# If there any benefit to using ridge regression in a simple linear regression problem where the aim is prediction?

Consider the following situation:

• We have a simple linear regression model (as opposed to a multiple regression model or a polynomial regression model).
• We are interested in prediction rather than inference.
• We have far more observations than predictors.

In this situation is there any benefit to using ridge regression? From what I understand ridge regression allows us to reduce overfitting in complex models at the cost of introducing some bias.

But in the above situation we don't have a complex model; in particular since we are using simple linear regression rather than polynomial regression we are modelling the relationship with a straight line as opposed to a curve.

So can ridge regression still provide us with any benefits in this scenario? Can we still obtain better predictions, by using ridge regression to reduce variance at the cost of introducing bias, when the model is so simple?

but regardless of scenario, as long as the task is prediction, ridge will actively work to reduce overfitting compared to OLS as can be seen in ridge's objective function. The better question is: what is the optimal $$\lambda^*$$ for a given prediction problem, and how different is it from $$\lambda=0$$, which is where ridge collapses to OLS? If you use cross-validation and find out $$\lambda^*$$ is far from $$\lambda=0$$ (the definition of "far" might be tricky for potentially very tiny values for $$\lambda$$, which depends on the units and scale of the data itself), then it will show ridge paid off. Otherwise, if $$\lambda^*\approx 0$$, then ridge might've only provided a small benefit over OLS.