1. What is Least Squares Regression?

Least Squares Regression is a statistical method for estimating the relationships among variables. It minimizes the sum of the squares of the differences between observed and predicted values.

2. How Does the Calculator Work?

The calculator uses the matrix formula:

Where:

• \( \beta \) — Vector of regression coefficients
• \( X \) — Design matrix of predictor variables
• \( y \) — Vector of response values
• \( X^T \) — Transpose of matrix X
• \( (X^T X)^{-1} \) — Inverse of X transpose times X

Explanation: The equation finds the coefficients that minimize the sum of squared residuals between observed and predicted values.

3. Importance of Regression Coefficients

Details: The coefficients quantify the relationship between each predictor variable and the response, controlling for other variables in the model.

4. Using the Calculator

Tips: Enter matrix X with rows separated by semicolons and columns by commas. Vector y should be comma-separated. Ensure dimensions match (X columns = number of predictors + 1 for intercept, rows = number of observations).

5. Frequently Asked Questions (FAQ)

Q1: What if my matrix is singular?
A: Singular matrices (non-invertible) occur with perfect multicollinearity. Remove redundant variables or use regularization.

Q2: How do I include an intercept?
A: Include a column of 1's in your X matrix for the intercept term.

Q3: What are the assumptions?
A: Linearity, independence, homoscedasticity, and normality of residuals.

Q4: When is least squares not appropriate?
A: With outliers, non-linear relationships, or when predictors outnumber observations.

Q5: How to interpret coefficients?
A: Each coefficient represents the change in y per unit change in that predictor, holding others constant.