Simple Linear Regression Model (Basics)

1) What is regression?

Regression is a statistical method used to study the relationship between a dependent variable and one or more independent variables, mainly for:

• explanation (how Y changes with X),
• prediction (forecast Y given X).

In simple linear regression, there is one independent variable.

2) Model specification

Let:

• $Y$ = dependent variable (explained variable)
• $X$ = independent variable (explanatory variable)

2.1 Population (theoretical) model

where:

• $\alpha$ = intercept
• $\beta$ = slope coefficient
• $u$ = error term (captures other factors affecting Y)

2.2 Estimated (sample) regression line

where:

• $a$ estimates $\alpha$
• $b$ estimates $\beta$

3) Interpretation of coefficients (high-yield)

3.1 Slope $b$

$b$ measures the change in Y for a one-unit change in X (on average).

Example: If $b = 0.8$, then increasing X by 1 unit increases Y by 0.8 units (on average).

3.2 Intercept $a$

Predicted value of Y when $X=0$. May or may not have meaningful economic interpretation depending on context.