Question 4 Consider the linear regression model 1. For estimates βί, 「= 1,.. . , n,...
5) Consider the simple linear regression model N(0, o2) i = 1,...,n Let g be the mean of the yi, and let â and ß be the MLES of a and B, respectively. Let yi = â-+ Bxi be the fitted values, and let e; = yi -yi be the residuals a) What is Cov(j, B) b) What is Cov(â, ß) c) Show that 1 ei = 0 d) Show that _1 x;e; = 0 e) Show that 1iei =...
Consider the least-squares residuals ei-yi-yi, 1, 2, . . . , linear regression model. Find the variance of the residuals Var(e). Is the vari- ance of the residuals a constant? Discuss. n,from the simple
.,n, 1.4. Show that in a linear regression model yt- B1t2, t-1, the squaredmultiple correlation coefficient based on the least squares estimates βί, β2 and Ut : Ảxt +A is necessarily between zero and one with R1 if and only if yt,t- 0,... , n (see L.12)) R2-1
Problem 1 Consider the simple linear regression model Ya-Rit Bix, + εί. Prove that when are the LS estimates the following holds: and βί im1 i-l
1. Consider the simple linear regression model: Ү, — Во + B а; + Ei, where 1, . . , En are i.i.d. N(0,02), for i1,2,... ,n. Let b1 = s^y/8r and bo = Y - b1 t be the least squared estimators of B1 and Bo, respectively. We showed in class, that N(B; 02/) Y~N(BoB1 T;o2/n) and bi ~ are uncorrelated, i.e. o{Y;b} We also showed in class that bi and Y 0. = (a) Show that bo is...
3. Consider the multiple linear regression model iid where Xi, . . . ,Xp-1 ,i are observed covariate values for observation i, and Ei ~N(0,ơ2) (a) What is the interpretation of B1 in this model? (b) Write the matrix form of the model. Label the response vector, design matrix, coefficient vector, and error vector, and specify the dimensions and elements for each. (c) Write the likelihood, log-likelihood, and in matrix form. aB (d) Solve : 0 for β, the MLE...
Exercise 2b please!
Exercise 1 Consider the regression model through the origin y.-β1zi-ci, where Ei ~ N(0,o). It is assumed that the regression line passes through the origin (0, 0) that for this model a: T N, is an unbiased estimator of o2. a. Show d. Show that (n-D2 ~X2-1, where se is the unbiased estimator of σ2 from question (a). Exercise2 Refer to exercise 1 a. Show that is BLUE (best linear unbiased estimator) b. Show that +1 has...
In the multiple linear regression model with estimation by ordinary least squares, why must we make an analysis of the scatter plot indices 1, 2,. . . , n and with the residuals ei for observations that are somehow ordered (for example, in time)? And what is the purpose of analyzing the sample autocorrelation function?
Please solve the question
Simulation: Assume the simple linear regression model i = 1,... , n Ул 3D Во + B1; + ei, N(0, o2) for i = 1,...,n. where e Let's set Bo = 10, B1 = -2.5, and n = 30 (a) Set a = 100, and x; = i for i = 1,...,n. (b) Your simulation will have 10,000 iterations. Before you start your iterations, set a random seed using your birthday date (MMDD) and report the...
Consider the simple linear regression model: Yi = Bo + Bilitei, i = 1,...,n. with the least squares estimates ỘT = (Bo ß1). We observe a new value of the predictor: x] = (1 xo). Show that the expression for the 100(1 - a)% prediction interval reduces to the following: . (xo – x2 Ēo + @130 Etap 11+ntan (x; – 7)2