2. Consider the simple regression modely =Bo+BiD + u,where D is a binary variable.a) Let ya be y; with D;= 1, yo be y; with Di= 0, and Σ₁ Yi = ¹₁ ₁₁ + 1 VioD₁Yioswhere N₁ + No = N. Show that y=+No=Dỹ₁+ (1 − Ď)ÿ¸·b) Show that the OLS estimator: B₁-Σ₁(Di-D)(yi - Y)Σ₁=1(Di-D)²[Hint: Note that Diy: = y₁1 and D² = D₁. Therefore B₁ =i=1Σ., Divi - Σ., Dyi - Σ15D, +Σ,53_ Σ 1 vi - 5Σ1 - 52, Di + NDyΣ1 D} - ND?=-2can be written as ₁ - YoΣ₁ Di-ND²i=1

Answered: Image /qna-images/answer/67c077a6-d237-445d-8025-d7bae60c2c19.jpg

Answered: 2. Consider the simple regression model…

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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3. [3] Consider a simple linear regression model for (y, X), Yi = Bo + xiß1+ €;, where the e;'s are iid N(0, o²). Show that T2 = F, where T = VMSres/Saa and MSR MSres are used to test the hypothesis Ho : B1 = 0 (in simple and multip
Given are five observations for two variables, x and y. xi 3 12 6 20 14 yi 55 45 50 15 20 #1) Develop the estimated regression equation by computing the values of b0 and b1 using b1 = Σ(xi − x)(yi − y) Σ(xi − x)2 and b0 = y − b1x. y= #2) Use the estimated regression equation to predict the value of y when x = 13.
Consider the multiple regression model with three independent variables, under the classical linear model assumptions MLR.1 through MLR.6: y = Bo + Bixi + Bzx2 + Bztz + u. You would like to test the null hypothesis H,: B1 – 3B2 = 1. (i) Let Bj and B denote the OLS estimators of ß1 and B,. Find Var(ß – 3B,) in terms of the vari- ances of Bj and Bz and the covariance between them. What is the standard error of Bi – 3B? |(ii) Write the t statistic for testing Họ: B1 – 3B2 = 1. |(iii) Define 0, = Bi – 3B2 and ô, = B1 – 3B. Write a regression equation involving Bo. 01. B2- and Bz that allows you to directly obtain ô, and its standard error.
Consider the data, X, 1 4 Y, 4 8 6 10 12 The estimated regression equation for these data is ý- 2.60 + 1.80x. (a) Compute SSE, SST, and SSR using equations SSE - I(r, - ), sST - E(y, - v), and SSR = E(, - . SSE 7.6 SST =40 SSR = 324 (b) Compute the coefficient of determination ?. Comment on the goodness of fit. (For purposes of this exercise, consider a proportion large if it is at least 0.55.) O The least squares line did not provide a good fit as a large proportion of the variability in y has been explained by the least squares line. O The least squares line did not provide a good fit as a small proportion of the variability in y has been explained by the least squares line. O The least squares line provided a good fit as a small proportion of the variability in y has been explained by the least squares line The least squares line provided a good fit as a large proportion of the variability in y has been explained by the least squares ine. (c) Compute the sample correlation…
4. Consider the regression model Y; = Bo + B1 X; + u;. Suppose that the estimated OLS regression line is Ÿ = 9.72 (there is no mistake in this equation). Which of the following is definitely true? There is insufficient variation in the variable X None of the other options is correct For every i, uį = 0 For every i, û; = 0 The regression R² is zero B would be estimated imprecisely
Consider the linear regression y, = B, + Bx, + u;, i=1,...,n,n+1,..,n+ p where E(u,) = 0. Is it possible to observe a result that E(B,)+ B,, while E(B) = B,-
4. In the estimation of regression equation of two variables X and Y the following results were obtained. X = 90, Y 70, n= 10, Ex -6360; Ey 2860, Exy 3900 Obtain the two regression equations. %3D
Given are five observations for two variables, x and y. xi 1 2 3 4 5 yi 3 8 6 10 13 (d) Develop the estimated regression equation by computing the values of b0 and b1 using b1 = Σ(xi − x)(yi − y) Σ(xi − x)2 and b0 = y − b1x. ŷ = (e) Use the estimated regression equation to predict the value of y when x = 2.
3
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.)
Suppose we have data on dependent variable y and explanatory variables x 1 and x_2, all measured in units. We estimate the following linear regression: Number of cbs = E( 2, 1656) Prob > ? R-squared Adj R-squared 1659 1123.68 0.0000 0.5757 %3D 0.5752 Coef. Std. Err. P>It| x_1 I x_2 1 cons I 1.003119 .68 80123 .2408667 .0252676 .0253665 .077143 39.70 27.12 3.12 0.000 0.000 0.002 Which of the following statements is correct? Select one: a. We can conclude that there is no nonlinear effect of x 2 on y. b. An increase of 1% in x_1 increases y by 0.3119%, when x 2 remains constant. c. The estimated level of y for someone with x_1 equal to 10 and x_2 equal to 8 is 15.535. d. The estimated level of y for someone with x_1 equal to 10 and x 2 equal to 8 is 15.776,

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