Exercise 7.15 Take the linear model E(el ) with n observations and xi is scalar (real-valued)....
Exercise 7.7 Of the variables (yi, xi) only the pair (yi, xi) are observed. In this case, we say that yi is a latent variable. Suppose where ui is a measurement error satisfying Let ß denote the OLS coefficient from the regression of yi on (a) Ís β the coefficient from the linear projection of yi on z? (b) Is β consistent for β as n oo? (c as n oo. e) Find the asymptotic distribution of yn(3-8 as
1. (20 points) Consider the linear regression model y = a + Bt + ut, ut id(0,%), (t = 1, ...,T). An estimator of B is b=1-1 YT- 41 (a) is estimator b consistent? (Hint: use Chebyshev's inequality) (b) If u i.i.d. N(0,1), what is the asymptotic distribution of b?
(1 point) Find the most general real-valued solution to the linear system of differential equationsx - xi(t) e^ (2t) -en(2t) =c1 + C2 x2(t) e^2t) -en (2t)
Define where S is the collection of all real valued sequences i.e. S = {x : N → R} and we denote xi for the ith element a the sequence x E S. Take for any x EL (i) Show that lic 12 (where recall 1-(x є s i Izel < oo)) (ii) Is l? Prove this or find a counterexample to show that these two sets do not coinside (iii) ls e c loc where recall looー(x є sl...
Exercise 5 Consider a linear model with n = 2m in which Yi = Bo + Bici + Eigi = 1,..., m, and Yi = Bo + B2X1 + Ei, i = m + 1, ...,n. Here €1,..., En are i.i.d. from N(0,0), B = (Bo, B1, B2)' and o2 are unknown parameters, X1, ..., Xn are known constants with X1 + ... + Xm = Xm+1 + ... + Xn = 0. 1. Write the model in vector form...
5. Consider independent observations Xi ~ N(μ, σ?) for i I, , n. Suppose the ,o, are known. Find the MLE of μ. Find the distribution variance parameters σ2, of the MLE.
Suppose we assume the usual linear relationship yibo byai e for n pairs of observations (xi, yi), (x2,32),. ,(En, yn). Now, show what we would calculate for the coefficients bo and bi under the following criteria. a. (8 pts) Suppose we insist that the average error is zero: e-0. What condition must b0 satisfy in order for that relationship to hold? b. (12 pts) Now further suppose that we insist upon the correlation between the inputs and the errors being...
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...
Question 1 Consider the following Multiple Regression Model yı BoB1B2 + El, y2 BIB2E2 y3 B2Es, and y4 Bo+BI4 Suppose that & 's are independent and identically distributed N(0, o2 ) a) Write down the model in the matrix form b) Show that 2 2 1 X'X2 3 2 1.67 -1.33 0.33 (X'X) 1.67 Note that -1.33 -0.67 1 2 3 0.33 -0.67 0.67 c) Find unbiased estimators for Bo, Bi, and B2 given that y 3, y2 1, y3-...
Exercise 9.5 Take the linear model Yi = x'iiß1 + x2iß2 + éj E (Xili) = 0 where both Xli and X2i are qx 1. Show how to test the hypotheses Ho: B, = B, against Hı:ß, # B2.