Problem 1. The Cook's distance is defined as Please show that Di = (Y - Y,))(Y −Y)) (p+1) MSE Di - (+²) (A) = 1 - hi where ri are internally studentized residuals.
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- 2. You have obtained measurements of height in inches of 29 female and 81 male students (Studenth) at your university. A regression of the height on a constant and a binary variable (BFemme), which takes a value of one for females and is zero otherwise, yields the following result: Studentl= 71.0– 4.84×BFemme , R2 = 0.40, SER = 2.0 (0.3) (0.57) a. What is the interpretation of the intercept? What is the interpretation of the slope? How tall are females, on average? b. Test the hypothesis that females, on average, are shorter than males, at the 1% level. c. Is it likely that the error term is homoskedastic here?Suppose the conditional mean function is Y = Bo + BiX + B2X² + B3X3 + U %3D where E[U]X]3D0. By mistake, a researcher omitted X^2 and X^3 terms in the regression and ran regression of Y on X and an intercept only. It turns out that covariance between X and B2X2 + B3X° is nonzero. Is the OLS estimator researcher computed consistent for beta 1? Yes, since the omitted variables are just functions of X. Yes, since the covariance between X and B2X2 + B3X is nonzero No, since the covariance between X and B2X4 + B3X° is nonzero No, since the omitted variables are negligibly small.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.
- What is the orthogonality assumption in OLS, taking Y = a + bX as the model, and the error term is e? (e is epsilon) Correlation(X, X) = 0 O Correlation(X, e) = 0 O Correlation(e, X)= 1 None of the above10. Suppose x = Exp(2). Show solution to answer the following: a) Use MGF technique to determine the distribution of Y = 10 X. b) Use CDF technique to determine the distribution of v = 5x- 4. c) Use the method of transformation to derive the PDF of w = 2-5 X.6. Differentiate: y = A. x²+1 x²+3x-1 -x²+1 (x²+3x-1)² x² + 3x - 1 2x²+3 aldetin-(x²+1) B. D. (x²+3x-1)² x²-1 x²+3x-1 xls 101 S= 8 Xllstat XS = (x)"}_0 D (o
- 5. Two ballpoint pens are selected at random from a box that contains 3 blue pens, 2 red pens and 3 green pens. If X is the number of the blue pens and Y is the number of the red pens selected, find the joint pdf f(x, y)Calculate the residual of P = (1.2, 2.4) with respect to the line 3x + 4y = 12.1. Consider a set of data (x₁, y₁), i = 1, 2, ..., n and an intercept model given by: Yi = Bo + εi with E (₁) = 0, Var(₁) = o² and all &'s are independent and normally distributed. Derive the standard error of the point estimator Bo.
- 9. Which one of the following is the OLS estimator of the variance of slope term for the regression Y = Bo + B1X, + ųę, t = 1,2, ..,T? Here, x = X, – X and y, = Y; – Y. %3D a) of. = (T-2)E-1 x b) o (T-2) E-1 *? c) o (T-2)E-1YẺ d) o Var (X) Var (Y) e) o3, Cov (X,Y) Var (Y)4. Suppose that n=100 i.i.d observations for (Y₁, X₁) yield the following results Ỹ = 32.1 +66.8X, SER=15.1, R² = 0.81 (15.1) (12.2) Another researcher is interested in the same regression, but he makes an error when he enters the data in his regression program: He enters each observation twice, so he has 200 observations. Using these 200 observations, what results will be produced by his program? Write it in the following format: Ỹ +---X, SER= (---) (---) R² =Q1. Consider the two different losses: (i) zero/one loss where Ai = 0 and dij = 1 and (ii) generic loss where A;i = 0 and A;j = A,. Will the discriminants under both cases be same? Justify your answer by showing all the steps involved.
