Simple Regression
A researcher finds: recycle = 20 + 1.8 × fee.
Interpret the slope 1.8.
A $1 increase in the waste fee is associated with a 1.8 percentage-point increase in recycling rate.
BMI = 29 − 0.15×exercise + 0.04×sugar. Interpret coefficients.
One more hour of exercise per week reduces BMI by 0.15, holding sugar constant. Each extra gram of daily sugar increases BMI by 0.04, holding exercise constant.
In ̂productivity = 50 + 2.4×sleep + 6×coffee − 0.5×sleep×coffee, interpret the coefficients for coffee
For zero sleep, each extra cup of coffee increases productivity by 6 tasks. Each extra hour of sleep reduces the productivity benefit of coffee by 0.5 tasks.
performance = 30 + 4×hours − 0.2×hours². What kind of shape does this imply?
Inverted U-shape (increasing then decreasing returns).
How does including the omitted variable affect the interpretation
Coefficients are now interpreted while holding that variable constant—resulting in less bias and clearer causality.
The predicted difference in recycling rate between $15 and $5 fees
18%
Adding a diet dummy changes the coefficient on exercise. Why?
Omitted variable bias: diet is correlated with exercise and affects BMI.
Predict productivity for 7 hours sleep and 0 cups coffee.Then, predict it when 2 cups of coffee are added
66.8 tasks. and 71.8 tasks.
Marginal effect of gaming hours?
∂performance/∂hours = 4 − 0.4×hours.
Predicted recycling when fee = $10
38%
State H₀ and Hₐ to test if β_income = 0.
H₀: β_income = 0 vs. Hₐ: β_income ≠ 0.
How many hours of sleep a non-coffee drinker would need to be as productive as a coffee drinker with 2 cups and 7 hours of sleep
50 + 2.4s = 71.8. A non-coffee drinker would need about 9.08 hours of sleep
Evaluate the marginal effect at 10 hours/week.
0; performance stops increasing after 10 hours.
log(price) = β₀ + β₁ age + β₂ log(mileage) + β₃ engineSize + β₄(age×engineSize). Why does omitting log(mileage) cause bias?
Mileage affects price and is correlated with age and engine size, so omitted variable bias arises.
Does this regression prove that fees cause more recycling?
No, correlation ≠ causation. Omitted variables like income, education, or environmental preferences could bias the estimate.
Which assumption is weakened when moving from unbiasedness to consistency?
Zero conditional mean; for consistency, we only need Cov(x, u) = 0 in the population.
Test whether coffee increases the effect of sleep. What is H₀?
H₀: β_sleep×coffee = 0; HA: β_sleep×coffee > 0.
Find the level of gaming that maximizes performance.
10 hours per week.
If log(mileage) ↓ price and ↑ age, what’s the bias on β_age? and on β_age×engineSize?
Downward bias; the effect appears more negative. Downward if older, large-engine cars have higher mileage.
State the 4 assumptions for unbiasedness
Linear in parameters, random sampling, no perfect multicollinearity, zero conditional mean of the errors
How do we calculate CI?
Obtain critical value in tables and then calculate: βhat +/- criticalvalue x SE(βhat)
At α = 1%, critical value ≈ 2.576. Do we reject H₀ if the t-statistic is 2.545. ?
No; |t| < 2.576, fail to reject at 1%.
Why can’t the gaming model be interpreted causally?
Omitted variables like motivation or ability might affect both gaming and performance.
Why does including all relevant variables improve causal interpretation?
It removes omitted variable bias, isolating true partial effects. Coefficients now represent effects holding mileage constant.