Let βˆ = (X′X)−1X′y where y ∼
N(Xβ,σ2I), X is an n×(k+1) matrix, and β is a (k+1)×1 vector. Are
βˆ′A′[A(X′X)−1A′]−1Aβˆ and y′[I − X(X′X)−1X′]y independent?
Let βˆ = (X′X)−1X′y where y ∼ N(Xβ,σ2I), X is an n×(k+1) matrix, and β is a (k+1)×1 vector. Are βˆ′A′[A(X′X)−1A′]−1Aβˆ a...
Let y = Xß + ε where ε ~ N(0, σ21). Let β = (XTX)-"XTy and let è-y-X β. (a) Show that è-(1-Pxje where Px (b) Compute Ee -e 2. X(XTX)-1x" (1 (c) Compute Varle-e.
Let y = Xß + ε where ε ~ N(0, σ21). Let β = (XTX)-"XTy and let è-y-X β. (a) Show that è-(1-Pxje where Px (b) Compute Ee -e 2. X(XTX)-1x" (1 (c) Compute Varle-e.
In the simple linear regression with zero-constant item for (xi , yi) where i = 1, 2, · · · , n, Yi = βxi + i where {i} n i=1 are i.i.d. N(0, σ2 ). (a) Derive the normal equation that the LS estimator, βˆ, satisfies. (b) Show that the LS estimator of β is given by βˆ = Pn i=1 P xiYi n i=1 x 2 i . (c) Show that E(βˆ) = β, V ar(βˆ) = σ...
Hello, please help solve problem and show all work thank
you.
(Linear models) Suppose we have a vector of n observations Y (response), which has distribution Nn(XB.ση where x is an n × p matrix of known values (indepedent variables), which has full column rank p, and β is a p x 1 vector of unknown parameters. The least squares estimator of ß is 4. a. Determine the distribution of β. xB. Determine the distribution of Y b. Let Y...
linear stat modeling & regression
1) Consider n data points with 3 covariates and observations {xn, ^i2, xi3,yid; i,,n, and you fit the following model, y Bi+Br2+Br+e that is yi A) +Ari,1 +Ari,2 +Buri,3 + єї where є,'s are independent normal distribution with mean zero and variance ơ2 . H the vectors of (Y1, . . . ,Yn). Assume the covariates are centered: Σίχί,,-0, k = 1,2,3. ere, n = 50, Let L are Assume, X'X is a diagonal matrix...
2. Consider the following model: y = XB + u where y is a (nx1) vector containing observations on the dependent variable, B = Bi , B X is a (n x 3) matrix. The first column of X is a column of ones whilst the second and third columns contain observations on two explanatory variables (x and x2 respectively). u is (n x 1) vector of error terms. The following are obtained: 1234.7181 1682.376 7345.581 192.0 259.6 1153.1) X'X...
Let Y = Xβ + ε be the linear model where X be an n × p matrix with orthonormal columns (columns of X are orthogonal to each other and each column has length 1) Let be the least-squares estimate of β, and let be the ridge regression estimate with tuning parameter λ. Prove that for each j, . Note: The ridge regression estimate is given by: The least squares estimate is given by: We were unable to transcribe this...
In the context of multiple regression, define the n X n matrix M =- X(X'X)-'X'. (i) Show that M is symmetric and idempotent. (ii) Prove that m, the diagonals of the matrix M, satisfy 0 sm s 1 for t = 1, 2, ..., n. (iii) Consider the linear model y = XB + u satisfies the Gauss-Markov Assumptions. Let û be the vector of OLS residuals. Show that Eſûù' x) = oʻM (iv) Conclude that while the errors {u:...
In the context of multiple regression, define the n X n matrix M =- X(X'X)-'X'. (i) Show that M is symmetric and idempotent. (ii) Prove that m, the diagonals of the matrix M, satisfy 0 sm s 1 for t = 1, 2, ..., n. (iii) Consider the linear model y = XB + u satisfies the Gauss-Markov Assumptions. Let û be the vector of OLS residuals. Show that Eſûù' x) = oʻM (iv) Conclude that while the errors {u:...
The random vector Y = (Y1, ...,
Yn)T is such that Y = Xβ + ε, where X is an n
× p full-rank matrix of known constants, β is a p-length vector of
unknown parameters, and ε is an n-length vector of random
variables. A multiple linear regression model is fitted to the
data.
(a) Write down the multiple linear regression model assumptions in
matrix format.
(b) Derive the least squares estimator β^ of β.
(c) Using the data:...
Q5. Suppose YON,(, oʻ1) and X is a n xp matrix of constants with rank p (<n). a) Show that A = X(X'X)'X' and I - A are idempotent and find the rank of each. b) If u is linear combination of columns of X i.e. u=Xb for some b find E(Y'AY) and E(Y'(I - A)Y) where A is an in (a) c) Find the distribution of Y'AY/? & Y'(I - A)Y/02 d) Show that Y'AY & Y'(I – A)Y...