Suppose thatY = Bo + BıXite, i-1,..., nwhere E(€;) = 0 and Var(e;) = o². Let 3o and 3₁ be the least squares estimates.(a) Find Var(Bo) (treating the X's as fixed).(b) The observed residuals are defined by ê; = Y; – Bo – Â₁ Xį. Show that Σ₁ ê; = 0.i=1(c) Show that ₁ ê¡¿Ÿ¿ = 0. What is the interpretation of this result?(d) Suppose you did a regression of Y's on the observed residuals. (In other words,treat the ê¿ as the X;'s.) What intercept and slope would you get?

Given the ith simple linear regression model asYi=β0+β1∗Xi+ϵi, i=1,...,n.Here Yi=ith dependent…

Answered: Suppose that Yi = Bo + B₁X₁ + €į, i =…

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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Each of 25 teenage girls with one brother was asked to provide her own height (y), in inches, and the height (x), in inches, of her brother. The scatterplot below displays the results. Only 22 of the 25 pairs are distinguishable because some of the (x,y) pairs were the same. The equation of the least-squares regression line is ŷ = 35.1 + 0.427x.*Girls are on the y-axis and brothers are on the x-axis* a.) Draw the least-squares regression line on the scatterplot above. b.) One brother’s height was x = 67 inches and his sister’s height was y = 61 inches. Circle the point on the scatterplot above that represents this pair and draw the segment on the scatterplot that corresponds to the residual for it. Give a numerical for the residual. c.) Suppose the point x = 84 , y = 71 is added to the data set. Would the slope of the least squares regression line increase, decrease, or remain about the same?Explain. Would the correlation increase, decrease, or remain about the same? Explain.
Imagine you have two estimators X and Y, for unknown parameter µ. Assume that they are estimated from independent datasets. Let Z = }(X+Y) be the average of these two estimators. To write the symbol µ, you can write mu. For other math notation, you can E[X], Var(X) and (X+Y)/2 or (1/2)(X+Y). Part a - If X and Y are unbiased estimators, E[X] = E[Y] = µ, then is Z unbiased? Explain why or why not. Your explanation should include a short derivation. Part bL Assume Var(X) = Var(Y). Is Z a lower variance or higher variance estimator than X? Explain your answer. Your explanation should include a short derivation.
Consider the linear model y=B,+B,x,+B,x,,+B,x+B,+u, You estimate the model y =B,+B,x,+B,x, observations and obtain the OLS residuals . You then estimate the auxiliary regression +u based on 123 31 3" 3i 41 The LM statistic +1 4/ you obtain to test the null hypothesis that H:B,=B,=0 is 20.91. What is the R2 of the auxiliary regression? It is not possible to say O 0.17 O 0.175714 0.177203
Suppose Y₁,..., Yn are independent random variables each with the Pareto distribution and E(Y) = (Bo + B1Xi) Is this a generalized linear model with a canonical link? Give reasons for your answer.
8) Consider the data (1,1), (3,9), (4,16), If f[1,3] = 4, f[3,4] = z, f[1,3,4] 1. find z %3D
Consider a set of data (x;, y;), i = 1,2, .,n and an intercept model given by: 1. Yi = Bo + Ei with E(s;) = 0, Var(ɛ;) = o? and all ɛ,'s are independent and normally distributed. Derive the standard error of the point estimator Bo.
Consider the following population model for household consumption: cons = a + b1 * inc+ b2 * educ+ b3 * hhsize + u where cons is consumption, inc is income, educ is the education level of household head, hhsize is the size of a household. Suppose a researcher estimates the model and gets the predicted value, cons_hat, and then runs a regression of cons_hat on educ, inc, and hhsize. Which of the following choice is correct and please explain why. A) be certain that R^2 = 1 B) be certain that R^2 = 0 C) be certain that R^2 is less than 1 but greater than 0. D) not be certain
Civil engineers often use the straight-line equation, y = bo+b ₁x, to model the relationship between the shear strength y of masonry joints and precompression stress, x. To test this theory, a series of stress tests were performed on solid bricks arranged in triplets and joined with mortar. The precompression stress was varied for each triplet and the ultimate shear load just before failure (called the shear strength) was recorded. The stress results for n = 7 resulted in a Coefficient of Determination of 0.7589. Given that r ²= 0.7589, give a practical interpretation of r 2, the coefficient of determination for the least squares model. In repeated sampling, approximately 75.89% of all similarly constructed regression lines will accurately predict shear strength. We expect about 75.89% of the observed shear strength values to lie on the least squares line. We expect to predict the shear strength of a triplet test to within about 0.7589 tons of its true value. About 75.89% of the total…
If beta1_hat = 0.4571 use the data below to find beta0_hat for the simple linear model using the method of Least Squares. Y 17 2 3 11 4 20 15 18 13 13.07 50.88 24.57 32.11
Below are bivariate data O each of twelve countries. For each of the countries, both x, the number of births per one thousand people in the population, and y, the female life expectancy (in years), are given. Also shown are the scatter plot for the data and the least-squares regression line. The equation for this line is ing birthrate and life expectancy information for y = 81.87 – 0.46x. Birthrate, x (number of births per 1000 pop.) Female life expectancy, y (in years) 85- 35.7 67.7 80- 41.5 63.9 75 31.9 63.3 19.9 73.0 70 50.5 60.4 65. 24.4 72.7 60- 50.1 63.2 55 13.8 72.5 50 50.3 54.6 45.6 57.9 15.9 76.2 Figure 1 26.6 71.9 Send data to Excel

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