What does the variance of the error term ε, denoted by σ2, indicate in regression analysis?

In regression analysis, the variance of the error term ε, denoted by σ², represents the spread or dispersion of the errors in the prediction made by the regression model. When we say that the variance of the error term is the same for all values of x, we are referring to the assumption of homoscedasticity. This means that the variance of the errors remains constant across all levels of the independent variable(s).

This assumption is crucial because it allows for valid inference on the regression coefficients. If the variance were to change with different values of x (which would be referred to as heteroscedasticity), it could lead to inefficient estimates and impact the reliability of hypothesis tests conducted on the regression coefficients, such as t-tests or F-tests. Consistency in the variance ensures that the estimated coefficients are robust and can be generalized across the range of data being analyzed.

In summary, the correct answer reflects the fundamental concept that, under the assumption of homoscedasticity in regression, the variance of the error term does not vary with different values of the independent variable(s), maintaining a constant level which is essential for the integrity of the regression analysis.