1. What is Least Squares Estimation?

Least squares is a statistical method used to find the line (or hyperplane in multiple dimensions) that minimizes the sum of squared differences between observed and predicted values. It's commonly used in linear regression analysis.

2. How Does the Calculator Work?

The calculator uses the matrix least squares equation:

Where:

• \( X \) — Design matrix of independent variables
• \( y \) — Vector of dependent variable values
• \( \hat{\beta} \) — Vector of estimated coefficients
• \( X^T \) — Transpose of matrix X
• \( (X^TX)^{-1} \) — Inverse of X^TX matrix

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

3. Importance of Least Squares

Details: Least squares estimation is fundamental in regression analysis, providing the best linear unbiased estimator (BLUE) under the Gauss-Markov theorem assumptions.

4. Using the Calculator

Tips:

• Enter matrix X with each row representing an observation
• Vector y should have same number of elements as rows in X
• First column of X is typically all 1s for the intercept term
• Separate values with spaces or commas

5. Frequently Asked Questions (FAQ)

Q1: What if X^TX is singular?
A: The calculator will show an error. This occurs when columns of X are linearly dependent (multicollinearity).

Q2: How many variables can I include?
A: Theoretically unlimited, but practical limits depend on computational resources.

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

Q4: Can I use this for polynomial regression?
A: Yes, by including polynomial terms as additional columns in X.

Q5: How accurate are the results?
A: Numerically accurate for well-conditioned problems, but subject to rounding errors with near-singular matrices.