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2. The linear regression model in matrix format is Y Χβ + e, with the usual definitions Let E(elX...
The linear regression model in matrix format is Y Xe, with the usual definitions. Let E(elX)- 0 a...
The linear regression model in matrix format is Y Xe, with the usual definitions. Let E(elX)- 0 and γ1 0 0 0 Y2 00 01 0 00 .0 0 0 00N 0 0 0'YN 0 0 0YNL Notice that as a covariance matrix, Σ is symmetric and nonnegative definite. ) Derive Var (BoLSX). (ii) Let A: = CY be any other linear unbiased estimator where C, is an N × K function of X. Prove Var (β|X) > (X'Σ-1X)-1.
The...
2. In the regression model Y-Χβ+ E, Xis a fixed n x k matrix of rank...
2. In the regression model Y-Χβ+ E, Xis a fixed n x k matrix of rank k S11, E(c)-0 and E(es')-σ2Ω where Ω is a known non-singular matrix. The GiLS estimator of B is given by the formula Consider the following data 16 31 2 3 51 4 10 Assuming that Ay a) b) Calculate the GLS estimate of β in the model Y,Xß + ε Calculate the OLS estimate c) Compare it the two estimates and comment on efficiency.
Q. 1 Consider the multiple linear regression model Y = x3 + €, where e indep...
Q. 1 Consider the multiple linear regression model Y = x3 + €, where e indep MV N(0,0²V) and V +In is a diagonal matrix. a) Derive the weighted least squares estimator for B, i.e., Owls. b) Show Bwis is an unbiased estimator for B. c) Derive the variances of w ls and the OLS estimator of 8. Is the OLS estimator of still the BLUE? In one sentence, explain why or why not.
1.Given the Multiple Linear regression model as Y-Po + β.X1 + β2X2 + β3Xs + which in matrix notation is written asy-xß +ε where -έ has a N(0,a21) distribution + + ßpXo +ε A. Show that the OLS estimat...
1.Given the Multiple Linear regression model as Y-Po + β.X1 + β2X2 + β3Xs + which in matrix notation is written asy-xß +ε where -έ has a N(0,a21) distribution + + ßpXo +ε A. Show that the OLS estimator of the parameter vector B is given by B. Show that the OLS in A above is an unbiased estimator of β Hint: E(β)-β C. Show that the variance of the estimator is Var(B)-o(Xx)-1 D. What is the distribution o the...
Consider the simple linear regression model y - e, where the errors €1, ,en are iid....
Consider the simple linear regression model y - e, where the errors €1, ,en are iid. random variables with Eki-0, var(G)-σ2, i-1, .. . ,n. Solve either one of the questions below. 1. Let Bi be the least squares estimator for B. Show that B is the best linear unbiased estimator for B1. (Note: you can read the proof in wikipedia, but you cannot use the matrix notation in this proof.) 2. Consider a new loss function Lx(A,%) 71 where...
Problem 3: Absence of Intercept Consider the regression model Y, = BX,+", where , and X,...
Problem 3: Absence of Intercept Consider the regression model Y, = BX,+", where , and X, satisfy Assumptions SLR1-SLR5. Y (i) Let B denote an estimator of B that is constructed as P where Y and X as are the sample means of Y,and X,, respectively. Show that B is conditionally unbiased. Derive the least squares estimator of B. Show that the estimator is conditionally unbiased. Derive the conditional variance of the estimator. (ii) (iii) (iv) 2
2. Let X and Y be jointly Gaussian random variables. Let ElX] = 0, E[Y] =...
2. Let X and Y be jointly Gaussian random variables. Let ElX] = 0, E[Y] = 0, ElX2-4. Ey2- 4, and PXY = [5] (a) Define W2x +3. Find the probability density function fw ( of W. [101 (b) Define Z 2X - 3Y. Find P(Z > 3) 5] (c) Find E[WZ], where W and Z are defined in parts (a) and (b), respectively.
3. Let y = (yi..... Yn) be a set of re- sponses, and consider the linear...
3. Let y = (yi..... Yn) be a set of re- sponses, and consider the linear model y= +E, where u = (1, ..., and e is a vector of zero mean, uncorrelated errors with variance o'. This is a linear model in which the responses have a constant but unknown mean . We will call this model the location model. (a) If we write the location model in the usual form of the linear model y = X 8+...
4. Consider the linear model Y = XB+e, where e MV N(0,021). (1) Derive the formula...
4. Consider the linear model Y = XB+e, where e MV N(0,021). (1) Derive the formula for , the least square estimate of B, using the matrix notation (2) Show that ß is an unbiased estimate for B. (3) Derive the formula for var(), using matrix notation.
