Q$1$ This problem is regarding writing your own code for choosing the regularization parameter $$ via cross

validation for a lasso estimator.

a$.$ Simulate a data set as follows

set.seed $(1)$

$n=300$;$p=200$;$s=5$

$x=$ matrix $(rnorm(n*p),n,p)$

$b=c(rep(1,s),$rep$(0,p-s))$

$y=1+x\%*\%b+$rnorm$(n)$

In this synthetic dataset there are $300$ observations, $200$ regression coefficients, of which only $5$ are non$-$zero.

The objective of the following steps is to recover the vector $b$ from the data in $x,$ and $y$

b$.$ Define a grid $L=\{0,$dots,$2\}$ of $100$ numbers between $0$ and $2,$ this grid shall serve as potential values for

the regularizer $.$ Hint: you may want to use the function seq$(),$ with the argument length.out$=100$

c$.$ Use the function glmnet $()$ to obtain lasso estimates for each value in the grid that you define in part

$($b$).$ Using the function coef $(),$ extract the estimated coefficient vector when $=L[10],$ i$.$e$.,$ the ${10}^{th}$

value in the grid $L.$

d$.$ For each value of $$ in the grid $L$ of part $($b$),$ compute the mean squared error on the entire dataset.

$($Use a for$()$ loop for this purpose$).$ This will provide a vector of $100$ values of mse, one for each value

of $.$ Plot $$ vs mse. What do you observe?

e$.$ Using a for$()$ loop. compute the cross vlaidation error, for each value of $$ in the grid $L,$ under a $k=5$

fold cross validation setup. $($you will need to repeatedly divide the data into testing and training, this

will require another for$()$ loop$).$

f$.$ Compile all your code into a custom function with input arguments $x,y,k$ and a grid $L.$ This function

should output the following results, $($i$)$ the best fit model with a k$-$fold cross validation, $($ii$)$ the vector

of cross validation errors $($one for each value of $),($iii$)$ the grid $L$ used for cross validation, $($iv$)$ the

value of lambda at which the best fit model is obtained.

g$.$ Finally, use the function you make in Part $($f$)$ with $k=5,$ then extract the vector of cross validation

errors $($say$,$ CVV$)$ and the grid $L$ that is used. Make a plot of L vs$.$ CVV$.$