How do you handle multicollinearity and heteroscedasticity in linear regression?

Multicollinearity is also highly relevant to equity concerns, as is a source of bias and is of particularly important in public-facing machine learning models. For example, a (hypothetical) model that determines credit-worthiness based on whether someone has held a managerial position would be remiss not to adjust for gender, as women tend to hold management positions less frequently than men, thus creating an inverse relationship between being a woman and being a manager. Overlooking this could disqualify otherwise worthy candidates from an important service.

Example, In a study of the factors affecting home prices, square footage and number of bedrooms are two commonly included variables in regression models. However, these variables can often be highly correlated, as larger homes typically have more bedrooms. This means that it may be difficult to disentangle the separate effects of each variable on home prices. One way to address multicollinearity in this case is to either remove one of the highly correlated variables (ex, number of beedrooms) or to use a different variable that is less correlated (ex, number of bathrooms) with the remaining variable.

By using machine learnining methods such as lasso or ridge regresssion you can deal with this problems. For multicolindarity rige and lasso are two statistical learning especially weighted lasso

Multicollinearity can be detected through 1. correlation matrix 2. variance inflation factor (VIF) 3. eigenvalues/condition number 4. tolerance 5. regression coefficient signs/magnitudes. No single method is sufficient; a combination is needed. Addressing multicollinearity may involve removing variables or using advanced techniques like PCA or PLS regression.

In my discipline, there are physical reasons why measurements are correlated. It is known before processing of any type that the data have cross correlations, some higher than others.

Multicollinearity can be handled by removing one or more of the highly correlated variables or by using advanced statistical techniques like principal component analysis (PCA) or partial least squares (PLS) regression. For example, in a study of employee salary, if both "years of experience" and "age" are highly correlated, one of these variables could be removed from the regression model. Alternatively, a new variable could be created that combines the information from both variables, such as "years of experience adjusted for age," which would reduce the multicollinearity problem.

Usually, we employ PCA. In some cases, taking the average of correlated variables. It depends on the physics of each measurement.

One of several stepwise regression procedures can mitigate multicollinarity by iteratively adding/removing predictors. Forward regression starts with an empty model and adds the predictor with the highest correlation with the DV. Subsequent predictors are added IFF they account for significant variance of the residualized DV score after each iteration. Because variance accounted for by a predictor is removed from the residualized DV, redundant predictors will fail to account for significant unique variance and will be excluded from the model.

For example, in a study of income and expenditure, the variance of the residuals may be larger for high-income earners than for low-income earners, indicating heteroscedasticity. This violates the assumption of homoscedasticity, which assumes constant variance across all levels of the independent variables.

For example, if we are studying the relationship between the temperature and sales of ice cream, and the Breusch-Pagan test shows a significant relationship between the squared residuals and temperature, then we may have heteroscedasticity. To address heteroscedasticity, one approach is to transform the data, such as taking the logarithm of the dependent variable or the independent variable, to make the variances more consistent across different levels of the independent variable. Another approach is to use weighted least squares, which gives more weight to observations with smaller variances. Finally, we can use robust standard errors, which adjust the standard errors of the estimates to account for heteroscedasticity.

Unfortunately, in my discipline heteroscedasticity is an unknown term and therefore completely ignored. Which is a bit dangerous since high dollar decisions are based in part on processing results output from my discipline.

I think that Priya's suggestions are helpful, but it's also important to recognize when forcing a model to fit does more harm than good. My example above, where women and leadership positions show negative collinearity, also shows a tendency that could be prone to heteroscedasticity as this relationship may change, weaken, or strengthen depending on the generation. This could cause differences in variance within the model, depending on the age of the applicant. Correcting strategies could be applied, but the model could also be split into smaller generational models, conditionally activated when someone inputs their age. This would probably be most accurate and would avoid the risk of throwing off other relationships in the model.

random effect model or mixed model can reduce the large variability in the model. You should a term to the model. This term shows the variability among the residuals