Why Do Residuals Need To Be Homoscedastic (Equal Variance)? I am MBA student and am taking a course on "Research Methods" - in this class, we are learning how to perform basic statistical analysis such as Hypothesis Testing and Regression Models.
Our prof is showing us how to use SAS software and fit a linear regression model to some data. For example, we are predicting how much money a company will earn based on the number of sales that they had.
Prior to finishing the regression analysis, we apparently have to check to make sure that certain assumptions are met. Our prof showed us this page over here which contains these assumptions: https://sphweb.bumc.bu.edu/otlt/MPH-Modules/BS/R/R5_Correlation-Regression/R5_Correlation-Regression4.html .
I can indirectly understand why these assumptions are necessary, but our prof isn't really capable to explaining the consequences of fitting regression models when these assumptions are not satisfied. For example, our prof simulated a fake data set in which some of the assumptions are not met. Then he fit a regression model to this data and showed how poor the performance was in hopes that this would convince us about the importance of these assumptions. While I do accept his authority on this matter, I can't help but wonder if this example that he showed us was a "one off" and not necessarily the general case (e.g. some datasets might not meet these assumptions yet still produce well-performing regression models).
For instance, suppose we considered the example of the "heteroscedastic" assumption : the residuals of the linear regression model MUST have equal variance (homoscedastic) - what happens if this is NOT the case? Is it possible to somehow prove that when the "heteroscedastic" assumption is not satisfied, this will statistically provide worse regression model in similar circumstances had the residuals satisfied the "heteroscedastic" assumption? Maybe the "heteroscedastic" assumption results in the regression coefficients having wider confidence intervals?

The assumptions of residual normality and homoscedaticity are used in the theoretical derivation of the Mean Squared Error. This loss is actually a scaled negative logarithmic likelihood for the following regression model:
$$y = f(\theta, x) + \epsilon$$
where $\epsilon$ is a zero mean normal random variable, independent of $x$.
If we use Mean Squared Error in our optimization task and the residuals turn out to be heteroscedatic, that means that our model was incorrect from the beginning.
If we want to retain residual normality, but capture heteroscedaticity, we can consider a more complex parametric model:
$$y = f(\theta, x) + g(\theta, x)\epsilon$$
where $\epsilon \sim N(0, 1)$ is independent of $x$.
The loss function to be optimized here will be different:
$$\sum_{i = 1}^N \ln(g(\theta, x_i)) + (\frac{y_i - f(\theta, x_i)}{g(\theta, x_i)})^2$$
