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Question 3 Recall the least square assumptions in Key Concept 4.3. Show that: i. E(u, |X)0...
Which of the following is not one of the least squares assumptions used in Stock and...
Which of the following is not one of the least squares assumptions used in Stock and Watson to show that the OLS estimators are unbiased and consistent and have approximately a normal distribution in large samples? 1) large outliers are unlikely 2) the error term is homoskedastic, i.e., Var(ui ∣ X=x) does not depend on x 3) the sample (Xi,Yi),i=1,…,n constitutes an i.i.d. random sample from the population joint distribution of X and Y 4) the conditional mean of the...
Simple linear regression model Assumptions: AI E[u] 0 for all i, i1, .., n On average,...
Simple linear regression model Assumptions: AI E[u] 0 for all i, i1, .., n On average, random component is zero Model runs through expected values of Yand Y A2 E[uaij]-0 for all i and j where i /j COV(IIİlh)- Unobserved component not related across observations E[14"]= for all i All observations have random component dravn from a distribution with the same variance σ2 , f(0,02) A3 var(11i)-σ (Homoskedasticitv) A4 E[Alli] = 0 for all i Random component and covariate not...
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y)...
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y) or each 1, and (11 CoV(X,Y) var(x)var(y) (Recall that p vararo
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J,...
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the...
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the OLS estimator for {00, Bo}, the minimizer of E. (Y: - a - 3x), is . (X.-X) (Y-Y) and a-Y-3X. - (Xi - x) When the equation (1) is the true data generating process, {X}- are non-stochastic, and {e} are random variables with B (ei) = 0, B(?) = 0, and Ele;e;) = 0 for any i, j = 1,2,...,n and i j, we...
can you help with these questions and briefly explain how you got to the answer. it...
can you help with these questions and briefly explain
how you got to the answer. it would be a big help thank for your
time.
Question 1 Suppose that the conditional variance is var(WXi) = (x), where i is a constant and h is a known function. The WLS estimator: O A. is the estimator obtained by first dividing the dependent variable and regressor by h and then regressing this modified dependent variable on the modified regressor using OLS. O...
1. Suppose t hat Xhas t he chi-square distribution on p1∈(0, ∞) degrees of f reedom...
1. Suppose t hat Xhas t he chi-square distribution on p1∈(0, ∞) degrees of f reedom and that, i ndependently, Y has t he chi-square distribution on p2∈(0, p1) degrees of f ree-dom. a. Use moment generating functions to find the distribution of X + Y . b. A naive guess might be that the distribution of X − Y is chi-square on p1− p2 degrees of freedom. Prove that such a guess is wrong by demonstrating that P (X...
1. Suppose t hat Xhas t he chi-square distribution on p1∈(0, ∞) degrees of f reedom...
1. Suppose t hat Xhas t he chi-square distribution on p1∈(0, ∞) degrees of f reedom and that, i ndependently, Y has t he chi-square distribution on p2∈(0, p1) degrees of f ree-dom. a. Use moment generating functions to find the distribution of X + Y . b. A naive guess might be that the distribution of X − Y is chi-square on p1− p2 degrees of freedom. Prove that such a guess is wrong by demonstrating that P (X...
3.10 (i) If X1, , Xn are i.i.d. according to the exponential density e-", r >0,...
3.10 (i) If X1, , Xn are i.i.d. according to the exponential density e-", r >0, show that (2.9.3) P [X(n)-log n < y]- e-e-v, -00 < y < oo. (ii) Show that the right side of (2.9.3) is a cumulative distribution function. (The distribution with this edf is called the ertreme value distribution.) (iii) Graph the cdf of X(n)-log n for n = 1, 2, 5 together with the mit e-e" (iv) Graph the densities corresponding to the cdf's...
is independent of X, and e Problem 3 Suppose X N(0, 1 -2) -1 <p< 1....
is independent of X, and e Problem 3 Suppose X N(0, 1 -2) -1 <p< 1. (1) Explain that the conditional distribution [Y|X = x] ~N(px, 1 - p2) (2) Calculate the joint density f(x, y) (3) Calculate E(Y) and Var(Y) (4) Calculate Cov(X, Y) N(0, 1), and Y = pX + €, where
Exercise 4 (Paired test, known normality of the difference). Let X, Y be RVs. Denote E[X]...
Exercise 4 (Paired test, known normality of the difference). Let X, Y be RVs. Denote E[X] = 4x and E[Y] = uy. Suppose we want to test the null hypothesis Houx = My against the alternative hy- pothesis H ux #uy. Suppose we have i.i.d. pairs (X1,Y),...,(X,Y) from the joint distribution of (X,Y). Further assume that we know the X - Y follows a normal distribution. (i) Noting that X1 - Y1,..., Xn-Ynare i.i.d. with normal distribution, show that (exactly)...
Simple linear regression model Assumptions: AI E[u] 0 for all i, i1, .., n On average, random component is zero Model runs through expected values of Yand Y A2 E[uaij]-0 for all i and j where i /j COV(IIİlh)- Unobserved component not related across observations E[14"]= for all i All observations have random component dravn from a distribution with the same variance σ2 , f(0,02) A3 var(11i)-σ (Homoskedasticitv) A4 E[Alli] = 0 for all i Random component and covariate not...
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y) or each 1, and (11 CoV(X,Y) var(x)var(y) (Recall that p vararo
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J,...
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the OLS estimator for {00, Bo}, the minimizer of E. (Y: - a - 3x), is . (X.-X) (Y-Y) and a-Y-3X. - (Xi - x) When the equation (1) is the true data generating process, {X}- are non-stochastic, and {e} are random variables with B (ei) = 0, B(?) = 0, and Ele;e;) = 0 for any i, j = 1,2,...,n and i j, we...
can you help with these questions and briefly explain
how you got to the answer. it would be a big help thank for your
time.
Question 1 Suppose that the conditional variance is var(WXi) = (x), where i is a constant and h is a known function. The WLS estimator: O A. is the estimator obtained by first dividing the dependent variable and regressor by h and then regressing this modified dependent variable on the modified regressor using OLS. O...
3.10 (i) If X1, , Xn are i.i.d. according to the exponential density e-", r >0, show that (2.9.3) P [X(n)-log n < y]- e-e-v, -00 < y < oo. (ii) Show that the right side of (2.9.3) is a cumulative distribution function. (The distribution with this edf is called the ertreme value distribution.) (iii) Graph the cdf of X(n)-log n for n = 1, 2, 5 together with the mit e-e" (iv) Graph the densities corresponding to the cdf's...
is independent of X, and e Problem 3 Suppose X N(0, 1 -2) -1 <p< 1. (1) Explain that the conditional distribution [Y|X = x] ~N(px, 1 - p2) (2) Calculate the joint density f(x, y) (3) Calculate E(Y) and Var(Y) (4) Calculate Cov(X, Y) N(0, 1), and Y = pX + €, where
Exercise 4 (Paired test, known normality of the difference). Let X, Y be RVs. Denote E[X] = 4x and E[Y] = uy. Suppose we want to test the null hypothesis Houx = My against the alternative hy- pothesis H ux #uy. Suppose we have i.i.d. pairs (X1,Y),...,(X,Y) from the joint distribution of (X,Y). Further assume that we know the X - Y follows a normal distribution. (i) Noting that X1 - Y1,..., Xn-Ynare i.i.d. with normal distribution, show that (exactly)...
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