Basics of Multiple Regression and Underlying Assumptions

Correct Answer: The error term is normally distributed with a mean of zero and constant variance.

The assumption about the distribution of the error term in a multiple regression model is that the error term is normally distributed with a mean of zero and constant variance. This assumption is important because it ensures that the estimates of the regression coefficients are unbiased and efficient. If the error term does not have a constant variance (i.e., heteroskedasticity), the standard errors of the coefficients may be biased, leading to incorrect inferences. The other two options are incorrect because: - A variance that increases as the value of the independent variables increases violates the assumption of constant variance (homoskedasticity). - A uniform distribution of the error term is not an assumption of the multiple regression model. The assumption is that the error term follows a normal distribution.

Correct Answer: The error term is uncorrelated with each independent variable.

The essential assumption is that the error term is uncorrelated with each independent variable. This assumption is crucial for the multiple regression model to produce unbiased and consistent estimates of the regression coefficients. If the error term is correlated with any of the independent variables, it would violate the assumption and lead to biased and inconsistent estimates. This is known as endogeneity, which can arise due to omitted variables, measurement errors, or simultaneity. The other two options are incorrect: - The error term should have a mean of zero, not a non-zero mean that is correlated with the independent variables. - The error term should have a constant variance (homoscedasticity), but this variance should not be dependent on the values of the independent variables. If the variance is dependent on the independent variables, it would indicate heteroscedasticity, which violates another assumption of the multiple regression model.

Correct Answer: The proportion of the variance in the dependent variable that is predictable from the independent variable(s)

The coefficient of determination (R-squared) in a multiple regression model measures the proportion of the variance in the dependent variable that can be explained by the independent variable(s). In other words, it indicates how well the regression model fits the observed data. The second answer option is incorrect because it describes the interpretation of the individual regression coefficients, not the overall R-squared value. The third answer option is partially correct, as R-squared does provide an indication of the goodness of fit of the model. However, it does not directly measure the statistical significance of the overall regression model, which is typically assessed using an F-test.