Enter Data
Quick Reference
Model Assumptions
For coefficient estimation (unbiasedness):
- Linear in parameters
- Random sampling (i.i.d.)
- No perfect multicollinearity
- Zero conditional mean: E(u|X) = 0
For standard errors and efficiency:
- Homoskedasticity: Var(u|X) = σ²
- Normality of errors (small-sample inference)
Standard errors shown are conventional (homoskedastic). For educational purposes only.
Regression Results
| Variable | Coefficient | Std. Error |
|---|---|---|
| Intercept | 2.2000 | 0.9381 |
| X1 | 0.6000 | 0.2828 |
Conventional (homoskedastic) standard errors
Scatter Plot with OLS Fitted Line
Residual Plot
Residuals vs. fitted values. Look for patterns: a random scatter suggests the linear model is appropriate; a fan shape suggests heteroskedasticity.
Calculation Steps
Understanding OLS Regression
What is OLS Regression?
Ordinary Least Squares (OLS) is the most widely used method for estimating a linear regression model. It works by finding the line (or hyperplane, in multiple regression) that minimizes the sum of squared residuals — the vertical distances between each observed data point and the fitted line.
Intercept: b0 = ȳ − b1 · x̄
The regression line always passes through the point (x̄, ȳ)
Simple vs. Multiple Regression
Simple Regression
One independent variable
y = b0 + b1x + u. Estimates the bivariate relationship between X and Y. Easy to visualize with a scatter plot and fitted line.
Multiple Regression
Two or more independent variables
y = b0 + b1x1 + b2x2 + ... + u. Controls for confounders, isolating each variable's ceteris paribus effect.
Interpreting Regression Output
A regression output table shows each variable's coefficient (estimated effect on Y per one-unit change in X, holding other variables constant) and standard error (precision of the estimate). Smaller standard errors indicate more precisely estimated coefficients.
- R-squared (R²) measures the proportion of variation in Y explained by the model (0 to 1). A value of 0.60 means 60% of variation is explained. Note: a low R² does not necessarily mean the model is bad, particularly in cross-sectional data.
- Adjusted R² penalizes for adding variables and is preferred when comparing models with different numbers of regressors.
- SER (Standard Error of Regression) estimates the standard deviation of the error term, measuring average prediction error.
OLS Assumptions (Gauss-Markov)
For OLS to produce unbiased, efficient estimates with valid standard errors, the following assumptions must hold (Wooldridge, Chapters 2–3):
- MLR.1 — Linearity: The population model is linear in parameters
- MLR.2 — Random Sampling: Observations are i.i.d. draws from the population
- MLR.3 — No Perfect Multicollinearity: No regressor is an exact linear function of others
- MLR.4 — Zero Conditional Mean: E(u|X) = 0 — omitted variables are not correlated with included regressors
- MLR.5 — Homoskedasticity: Var(u|X) = σ² — error variance is constant (needed for valid conventional SEs)
- MLR.6 — Normality: u|X ~ N(0, σ²) — only needed for exact small-sample t and F distributions
Frequently Asked Questions
Disclaimer
This calculator is for educational purposes only and assumes the classical linear model. Actual econometric analysis requires careful consideration of model specification, data quality, and assumption violations. Standard errors reported are conventional (homoskedastic) and may be invalid under heteroskedasticity. This tool should not be used as the sole basis for research conclusions or investment decisions.
