Biases and Standard Errors of Standardized Regression Coefficients.

Area of Science:

• Statistics
• Econometrics

Background:

• Standardized regression coefficients are widely used in statistical analysis.
• Existing formulas for standard errors (SE) and biases may lack general consistency.
• The behavior of these coefficients under different predictor assumptions is crucial.

Purpose of the Study:

• To derive consistent standard errors (SE) and biases for sample standardized regression coefficients.
• To evaluate the accuracy of commonly used SE formulas.
• To assess the practical implications of bias in regression analysis.

Main Methods:

• Analytical derivation of SE and bias for standardized regression coefficients.
• Theoretical analysis of formula consistency under varying population parameters.
• Monte Carlo simulations to compare asymptotic and empirical SE estimates.

Main Results:

• Consistent SE and O(1/n) biases are obtained for standardized regression coefficients.
• Textbook SE formulas are shown to be consistent only when the population regression coefficient is zero.
• Sample standardized regression coefficients exhibit general bias, but it's often negligible in practice with sufficient sample size.
• Asymptotic SE estimates tend to under-predict empirical SEs at smaller sample sizes.

Conclusions:

• The study provides accurate methods for calculating SE and bias in standardized regression.
• Commonly cited SE formulas require careful application, particularly when the population coefficient is non-zero.
• Researchers should be aware of potential under-prediction of SEs in smaller samples.