17-ProblemSet8.Rmd

---
title: "Problem Set 8"
output:
  pdf_document: default
  html_document: default
---
# Ridge Regression

\noindent Consider the following data generating process with $n=500$ observations and $p$ covariates. Initially set the number of predictors $p=10$ and  $\mathbf{X} \sim \mathcal{N}_p(\mathbf{0},\boldsymbol{\Sigma})$. $\boldsymbol{\Sigma}$ is diagonal and contains values ranging from 1 to 10 in some sequence. The true coefficients range from $\boldsymbol{\beta}=$ $0-1$ (you can sample values from that range or use equi-spaced values on that interval) and the errors are drawn from a normal distribution $\boldsymbol{\epsilon} \sim \mathcal{N}(0,1).$
  
The aim of this exercise: compare the predictions generated by OLS and Ridge regression.   
  
- Implement the ridge regression estimator using glmnet for a grid of different penalty parameters. 
- Draw a test data set and plot the test error for the different values of $\lambda$.   
- Choose the optimal $\lambda$ using the build in cross-validation function from glmnet and calculate the mean squared test error drawing a new test data set and compare Ridge regression and OLS.



```{r ridgle, include=T, error=TRUE,eval=F}
###For calculating the ridge regression coefficients you need to install
###the glmnet package and call the library#############################
library(glmnet)
#####For calculating the prediction error you can use the predict() function##
````