PE_L2_(without Appendix)

Question 1

What is the Simple Linear Regression (SLR) population model?

Answer 1

• Model form: y = β₀ + β₁x + u
• y: dependent (outcome) variable, consequence
• x: independent (explanatory) variable, cause
• u: error (disturbance) term

“population model”
We want to find causal relations (“! causes “”), not (mere) correlations!

Question 2

Which assumptions define the SLR framework?

Answer 2

• SLR.1: Model is linear in parameters (beta)
• SLR.2: Random sampling
• SLR.3: Sample variation in x
• SLR.4: Zero conditional mean (E(u|x)= E(u)=0)
  E ist der Erwartungswert
• SLR.5: Homoskedasticity (Var(u|x)=σ²)

Question 3

What does ‘zero conditional mean’ imply?

Answer 3

• E(u|x) = 0
• The error term u does not systematically depend on x
• Guarantees no omitted-variable bias if included variables capture relevant effects

Question 4

Why is sample variation in x important?

Answer 4

• Prevents the denominator in slope formulas from being zero
• Ensures x has enough variability to estimate β₁
• Without it, the slope cannot be computed

Question 5

How do we interpret the OLS estimates?

Answer 5

• β₁ (slope): Estimated change in y for a one-unit change in x
• β₀ (intercept (value of : when & is zero)): Estimated value of y when x=0
• Both are unbiased if assumptions hold

Question 6

What is the idea of ‘least squares’?

Answer 6

• Minimise the sum of squared residuals: ∑(y_i − ŷ_i)²
• Ensures the fitted line is as close as possible to the data points (in a vertical sense)

Question 7

What is R²?

Answer 7

• Coefficient of determination
• R² = SSE / SST = 1 − SSR / SST
• Measures the proportion of the total variation in y explained by the model
• Goodness of fit

Question 8

What does homoskedasticity mean?

Answer 8

• Var(u|x) = σ² (constant)
• The spread of the error term does not depend on x
• A key assumption for deriving simple variance formulas

Question 9

How is the error variance estimated?

Answer 9

• Using the residuals: û_i = y_i − ŷ_i
• σ̂² = SSR / (n − 2)
• (n − 2) because two parameters (β₀, β₁) are estimated from the data

Question 10

How can we handle non-linear relationships?

Answer 10

• ‘Linear in parameters’ does not require a strictly straight-line in x
• Can use log forms: ln(y), ln(x)
• Interpretation changes: e.g., log-log model implies elasticity interpretation

Question 11

What is the unobserved error term?

Answer 11

1. Strictly unpredictable random behaviour that may be unique to that observation
2. Unspecified (unobserved) factors / explanatory variables, which are not in the model
3. An approximation error if the relationship between : and & is not exactly linear

Question 12

OLS

Answer 12

Ordinary Least Squares

Question 13

Estimate β0 + β1

Answer 13

Step 1:
E(u) = 0
E(u|x) = 0
??
Step 2:
??

Question 14

Implications of Simple Linear Regression (SLR) Model

Answer 14

Implication 1: OLS is unbiased (because of SLR1-4)
Implication 2: Properties of variance of the OLS Estimators