What does a higher value of the penalty parameter in ridge regression achieve?

A higher value of the penalty parameter in ridge regression specifically decreases model complexity. This is achieved by adding a penalty term to the loss function, which is proportional to the square of the magnitude of the coefficients. As the penalty parameter increases, it discourages large coefficient values, effectively shrinking the coefficients of less significant predictors towards zero.

By doing this, the model becomes less sensitive to noise in the training data, which can often lead to overfitting if an overly complex model is used. By constraining the coefficients, ridge regression maintains a balance between fitting the training data well and ensuring that the model remains generalizable to new, unseen data. This is particularly important when dealing with multicollinearity, as it helps to stabilize the estimates of coefficients that would otherwise be highly variable.

This concept is essential in risk modeling because it enables the creation of robust predictive models even in the presence of multicollinearity among the predictors.