Let Y~ N(YXO, o²I), where X equals: X = TTLE 1 1 1 -1 22 0 21 0-3 ܚ ܝ And is the 3 x 1 regression coefficient matrix. If Y= [6 1 11 3]', compute the maximum likelihood estimate 0ML, and o- Show all matrix calculations. ML
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![6. Let Y~ N(YIXO, ²1), where X equals:
X =
HHHL
1
[1
1
-1
2
0 -3
0
And is the 3 x 1 regression coefficient matrix. If Y- [6 1 11 3]', compute the
maximum likelihood estimate 0ML, and o- Show all matrix calculations.
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- Consider the following population model for household consumption: cons = a + b1 * inc+ b2 * educ+ b3 * hhsize + u where cons is consumption, inc is income, educ is the education level of household head, hhsize is the size of a household. Suppose a researcher estimates the model and gets the predicted value, cons_hat, and then runs a regression of cons_hat on educ, inc, and hhsize. Which of the following choice is correct and please explain why. A) be certain that R^2 = 1 B) be certain that R^2 = 0 C) be certain that R^2 is less than 1 but greater than 0. D) not be certainWrite out the design matrix 1. Polynomial regression : 2. Data transformation: X and find its rank for the following Y₁ = Bot fixi + B₂X²³² + ... + B₂-, X; log7; = B₁ + B₁ (-log D₁) + E; P-1 > +E; " i=1.- `n models. n>p. i=1...nConsider the linear regression model (with deterministic regressors) y = Bi + P2x, + €,, G - (0,0) i.i.d., i = 1.,n. Let ô = (,. BY be the OLS estimator of B = (B. P2) and (x,-x) be another estimator of B. True False Elyı] = Ely.) The estimator , is biased as EIA,) = B (x, - x1) The estimator , is an efficient estimator of Pi as only two observations are needed for its calculation. The estimator B, is a linear estimator of Pi .
- Let B₁ and 3₂ denote the OLS estimators of B₁ and 2 in the regression: yi= Bo + Bixi + B2x₁2 + Ui. Let B₁ denote the OLS estimators of B₁ in the regression Vi = Bo + Bixil + Vi. Let 8₁=1(x₁ - x₁)(X₁2 - X₂) / Σ²-₁(x₁₁ — X₁)². a) ₁ is the OLS estimator of a slope parameter from what regression model? b) Show that 1(Vi-Bo - B₁x₁₁ - B2x₁2) (Xi1 — x₁) = 0. c) Prove that B₁ =B1 + Ổ₁ß2. Explain the situations in which omitting x2 from the model will NOT bias the OLS estimate of B₁.Express the model Y₁ = Bo + Bixi + B₁x² + €i i=1,2,..., 5 where the &; have mean zero, variance o² and are uncorrelated, as a general linear model in matrix form by specifying Y, X, 3 and e.Find the least-squares regression line ŷ bo+ b₁z through the points For what value of is y = 0? H= (-2,0), (1, 6), (4, 14), (8, 19), (10, 27).
- Q2 / A rectangular plot of land is measured as follow using Jacobian Matrix to compute area and standard deviations of this land. length= 800 m +0.12 m width= 550m±0.7m .Find the least-squares regression line ý = b + b₁ through the points (-2,0), (0,9), (5, 15), (8,20), (12,24). For what value of x is y = 0Assume: qi= β0 + β1pi + ui Please derive the estimated values (b0 and b1) based on the equation using OLS (ordinary least squares regression). Show that they are unbiased estimators of β0 and β1.
- Find the least-squares regression line ŷ = bo + bjx through the points (-2, 0), (3, 7), (5, 14), (9, 19), (12, 26), and then use it to find point estimates ŷ corresponding to x = 4 and x = 10. For x = 4, y = For x = 10, y =Consider the linear model y=B,+B,x,+B,x,,+B,x+B,+u, You estimate the model y =B,+B,x,+B,x, observations and obtain the OLS residuals . You then estimate the auxiliary regression +u based on 123 31 3" 3i 41 The LM statistic +1 4/ you obtain to test the null hypothesis that H:B,=B,=0 is 20.91. What is the R2 of the auxiliary regression? It is not possible to say O 0.17 O 0.175714 0.177203What is the significance of R and R2 in gression model?