📖 简单线性回归模型

Quantitative Methods — Simple Linear Regression · Lesson 1

I. Where This Lesson Fits

🎯 L137 is the foundation of the entire regression module — understand what the model says before we learn to calculate and evaluate.

II. Why Learn Regression?

Real-world finance applications:

Regression = using information from one variable to predict/explain another. The most essential "prediction" tool in finance.

III. The Simple Linear Regression Model

3.1 Basic Formula

$$ Y_i = b_0 + b_1 X_i + \varepsilon_i $$

3.2 "Simple" vs. "Multiple"

3.3 Model Structure — Deconstructed

Decompose every Y value into two parts:

$$\text{Actual Value} = \underbrace{b_0 + b_1X_i}{\text{Systematic Part (Explainable)}} + \underbrace{\varepsilon_i}{\text{Random Part (Unexplainable)}}$$

Systematic Part = Regression Line: the ideal straight line that all (Xᵢ, Yᵢ) points follow

Random Part = Error: the vertical distance from each point to the regression line

Y ↑
 |    ·  ← actual point
 |   /|   ↑ vertical line = εᵢ (residual)
 |  / | 
 | ·——· ← point on regression line (fitted value)
 |/
 +──────────→ X

IV. Economic Interpretation of Regression Coefficients

4.1 b₁ (Slope) — The Most Critical Coefficient

b₁ = for each 1-unit increase in X, Y is expected to change by b₁ units

Finance examples:

4.2 b₀ (Intercept) — Starting Point When Nothing Else Matters

b₀ = Expected value of Y when X = 0

⚠️ Caution: X = 0 may not be meaningful! - House Price ~ Area: b₀ = price of a 0 m² house → meaningless, do not interpret economically - Excess Return ~ Market Excess Return: b₀ = stock's excess return when market excess is 0 → has real meaning (Alpha!)

4.3 Practical Judgment of b₀ and b₁

CAPM Regression: R_stock − Rf = b₀ + b₁(R_market − Rf) + ε

V. The Six Classical Assumptions of the Error Term ε (Gauss-Markov Foundation)

These assumptions will be explored in detail in L140. In L137, grasp the basic concepts.

🔑 Satisfying ①–④ → OLS is the Best Linear Unbiased Estimator (BLUE)

VI. Standard Regression Output — How to Read a Regression Table

The CFA Level 1 exam frequently presents regression output tables. You must know how to read them:

Dependent Variable: Stock_Return
Method: Least Squares
Sample: 60 months

Variable      Coefficient   Std. Error   t-Statistic   Prob.
────────────────────────────────────────────────────────────
Intercept      0.0025        0.0038        0.658        0.513
Mkt_Return     1.2000        0.1500        8.000        0.000
────────────────────────────────────────────────────────────
R-squared      0.5200        Durbin-Watson: 1.95
F-statistic    64.00        Prob(F): 0.000

How to read the table:

In this lesson, the goal is only to read the table. How OLS computes these numbers → L138.

VII. Practice Questions (10 Questions)

Basic Concepts (Q1–Q5)

Q1. In a simple linear regression model, the independent variable is: A. The variable being predicted B. The error term C. The variable used for prediction D. The intercept

Q2. In the regression model Y = 3 + 2X + ε, the slope coefficient indicates: A. The expected value of Y is 3 B. For each 1-unit increase in X, Y is expected to increase by 2 units C. For each 1-unit increase in X, Y will definitely increase by 2 units D. The expected value of the error term is 3

Q3. Regarding the intercept b₀, which statement is correct? A. In a CAPM regression, b₀ represents Beta B. b₀ always has an economic interpretation C. It is the expected value of Y when X = 0 D. b₀ is usually equal to the mean of the dependent variable

Q4. The error term ε represents: A. The value predicted by the regression line B. Measurement error in the independent variable X C. The difference between the actual Y value and the regression line prediction D. The standard error of the regression coefficient

Q5. The term "simple" in simple linear regression refers to: A. The model is easy to understand B. There is only one independent variable C. No hypothesis testing is required D. A linear relationship is guaranteed to hold

Applied Understanding (Q6–Q8)

Q6. An analyst runs the regression: Monthly Stock Return = 0.001 + 1.5 × Monthly Market Return. The Beta of this stock is: A. 0.001 B. 1.0 C. 1.5 D. Cannot be determined

Q7. Continuing from Q6, if the market rises 2% this month, the expected return of this stock is: A. 0.001% B. 2.0% C. 3.0% D. 3.001%

Q8. An analyst estimates b₁ = −0.8 (p = 0.03), Ha: b₁ ≠ 0, α = 0.05. The conclusion is: A. b₁ is significantly positive B. b₁ is significantly negative C. b₁ is not significantly different from 0 D. Sample size is insufficient to draw a conclusion

Integrated Reasoning (Q9–Q10)

Q9. Regression output: b₁ = 2.5, Std. Error = 1.0. The t-statistic for testing H₀: b₁ = 0 is: A. 0.4 B. 2.5 C. 1.5 D. 3.5

Q10. In a CAPM regression, b₀ (Alpha) has p = 0.42, and b₁ (Beta) has p = 0.001. With α = 0.05, the correct conclusion is: A. The fund manager has significant stock-picking ability B. The stock's Beta is significant, but Alpha is not significant C. Neither Alpha nor Beta is significant D. The overall model is invalid

VIII. Answers and Explanations

IX. CFA Level 1 Key Takeaways

📌 Next lesson L138: Ordinary Least Squares (OLS) — How do we compute b₀ and b₁? We will learn to minimize the sum of squared errors.