0, 6.19)$.
🔬 With $n=500$, the OLS estimates ($\hat{\beta}_0 \approx 0.139, \hat{\beta}_1 \approx 2.213$) were far from true values, but estimates became very close to population values when $n$ was increased to 5000.
✅ The sum of residuals ($\hat{\mu}$) was nearly zero, and the sum of $(\text{X} \times \hat{\mu})$ was approximately zero, confirming core OLS algebraic properties.

Simple Linear Regression Example 9: Murders vs. Executions (Capital Punishment Debate)
🔪 In 1996, 1051 counties had zero murders.
🔗 The estimated positive correlation (slope $\approx 58.56$) suggests one more execution is associated with about 59 more murders, not a deterrent effect.
❌ This estimate is likely biased due to omitted variables (population, economic development) and simultaneity (more murders lead to more executions).

Simple Linear Regression Example 11: GPA vs. PC Ownership
💻 Among 141 students, 56 owned a PC. The estimated GPA for students without a PC was 3.0, and for those with a PC, it was $3.0 + 0.17 = 3.17$.
❗️ The $\text{R}^2$ was only 0.05, confirming that PC ownership explains very little of the GPA variation.
🚫 The conclusion is that causality cannot be established due to omitted variables (e.g., student ability, diligence) correlated with both PC ownership and GPA, violating the zero conditional mean assumption.

Key Points & Insights
➡️ Simple linear regression models often yield unreasonable predictions (e.g., output $> 100\%$) or low explanatory power ($\text{R}^2$ below 10%).
➡️ When the outcome variable is log-transformed, the slope coefficient represents the percentage change for a one-unit change in the level explanatory variable (e.g., tenure example).
➡️ In models with potential omitted variables or simultaneity (like the capital punishment example), the OLS estimates are biased because the zero conditional mean assumption is violated.
➡️ For highly correlated variables (like math and written scores), simple regression yields correlation measures, not necessarily causal effects.

📸 Video summarized with SummaryTube.com on Oct 10, 2025, 13:43 UTC