Simple Regressionbeginner

The Simple Linear Regression Model

The simple linear regression (SLR) model writes the dependent variable as $y = \beta_0 + \beta_1 x + u$ , where $\beta_1$ is the slope and $\beta_0$ the intercept. The slope gives the change in $y$ for a one-unit change in $x$ , holding the unobserved factors $u$ fixed, so $\Delta y = \beta_1 \Delta x$ when $\Delta u = 0$ . The error term $u$ collects every other influence on $y$ not captured by $x$ . The model is linear in the parameters, which permits curved relationships in the variables themselves through transformations such as logs.

Try it yourself

Least squares — minimising SSR

OLS picks the line that minimises the sum of squared residuals, SSR = Σ(yᵢ − ŷᵢ)². Residuals are the vertical gaps from each point to the line. Drag your blue line and try to beat the gold OLS line on SSR.

SSR — your line vs OLS26 vs 26

OLS line ŷ = 2.1 + 1.55xSSR (OLS, min) 26R² (OLS) 93%

Your intercept b₀2.1

Your slope b₁1.55

Your line sits exactly on the OLS line, so the two SSRs are equal at the minimum 26. Nudge a slider and the loss can only go up.

Why it matters

Think of $y$ as the outcome you care about, wage, and $x$ as the one factor you put on the right-hand side, years of schooling. The line $\beta_0 + \beta_1 x$ is the systematic part you can explain with $x$ , and $u$ is everything else, ability, family background, luck, rolled into a single bucket. The slope answers the practical question of how much $y$ moves when $x$ moves by one unit. The intercept is just where the line crosses at $x = 0$ , which is only meaningful when $x = 0$ is itself sensible.

Formulas

Population SLR model

y = \beta_0 + \beta_1 x + u

\beta_0

is the intercept,

\beta_1

the slope, and

u

the error (disturbance) capturing all unobserved factors affecting

y

Slope as a marginal effect

\beta_1 = \frac{\Delta y}{\Delta x} \quad \text{when} \quad \Delta u = 0

The slope is the effect of

x

y

holding the unobservables fixed. Without that ceteris paribus condition it is not a causal effect.

Worked examples

Scenario

An applied labour economist models hourly wage on years of education using a sample from the U.S. labour force and runs `regress wage educ` in Stata.

Solution

Stata estimates $\widehat{\text{wage}} = \hat{\beta}_0 + \hat{\beta}_1\,\text{educ}$ . If $\hat{\beta}_1 = 0.54$ , each additional year of schooling is associated with a wage that is higher by about 0.54 dollars per hour. The intercept $\hat{\beta}_0$ is the predicted wage at zero years of education, which here is an out-of-sample extrapolation and should not be over-interpreted.

NoteThe phrase "is associated with" is deliberate. Causal language requires the zero conditional mean assumption to hold.

Common mistakes

✗Linear regression requires a straight-line relationship between the raw variables. The model only needs to be linear in the parameters $\beta_0$ and $\beta_1$ . Using $\log(y)$ or adding $x^2$ still fits inside the linear-in-parameters framework while producing a curved fit in the original units.
✗The intercept is always economically meaningful. $\beta_0$ is the value of $y$ when $x = 0$ . If $x = 0$ never occurs in the data (for example zero years of education), the intercept is a mathematical anchor for the line, not a quantity to interpret on its own.
✗The error term $u$ means the model is wrong or poorly specified. Every regression has an error term by construction. $u$ represents the many factors other than $x$ that influence $y$ . Its presence is normal and expected, not a sign of failure.
✗ $x$ must cause $y$ for the slope to be defined. The slope is a feature of the joint distribution of $x$ and $y$ and is always defined. Whether it carries a causal meaning is a separate question that depends on assumptions about $u$ .

Revision bullets

•Model is $y = \beta_0 + \beta_1 x + u$ , linear in the parameters
• $\beta_1$ = change in $y$ per one-unit change in $x$ , holding $u$ fixed
• $\beta_0$ = value of $y$ when $x = 0$ (interpret only if $x = 0$ is sensible)
• $u$ collects all unobserved factors affecting $y$
•Linearity restricts the parameters, not the functional form of the variables

Quick check

In the model $\text{wage} = \beta_0 + \beta_1\,\text{educ} + u$ , what does $\beta_1$ represent?

Which statement about the linearity of the simple regression model is correct?

Connected topics

Population Model OLS Derivation E(u|x)=0 MLR model Logs & Elastic.

Sources

Wooldridge (2019), Ch. 2.1
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage Learning, 2019. ISBN 978-1-337-55886-0.
Section 2.1 defines the simple regression model, the interpretation of the slope and intercept, and the role of the error term.
Wooldridge (2019), §2.1 (linearity)
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage Learning, 2019.
Discusses why the model is linear in parameters and how transformations expand the class of relationships it can capture.

How to cite this page

Dr. Phil's Quant Lab. (2026). The Simple Linear Regression Model. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-slr-model

← Back to the atlas See in the network →