3. Consider simple linear regression model yi = Bo + B12; + &; and B. parameter...
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...
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
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...
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.
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
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...
6. This problem considers the simple linear regression model, that is, a model with a single covariate r that has a linear relationship with a response y. This simple linear regression model is y = Bo + Bix +, where Bo and Bi are unknown constants, and a random error has normal distribution with mean 0 and unknown variance o' The covariate a is often controlled by data analyst and measured with negligible error, while y is a random variable....
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
7. In a simple regression model, suppose all of the assumptions of the classical linear regression morel apply, except that rather than assume E (ui | Xi) = 0, you assume that E (Ui / X;) = ali and E (xi) = 0 where a > 0 is a constant. (a) What is the conditional expectation of the OLS slope coefficient, i.e. E (B1 | 21, ..., XN)? (b) In this case, is ß1 an unbiased estimator of B1 or...
Consider the following slope estimator: b=2i=1 Yi Suppose the true model is ki + Bo + Bicite and the model satisfies the Gauss-Markov conditions. Answer the following questions: (a) What assumption in addition to the Gauss-Markov assumptions is required to estimate the model? (b) Show that in general, b is a biased estimator of B1. (c) Outline the special condition(s) under which b is an unbiased estimator of B1.