Regression coefficients are key components of regression analysis, and they provide important information about the relationship between independent and dependent variables in a regression model. Understanding the properties of regression coefficients is essential for interpreting the results of regression analysis accurately. Here are the main properties of regression coefficients:

1. Interpretability:
   • Regression coefficients are interpretable in the context of the specific variables they represent. For example, in simple linear regression, the coefficient represents the change in the dependent variable for a one-unit change in the independent variable.
2. Direction:
   • The sign (positive or negative) of the coefficient indicates the direction of the relationship between the independent variable and the dependent variable.
     • A positive coefficient suggests a positive relationship: As the independent variable increases, the dependent variable also increases.
     • A negative coefficient suggests a negative relationship: As the independent variable increases, the dependent variable decreases.
3. Magnitude (Absolute Value):
   • The absolute value of the coefficient quantifies the strength of the relationship. Larger absolute values indicate a stronger effect of the independent variable on the dependent variable.
4. Units:
   • The coefficient’s units depend on the units of the independent and dependent variables. For example, if the independent variable is measured in dollars and the dependent variable in units sold, the coefficient represents the change in units sold per dollar change in the independent variable.
5. Linearity Assumption:
   • Regression coefficients assume a linear relationship between the independent variable(s) and the dependent variable. They measure the change in the dependent variable for a one-unit change in the independent variable, assuming the relationship is linear.
6. Independence:
   • In multiple regression, each coefficient represents the change in the dependent variable when the corresponding independent variable changes while holding all other independent variables constant. This assumes that the independent variables are not highly correlated with each other (no multicollinearity).
7. Ordinary Least Squares (OLS) Property:
   • In the context of OLS regression, the coefficients are estimated to minimize the sum of squared differences between the observed and predicted values of the dependent variable. These estimates provide the “best-fitting” linear relationship.
8. Hypothesis Testing:
   • Hypothesis tests, such as t-tests, can be conducted to determine whether the estimated coefficients are statistically significant. A significant coefficient implies that the independent variable has a significant effect on the dependent variable.
9. Confidence Intervals:
   • Confidence intervals can be constructed around the estimated coefficients, providing a range of values within which the true population coefficient is likely to lie with a certain level of confidence.
10. R-squared (R²):
    • R-squared measures the proportion of variance in the dependent variable explained by the independent variable(s). Higher R-squared values indicate that the independent variable(s) explain a larger portion of the variation in the dependent variable.
11. Residuals and Error Term:
    • The coefficients relate the independent variable(s) to the mean of the dependent variable, while the residuals (the differences between observed and predicted values) represent the random error or unexplained variation in the dependent variable.
12. Assumption of Causality:
    • While regression coefficients measure associations, they do not establish causality. Establishing causality often requires further experimental or causal inference methods.

Understanding these properties helps analysts and researchers make informed interpretations of regression results, assess the strength and direction of relationships, and evaluate the significance of independent variables in explaining variations in the dependent variable.