In a multiple regression model, what is assumed about the variance of the error term, ε?

In a multiple regression model, one of the fundamental assumptions is that the variance of the error term, denoted as ε, remains constant across all levels of the independent variables (x1, x2, ..., xp). This condition is known as homoscedasticity. Assuming that the variance of the error term is consistent allows the regression model to provide reliable estimates of the coefficients and accurate predictions.

When this assumption holds true, it means that the spread or dispersion of the residuals (the differences between observed and predicted values) does not change regardless of the values of the independent variables. This is important for conducting valid statistical inference, including hypothesis testing and constructing confidence intervals. If the variance were to change with different values of the predictors (known as heteroscedasticity), it could lead to inefficient estimates and incorrect conclusions regarding the significance of the predictors.

The other options suggest various forms of variability in the error term that are not characteristic of the standard assumptions in regression analysis, which is why they do not align with the correct understanding of the model's assumptions.