STATISTICAL AND MACHINE LEARNING ECON576 Problem Set 2 1. (2 points) Consider a special linear regression model in which each observation has its own slope coefficient: yi = XiBitei, , xi70, i = 1, . ..,n. What is the least squares estimator of B4? 2. (5 points) Consider the following linear regression model with only one covariate: yi = CiB tei, i = 1, ..., n. One popular shrinkage estimator of B is the ridge estimator, defined as: Bridge = argmin (1) B i=1 where > > 0 is a penalty term. That is, Bridge penalizes any deviation from 0. (a) Find the derivative of the objective function with respect to B. b) Derive an explicit formula for Bridge. (c) What happens if ) = 0? What happens if 1 = co?3. (8 points) This exercise uses gradient descent to solve the ridge minimization prob- lem in (1). Specifically, let f(B) denote the objective function in (1), i.e., n f ( B) = _(yi - IiB)2 + 182. i= 1 Next, let Vf(B) denote the gradient of f, i.e., Vf(B) = df/dB. Then, given a positive learning rate o > 0 and an initial value Bo, the gradient descent iteratively updates Bi+1 using: Bit1 = Be - OVf(B)| B=BE' The iteration stops until the successive values of Be+1 remain about the same, say, when It+1 - Bel 0. Now suppose we have the data 2.5 5.8 6 3.7 X = y = -2 -1.1 -3.7 -5Use gradient descent to find the ridge estimate with A = 10. Use the initial value Bo =0, a =0.001 and = 10~*. What is the ridge estimate? Plot the values of the intermediate [3;. 4. (5 points) This question involves the Auto dataset, which is included in the package ISLR2. (a) Fit a multiple linear regression with mpg as the response and all other variables except name as predictors. Use the summary() to print the results. Is there a relationship between the predictors and the response? For which of the predictors can you reject the null hypothesis Hy : 3; = 07 If I obtain a p-value of 0.03 for the null hypothesis Hy : 3; = 0, is it correct to interpret it as: given our sample, ; = 0 with probability 0.037 Explain