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Question 2. (2.5 points. You are considering the model Y = XB + X2B,...
II. Derivations (You must show all your work for full credit.) i. Given the model y=XB+ɛ,...
II. Derivations (You must show all your work for full credit.) i. Given the model y=XB+ɛ, derive the least squares estimate for ß? (10 points) ii. Show that B=(x+x)"x"y is an unbiased estimate for B.(10 points) ii. Given vlə) = E[(@–B\–B)], derive the variance- covariance matrix for the least squares estimator (10 points). iv. Given the model y=XB+ɛ, the transformation matrix T, and TTT=22-1, derive the GLS estimator (10 points).
Do I get the right answers? If not, can someone please explain? (a) 2 points possible (graded, results hidden) Conside...
Do I get the right answers? If not, can someone please
explain?
(a) 2 points possible (graded, results hidden) Consider a Gaussian linear model Y = aX + e in a Bayesian view. Consider the prior (a) = 1 for all a eR. Determine whether each of the following statements is true or false. (a) is a uniform prior. O True C False n(a) is a Jeffreys prior when we consider the likelihood L (Y = y|A = a, X...
in a Bayesian view. Consider the prior π(a)-1 for all a e R Consider a Gaussian linear model Y = aX+ E Determine whether each of the following statements is true or false. π(a) a uniform prior. (1) (...
in a Bayesian view. Consider the prior π(a)-1 for all a e R Consider a Gaussian linear model Y = aX+ E Determine whether each of the following statements is true or false. π(a) a uniform prior. (1) (a) True (b) False L(Y=y14=a,X=x) (2) π(a) is a jeffreys prior when we consider the likelihood (where we assume xis known) (a) True (b)False Y-XB+ σε where ε E R" is a random vector with Consider a linear regression model E[ε1-0, E[eErJ-1....
2. Consider a simple linear regression i ion model for a response variable Y, a single...
2. Consider a simple linear regression i ion model for a response variable Y, a single predictor variable ,i1.., n, and having Gaussian (i.e. normally distributed) errors: This model is often called "regression through the origin" since E(X) = 0 if xi = 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function Hint: The function g(x)log(x) +1-x...
Consider the following linear regression model 1. For any X = x, let Y = xB+U,...
Consider the following linear regression model 1. For any X = x, let Y = xB+U, where B erk. 2. X is exogenous. 3. The probability model is {f(u; ) is a distribution on R: Ef [U] = 0, VAR; [U] = 62,0 >0}. 4. Sampling model: {Y}}}=1 is an independent sample, sequentially generated using Y; = xiß +Ui, where the U; are IID(0,62). (i) Let K > 0 be a given number. We wish to estimate B using least-squares...
Suppose we have the full rank linear model y = XA+ Ewiun xp design matrix X,...
Suppose we have the full rank linear model y = XA+ Ewiun xp design matrix X, normal errors E N (0,0?Inxn). Let b be the least squares estimator of B. (C) Prove that (b-B)? XT X(6-8) o2 follows the x? distribution. Hint: Write Xb in terms of X, B and e. (d) Hence derive a 100(1 - a)% joint confidence region of ß given in notes (b - B) TXTX(b-)/po<Fa:pon-p, where Faip,n-p denotes the upper ath quantile of the Fpin-p...
QUESTION 3 Suppose that Y, Y2, ., Y, are independent variables such that Y, =Bx? +€,,...
QUESTION 3 Suppose that Y, Y2, ., Y, are independent variables such that Y, =Bx? +€,, != 1,2,,n, where B is an unknown parameter, X1, X2, X, are known real numbers (+0), and €1. €2. ,€, are independent random errors each with a normal distribution with mean 0 and variance o (a) Show that is an unbiased estimator of B What is the variance of the estimator? (b) Show that the least squares estimator of B is not the same...
2. The linear regression model in matrix format is Y Χβ + e, with the usual definitions Let E(elX...
2. The linear regression model in matrix format is Y Χβ + e, with the usual definitions Let E(elX) 0 and T1 0 0 01 0 r2 00 0 0 0 0.0 0 γΝ 0 00 Notice that as a covariance matrix, Σ is bymmetric and nonnegative definite () Derive Var (0LS|x). (ii) Let B- CY be any other linear unbiased estimator where C' is an N x K function of X. Prove Var (BIX) 2 (X-x)-1 3. An oracle...
2. Consider the simple linear regression model: where e1, .. . , es, are i.i.d. N (0, o2), for i= 1,2,... , n. Suppose...
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...
