An important assumption we make with multiple regression that of homoskedasticity. This means:
a. the population regression line passes through the conditional means.
b. the errors have a constant variance.
c. the residuals are normally distributed.
d. Th errors are independent of one another.
e. None of the above.
The correct answer and explanation is:
The correct answer is b. the errors have a constant variance.
In multiple regression analysis, homoskedasticity refers to the assumption that the variance of the errors (or residuals) is constant across all levels of the independent variables. This is important because if the variance of the residuals changes at different values of the independent variables (a condition known as heteroskedasticity), it can lead to inefficient estimates of the regression coefficients, biased statistical tests, and invalid inferences.
When homoskedasticity holds, the spread of the residuals remains roughly the same across all fitted values of the dependent variable. This means that no matter what value the independent variables take, the residuals will not show patterns of increasing or decreasing variability. If the residuals display this kind of non-constant variance, it signals a problem in the model and may require corrective measures such as transforming the data or using weighted least squares regression.
The assumption of homoskedasticity is one of the classical assumptions in ordinary least squares (OLS) regression. If violated, it can undermine the efficiency of the regression estimates, even though they may remain unbiased. More importantly, it affects the standard errors of the regression coefficients, leading to misleading t-tests and confidence intervals.
There are several ways to check for homoskedasticity. One common method is to create a scatter plot of the residuals against the fitted values. If the plot shows a random scatter without any clear patterns or trends, it suggests that homoskedasticity holds. Alternatively, statistical tests such as Breusch-Pagan or White’s test can be used to formally assess the presence of heteroskedasticity.