2 added 150 characters in body edited Mar 12 '18 at 21:23 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 $$MSE(\widehat{\beta_{\lambda}}) < MSE(\widehat{\beta}^{OLS})$$ 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  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 $$MSE(\widehat{\beta_{\lambda}}) < MSE(\widehat{\beta}^{OLS})$$ 1 answered Mar 12 '18 at 16:52 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