Consider the multiple regression model y = X3 + €, with E(€)=0 and var(€)=oʻI. Problem 1...
Consider the multiple regression model y = X3 + €, with E(€)=0 and var(€)=oʻI. Problem 1 Gauss-Mrkov theorem (revisited). We already know that E = B and var() = '(X'X)". Consider now another unbiased estimator of 3, say b = AY. Since we are assuming that b is unbiased we reach the conclusion that AX = I (why?). The Gauss-Markov theorem claims that var(b) - var() is positive semi-definite which asks that we investigate q' var(b) - var() q. Show...
The linear regression model in matrix format is Y Xe, with the usual definitions. Let E(elX)- 0 and γ1 0 0 0 Y2 00 01 0 00 .0 0 0 00N 0 0 0'YN 0 0 0YNL Notice that as a covariance matrix, Σ is symmetric and nonnegative definite. ) Derive Var (BoLSX). (ii) Let A: = CY be any other linear unbiased estimator where C, is an N × K function of X. Prove Var (β|X) > (X'Σ-1X)-1.
The...
2. In the regression model Y-Χβ+ E, Xis a fixed n x k matrix of rank k S11, E(c)-0 and E(es')-σ2Ω where Ω is a known non-singular matrix. The GiLS estimator of B is given by the formula Consider the following data 16 31 2 3 51 4 10 Assuming that Ay a) b) Calculate the GLS estimate of β in the model Y,Xß + ε Calculate the OLS estimate c) Compare it the two estimates and comment on efficiency.
Q. 1 Consider the multiple linear regression model Y = x3 + €, where e indep MV N(0,0²V) and V +In is a diagonal matrix. a) Derive the weighted least squares estimator for B, i.e., Owls. b) Show Bwis is an unbiased estimator for B. c) Derive the variances of w ls and the OLS estimator of 8. Is the OLS estimator of still the BLUE? In one sentence, explain why or why not.
1.Given the Multiple Linear regression model as Y-Po + β.X1 + β2X2 + β3Xs + which in matrix notation is written asy-xß +ε where -έ has a N(0,a21) distribution + + ßpXo +ε A. Show that the OLS estimator of the parameter vector B is given by B. Show that the OLS in A above is an unbiased estimator of β Hint: E(β)-β C. Show that the variance of the estimator is Var(B)-o(Xx)-1 D. What is the distribution o the...
Consider the simple linear regression model y - e, where the errors €1, ,en are iid. random variables with Eki-0, var(G)-σ2, i-1, .. . ,n. Solve either one of the questions below. 1. Let Bi be the least squares estimator for B. Show that B is the best linear unbiased estimator for B1. (Note: you can read the proof in wikipedia, but you cannot use the matrix notation in this proof.) 2. Consider a new loss function Lx(A,%) 71 where...
Problem 3: Absence of Intercept Consider the regression model Y, = BX,+", where , and X, satisfy Assumptions SLR1-SLR5. Y (i) Let B denote an estimator of B that is constructed as P where Y and X as are the sample means of Y,and X,, respectively. Show that B is conditionally unbiased. Derive the least squares estimator of B. Show that the estimator is conditionally unbiased. Derive the conditional variance of the estimator. (ii) (iii) (iv) 2
2. Let X and Y be jointly Gaussian random variables. Let ElX] = 0, E[Y] = 0, ElX2-4. Ey2- 4, and PXY = [5] (a) Define W2x +3. Find the probability density function fw ( of W. [101 (b) Define Z 2X - 3Y. Find P(Z > 3) 5] (c) Find E[WZ], where W and Z are defined in parts (a) and (b), respectively.
3. Let y = (yi..... Yn) be a set of re- sponses, and consider the linear model y= +E, where u = (1, ..., and e is a vector of zero mean, uncorrelated errors with variance o'. This is a linear model in which the responses have a constant but unknown mean . We will call this model the location model. (a) If we write the location model in the usual form of the linear model y = X 8+...
4. Consider the linear model Y = XB+e, where e MV N(0,021). (1) Derive the formula for , the least square estimate of B, using the matrix notation (2) Show that ß is an unbiased estimate for B. (3) Derive the formula for var(), using matrix notation.
Consider the multiple regression model y = X3 + €, with E(€)=0 and var(€)=oʻI. Problem 1 Gauss-Mrkov theorem (revisited). We already know that E = B and var() = '(X'X)". Consider now another unbiased estimator of 3, say b = AY. Since we are assuming that b is unbiased we reach the conclusion that AX = I (why?). The Gauss-Markov theorem claims that var(b) - var() is positive semi-definite which asks that we investigate q' var(b) - var() q. Show...
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