When perform multiple regression with OLS instead of ridge regression? I have read a lot of advantages of regularisation methods (e.g. ridge, lasso), but in what cases might multiple regression with Ordinary Least Squares be more desirable? For example is there a downside to using ridge regression for n >> p problems (where n is the number of samples and p is the number of features), or when there is little multicollinearity between predictor variables? If not then I'm wondering what OLS is good for (except in cases where there's only one predictor variable of course). 
 A: Ridge regression helps multicollinearity and overfitting. But sometimes you just don't care about overfitting or multicollinearity:


*

*Your want unbiased coefficients and you don't care about their standard errors because you don't want to do any statistical test

*Your don't need to do predictive modelling - you don't have a test set anyway

*You don't have that many predictors 


Ridge regression might prevent you understanding the details in your data set.
A: The downside of ridge regression is that the parameter estimates are biased.  When the variances of the expectations aren't large, then unbiasedness is a lot to give up. 
A: I hope this helps :
> x1 <- rnorm(30)
> x2<- rnorm(30, mean = x1, sd=0.01)
> y <- rnorm(30, mean = 5+x1+x2)
> cor(x1,x2)
[1] 0.9999316
> fit <- lm(y~x1+x2)$coeff;fit
(Intercept)          x1          x2 
   5.175611    3.859883   -1.452410 
   > 
   > 
   > library(MASS) 
   > lm.ridge(y~x1+x2,lambda = 1)
           x1       x2 
5.171224 1.187841 1.174328 
   > 
   > 
   > 
   > 
   > x1 <- rnorm(30)
   > x2<- runif(30, min = 0, max = 1)
   > cor(x1,x2)
[1] 0.3348988
   > fit <- lm(y~x1+x2)$coeff;fit
(Intercept)          x1          x2 
  5.0691119   0.2017907  -0.6022160 
> lm.ridge(y~x1+x2,lambda = 1)
               x1         x2 
  5.0507017  0.1928616 -0.5704941  

Also, there is a theorem stated that " There always exists a value $\lambda$ such that $$MSE(\widehat{\beta_{\lambda}}) < MSE(\widehat{\beta}^{OLS})$$
