1. What is Standard Error in Linear Regression?

The standard error in linear regression measures the accuracy with which the coefficients are estimated from the data. It represents the standard deviation of the sampling distribution of the coefficient estimates.

2. How Does the Calculator Work?

The calculator uses the matrix formula:

Where:

• \( \sigma^2 \) — Variance of the residuals
• \( X \) — Design matrix of predictors
• \( X^T X \) — Cross-product matrix
• \( (X^T X)^{-1} \) — Inverse of cross-product matrix
• \( \text{diag} \) — Diagonal elements of the matrix

Explanation: The formula calculates how much the coefficient estimates would vary across different samples from the same population.

3. Importance of Standard Error Calculation

Details: Standard errors are crucial for constructing confidence intervals and conducting hypothesis tests about the regression coefficients. They help determine the precision of the estimated coefficients.

4. Using the Calculator

Tips:

• Enter the residual variance (σ²)
• Enter the XᵀX matrix in comma-separated format (rows separated by semicolons)
• Example: For matrix [[1,2],[3,4]], enter "1,2;3,4"
• The matrix must be square and invertible

5. Frequently Asked Questions (FAQ)

Q1: What is σ² in the formula?
A: σ² is the variance of the residuals, calculated as SSE/(n-p) where SSE is sum of squared errors, n is sample size, and p is number of parameters.

Q2: Why do we need (XᵀX)⁻¹?
A: The (XᵀX)⁻¹ matrix captures the covariance structure of the predictors. It's essential for understanding how predictor correlations affect coefficient precision.

Q3: How are standard errors used in hypothesis testing?
A: The t-statistic for testing H₀: βⱼ = 0 is calculated as βⱼ/SE(βⱼ). Larger SE leads to smaller t-statistics and less significant results.

Q4: What affects the size of standard errors?
A: SE increases with residual variance (σ²) and decreases with sample size and predictor variability. Correlated predictors also increase SE.

Q5: Can this be used for weighted least squares?
A: For WLS, the formula changes to SE = √diag(σ²(XᵀWX)⁻¹) where W is the weight matrix.