**Understanding Linear Regression and Stochastic Gradient Descent**Linear regression aims to fit the parameters \(\hat{\boldsymbol{\theta}}\) based on the training set:\[ D = \{ (\vec{x}^{(i)}, y^{(i)}), i = 1, 2, \ldots, m \} \]The goal is to ensure that the hypothesis function:\[ h_{\theta}(\vec{x}) = \hat{\boldsymbol{\theta}}^T \cdot \vec{x} = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \ldots + \theta_n x_n \]can accurately predict the output \( y \) for a new input vector \(\vec{x}\).For effective parameter fitting, we derive the stochastic gradient descent update rule. This rule helps in updating \(\hat{\boldsymbol{\theta}}\) iteratively to minimize the least squares cost function \( J(\hat{\boldsymbol{\theta}}) \).

Given that, D=x⇀i,y⇀i, i=1,2,...,m The hypothesis function is given as: hθx⇀=θ⇀T·x⇀…

Answered: Linear regression aims to fit the…

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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Heteroscedasticity Stigler and Friedland (1983) conducted a study to determine whether the separation of company control from company ownership affects company profits. Using data from 69 companies in the United States, the authors estimate the following model: profiti = α + β1asseti + β2management_controli + ei Where i is company and: profiti = annual profit (in million dollar) asseti = company asset (in million dollar) management_controli = dummy variable that is worth one if the control of the company is held by the manager The regression results are presented in the table in the picture a. Explain whether the statements below are TRUE, FALSE, or CANNOT BE DETERMINED. "If there is a heteroscedasticity problem, the confidence interval of the OLS estimator is not valid." b. Determine the 95% confidence interval for the parameter 2, what can you conclude?
Detail all the steps involved in testing the hypothesis below for the linear regression model y = XB + e, where X = (50 x 6) for two cases. Ho: X3B3 + x4B4 = 0 На: x3B3 + x4B4 # 0
Consider a polynomial regression model Y = Bo + BrX + B2X +...+BX + Ui, where E (u₁|X₁, X²,..., X) = 0, observations (Y₁, X;) are independent and identically distributed and 0 < E (Y₂¹) < ∞, 0 < E (X¹¹) < ∞. How is the OLS estimation affected if we estimate a linear regression 1) when the true form of the regression function is quadratic (r = 2) or cubic (r = 3)? (r =
3
Please answer question (c)
Consider the regression model y = a + ßx+u, where x is an endogenous variable. a) Write x₁ = x + ê; and show that = x. b) Suppose there are k valid instruments: Z1, Z2, ..., Zk. Let x₁= πo+f1zil + π2Zi2+...+îkzik. Show that the 2SLS estimator is equivalent to an IV estimator using as an instrument: Σ₁₁(&i- Ñ)(vi- V) _ Σ₁(Ri—Ñ)(Yi-Y) Ei=1(xi-x)² 21(®i- Đ)(xi-x) = [Hint: OLS first order conditions imply №₁ ê¡Â¡ = 0 and Σ¼_₁ ê¡= 0.] c) Suppose there is only one instrument z and ✰; = fo+ fizi. Show that the 2SLS estimator is numerically identical to the IV estimator: = Σ₁ (Îi—Ñ)(vi-V) – Σ₁(²₁-7)(vi-V) Σ=1(ât− x)2 Σ₁ (Z₁-7)(xi-x)* n [Hint: Use the result in (b) and plug in â; and ☎ = ñîo+â¦Z.]
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6. Suppose we estimate a linear regression equation Y; = Bo + B1X; + u; by OLS. (a) Show thatE ûi 0, where the û;'s are the regression residuals. i3D1 (b) Suppose we regress X; on û; by OLS, including a constant term in the regression. Show that the estimated coefficient on û; is equal to zero. (c) Suppose we regress Y; on the predicted value Y; by OLS, including a constant term in the regression. Show that the estimated coefficient on Y; is equal to one. (Hint: the regression residual is defined by û; = Y; – Y; where the predicted value Y; = Bo + B,X;. Here, Bo and B1 are the OLS estimators.)
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