2. Suppose we are given data on n observations (x,Y), i 1,... , n, and we...
2. Suppose we are given data on n observations (x,Y), i 1,... , n, and we have a linear model, = SXY/SXX and A,-ㄚ-Ax be the least-square estimates so that E(X) = β0 +ATp Let given in lecture. (a) Show that E(5xx)-A5xx and E(Y)-Ao +A2. (b) Use (a) to show that E(A)-A and E(A)-A. În other words, these are unbiased estimators (c) The fitted values Yi = ArtAz; are used as estimates of E(K), and the residuals ei = Y-...
II. Derivations (You must show all your work for full credit.) i. Given the model y=XB+ɛ, derive the least squares estimate for ß? (10 points) ii. Show that B=(x+x)"x"y is an unbiased estimate for B.(10 points) ii. Given vlə) = E[(@–B\–B)], derive the variance- covariance matrix for the least squares estimator (10 points). iv. Given the model y=XB+ɛ, the transformation matrix T, and TTT=22-1, derive the GLS estimator (10 points).
Do I get the right answers? If not, can someone please
explain?
(a) 2 points possible (graded, results hidden) Consider a Gaussian linear model Y = aX + e in a Bayesian view. Consider the prior (a) = 1 for all a eR. Determine whether each of the following statements is true or false. (a) is a uniform prior. O True C False n(a) is a Jeffreys prior when we consider the likelihood L (Y = y|A = a, X...
in a Bayesian view. Consider the prior π(a)-1 for all a e R Consider a Gaussian linear model Y = aX+ E Determine whether each of the following statements is true or false. π(a) a uniform prior. (1) (a) True (b) False L(Y=y14=a,X=x) (2) π(a) is a jeffreys prior when we consider the likelihood (where we assume xis known) (a) True (b)False Y-XB+ σε where ε E R" is a random vector with Consider a linear regression model E[ε1-0, E[eErJ-1....
2. Consider a simple linear regression i ion model for a response variable Y, a single predictor variable ,i1.., n, and having Gaussian (i.e. normally distributed) errors: This model is often called "regression through the origin" since E(X) = 0 if xi = 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function Hint: The function g(x)log(x) +1-x...
Consider the following linear regression model 1. For any X = x, let Y = xB+U, where B erk. 2. X is exogenous. 3. The probability model is {f(u; ) is a distribution on R: Ef [U] = 0, VAR; [U] = 62,0 >0}. 4. Sampling model: {Y}}}=1 is an independent sample, sequentially generated using Y; = xiß +Ui, where the U; are IID(0,62). (i) Let K > 0 be a given number. We wish to estimate B using least-squares...
Suppose we have the full rank linear model y = XA+ Ewiun xp design matrix X, normal errors E N (0,0?Inxn). Let b be the least squares estimator of B. (C) Prove that (b-B)? XT X(6-8) o2 follows the x? distribution. Hint: Write Xb in terms of X, B and e. (d) Hence derive a 100(1 - a)% joint confidence region of ß given in notes (b - B) TXTX(b-)/po<Fa:pon-p, where Faip,n-p denotes the upper ath quantile of the Fpin-p...
QUESTION 3 Suppose that Y, Y2, ., Y, are independent variables such that Y, =Bx? +€,, != 1,2,,n, where B is an unknown parameter, X1, X2, X, are known real numbers (+0), and €1. €2. ,€, are independent random errors each with a normal distribution with mean 0 and variance o (a) Show that is an unbiased estimator of B What is the variance of the estimator? (b) Show that the least squares estimator of B is not the same...
2. The linear regression model in matrix format is Y Χβ + e, with the usual definitions Let E(elX) 0 and T1 0 0 01 0 r2 00 0 0 0 0.0 0 γΝ 0 00 Notice that as a covariance matrix, Σ is bymmetric and nonnegative definite () Derive Var (0LS|x). (ii) Let B- CY be any other linear unbiased estimator where C' is an N x K function of X. Prove Var (BIX) 2 (X-x)-1 3. An oracle...
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...
2. Suppose we are given data on n observations (x,Y), i 1,... , n, and we have a linear model, = SXY/SXX and A,-ㄚ-Ax be the least-square estimates so that E(X) = β0 +ATp Let given in lecture. (a) Show that E(5xx)-A5xx and E(Y)-Ao +A2. (b) Use (a) to show that E(A)-A and E(A)-A. În other words, these are unbiased estimators (c) The fitted values Yi = ArtAz; are used as estimates of E(K), and the residuals ei = Y-...
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