Consider the simple linear regression model: Yi = Bo + Bilitei, i = 1,...,n. with the...
A simple linear regression model is given as follows Yi = Bo + B1Xi+ €i, for i = 1, ...,n, where are i.i.d. following N (0, o2) distribution. It is known that x4 n, and x = 0, otherwise. Denote by n2 = n - ni, Ji = 1 yi, and j2 = 1 1. for i = 1, ... ,n1 < n2 Lizn1+1 Yi. n1 Zi=1 1. Find the least squares estimators of Bo and 31, in terms of...
4. In the simple linear regression model yi = Bo + B, 21 +, a. Bcannot be estimated without first assuming (EU) = 0 b. B, could represent the average marginal association between 2 and y or the average effect of x on y c. we can directly observe e d. the B, estimate is unbiased only if E(€) = 0 e. None of the above
1. Consider the simple linear regression model where Bo is known. a) Find the least squares estimator bi of B (b) Is this estimator unbiased? Prove your result. (c) Find an expression for Var(b1x1, ,xn) in terms of x1, ,xn and σ2.
2. Consider the simple linear regression model: where e1, .. . , es, are i.i.d. N (0, o2), for i= 1,2,... , n. Suppose that we would like to estimate the mean response at x = x*, that is we want to estimate lyx=* = Bo + B1 x*. The least squares estimator for /uyx* is = bo bi x*, where bo, b1 are the least squares estimators for Bo, Bi. ayx= (a) Show that the least squares estimator for...
3. Consider simple linear regression model yi = Bo + B12; + &; and B. parameter estimate of the slope coefficient Bi: Find the expectation and variance of 31. Is parameter estimate B1 a) unbiased? b) linear on y? c) effective optimal in terms of variance)? What will be your answers if you know that there is no intercept coefficient in your model?
Problem 7. Consider the simple linear regression model Y1 = Bo + BiX; +€; for i=1,2,...,n where the errors Eį are uncorrelated, have mean zero and common variance Varſei] = 02. Suppose that the Xį are in centimeters and we want to write the model in inches. If one centimeter = c inch with c known, we can write the above model as Yį = y +71 Zitki where Zi is Xi converted to inches. Can you obtain the least-squared...
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
Consider the simple linear regression model where Bo is known. (a) Find the least squares estimator bi of β1- (b) Is this estimator unbiased? Prove your result
Exercise 5 Consider a linear model with n = 2m in which Yi = Bo + Bici + Eigi = 1,..., m, and Yi = Bo + B2X1 + Ei, i = m + 1, ...,n. Here €1,..., En are i.i.d. from N(0,0), B = (Bo, B1, B2)' and o2 are unknown parameters, X1, ..., Xn are known constants with X1 + ... + Xm = Xm+1 + ... + Xn = 0. 1. Write the model in vector form...
(Do this problem without using R) Consider the simple linear regression model y =β0 + β1x + ε, where the errors are independent and normally distributed, with mean zero and constant variance σ2. Suppose we observe 4 observations x = (1, 1, −1, −1) and y = (5, 3, 4, 0). (a) Fit the simple linear regression model to this data and report the fitted regression line. (b) Carry out a test of hypotheses using α = 0.05 to determine...