You have a sample of size n = 1 with data y₁ = 2 and x₁ = 1. You are interested in the value of 3 in the regression Y = XB+ u. (Note there is no intercept.) (a) Plot the sum of squared residuals (y₁ - bx₁)² as function of b. You can choose your own range for b, one reasonable choice is b = [-2,5]. (Use any software you prefer. Excel is one option.) (b) Show that the least squares estimate of ß is BOLS = 2. (c) Using Lasso = 1, plot the Lasso penalty term XLasso b as a function of b.
You have a sample of size n = 1 with data y₁ = 2 and x₁ = 1. You are interested in the value of 3 in the regression Y = XB+ u. (Note there is no intercept.) (a) Plot the sum of squared residuals (y₁ - bx₁)² as function of b. You can choose your own range for b, one reasonable choice is b = [-2,5]. (Use any software you prefer. Excel is one option.) (b) Show that the least squares estimate of ß is BOLS = 2. (c) Using Lasso = 1, plot the Lasso penalty term XLasso b as a function of b.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Part f only. It should be one figure with two lines.
![## Problem 3. [SW 14.10]
You have a sample of size \( n = 1 \) with data \( y_1 = 2 \) and \( x_1 = 1 \). You are interested in the value of \( \beta \) in the regression \( Y = X \beta + u \). (Note there is no intercept.)
(a) Plot the sum of squared residuals \( (y_1 - b x_1)^2 \) as a function of \( b \). You can choose your own range for \( b \), one reasonable choice is \( b \in [-2, 5] \). *(Use any software you prefer. Excel is one option.)*
(b) Show that the least squares estimate of \( \beta \) is \( \hat{\beta}^{OLS} = 2 \).
(c) Using \( \lambda_{Lasso} = 1 \), plot the Lasso penalty term \( \lambda_{Lasso} |b| \) as a function of \( b \).
(d) Using \( \lambda_{Lasso} = 1 \), plot the Lasso penalized sum of squared residuals \( (y_1 - b x_1)^2 + \lambda_{Lasso} |b| \). Please put all three lines in one plot.
(e) Find the value of \( \hat{\beta}^{Lasso} \).
(f) Using \( \lambda_{Lasso} = 0.5 \), repeat (c) and (d) (put both lines in one plot). Find the value of \( \hat{\beta}^{Lasso} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61ff295b-afd4-43c3-8ed2-2cc17b9c2249%2Ff2e2ece6-7450-482d-98c5-3e6254b5429c%2Fkyjtkx_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem 3. [SW 14.10]
You have a sample of size \( n = 1 \) with data \( y_1 = 2 \) and \( x_1 = 1 \). You are interested in the value of \( \beta \) in the regression \( Y = X \beta + u \). (Note there is no intercept.)
(a) Plot the sum of squared residuals \( (y_1 - b x_1)^2 \) as a function of \( b \). You can choose your own range for \( b \), one reasonable choice is \( b \in [-2, 5] \). *(Use any software you prefer. Excel is one option.)*
(b) Show that the least squares estimate of \( \beta \) is \( \hat{\beta}^{OLS} = 2 \).
(c) Using \( \lambda_{Lasso} = 1 \), plot the Lasso penalty term \( \lambda_{Lasso} |b| \) as a function of \( b \).
(d) Using \( \lambda_{Lasso} = 1 \), plot the Lasso penalized sum of squared residuals \( (y_1 - b x_1)^2 + \lambda_{Lasso} |b| \). Please put all three lines in one plot.
(e) Find the value of \( \hat{\beta}^{Lasso} \).
(f) Using \( \lambda_{Lasso} = 0.5 \), repeat (c) and (d) (put both lines in one plot). Find the value of \( \hat{\beta}^{Lasso} \).
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