Proving Refinement: DLF of m d?N d>0 COLOUR = {green, red} green = red n?N a?N b?N c?N a+b+c=n a=0?c=0 ml.tl?COLOUR il.tl?COLOUR ml.tl=green ? a+b<d?c=0 il.tl=green ? b>0?a=0 ml.tl=red ? il.tl=red ml.pass?{0,1} il.pass?{0,1} ml.tl=red ? ml.pass=1 il.tl=red ? il.pass=1 a+b<d?c=0 c>0 a>0 b>0?a=0 ml.tl=red?a+b<d?c=0?il.pass=1 il.tl=red?b>0?a=0?ml.pass=1 ml.tl=green il.tl=green a>0 c>0 d?N d>0 c>0 b?N ml.tl=red il.tl=red ml.tl=red ? ml.pass=1 il.tl=red ? il.pass=1 b<d?ml.pass=1?il.pass=1 b>0?ml.pass=1?il.pass=1 d?N d>0 b>0 ml.tl=red il.tl=red ml.pass=1 il.pass=1 d>0 b>0 OR R2 HYP d>0 b>0 ARI OR L d>0 b>0?b=0 b<d?b>0 d>0 b>0 EQ LR MON OR.R1 HYP d>0 b<d?b>0 0<d?0>0 0<d
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Homework 1. Let S = {[x 0; y 0], x,y ∈ ℤ} a) Show that <S, +, > is a ring. b) Show that ϕ: S → ℤ defined by ϕ([x 0; y 0]) = x is a ring homomorphism. c) Describe ker ϕ. 2. Let R be a ring. The center of R is the set {x ∈ R : ax = xa ∀ a ∈ R}. Prove that the center of a ring is a subring. 3. Show that every Boolean ring is commutative. 4. Let R be the set of all real-valued functions defined for all real numbers under function addition & multiplication. a. Determine all zero-divisors of R. b. Determine all nilpotent elements of R. c. Show that every non zero element is a zero-divisor or a unit. 5. Prove that the intersection of any set of ideals of a ring is an ideal. Hint: Let I = ∩ (α ∈ Δ) Iα be an intersection of ideals Iα s.t Δ is an indexing set 6. In ℤ ⊕ ℤ, let I = {(a,0) : a ∈ ℤ}. Show that I is a prime ideal but not a maximal ideal.
Sri K.
Consider the general linear regression model Y = XB + ε where Y is n x 1, X is n x p and of rank 8 is p x 1, ε is n x 1, and ε is N(0, 021). (a) The hat matrix H is given by H = X(X'X)^(-1)X'. Show that (I - H) is idempotent where I is the identity matrix. (b) Using the least squares method, we minimize RSS = ε'ε = (Y - Xβ)'(Y - Xβ) to obtain β = (X'X)^(-1)XY. Show that RSS can also be written as Y'(I - H)Y. Obtain an expression for the variance-covariance matrix of the fitted values Yi, i = 1, 2, in terms of the hat matrix H: ε = Y - Xβ is the vector of residuals. Are the residuals statistically independent? Justify your answer with an explanation. Show that Hε = 0. Suppose we denote by hj the (i, j) element of the hat matrix H. Thus, ε can be written as ε = Xβ - Xh, where h = [h1, h2, ..., hn]' is the vector of leverage values. What does this equation of ε show? Suppose that we partition X and β as B1 β = B2 X = [X1 X2] where X1 is n x p1, X2 is n x p2, and p1 + p2 = p. B1 is p1 x 1, and B2 is p2 x 1. If the true model is Y = XB + ε, and we fit the model Y = X1B + ε, have we underspecified or overspecified the model? For the least squares part (E), β1 = (X1'X1)^(-1)X1'Y. If the true model is Y = XB + ε, compute E(β1). As a result of model misspecification in part (C), we could obtain an estimator of β which is larger than it should be. Does this affect inferences made about the model? Explain.
Question 3 Overlapping Generations Model Consider an economy in which 2 generations overlap. Time is discrete and people live two periods, people start without assets and are not able to work in the second period. The government imposes a Social Security scheme in which individuals must pay a lump sum amount φ when young and they receive an amount (1+rt+1)φ when old. The economy has a Cobb-Douglas production function yt = kαt (y and k are in per effective units) and the utility function for someone born at time t is given by: Ut = (C1,t^(1-θ))/(1-θ) + (1/(1+ρ)) * (C2,t+1^(1-θ))/(1-θ) The growth rate of the population and technology are Lt = (1 + n)Lt–1 and : At = (1 + g)At–1, respectively. Markets are competitive and there is no capital depreciation. a) [5pts] Write down the intertemporal budget constraint, the Lagrangian and Euler Equation. b) [5pts] Write down the optimal consumption choices C1,t and C2,t+1. c) [5pts] Calculate the wage rate (intensive form) and law of motion of capital (intensive form) assuming s = 1/(2+ρ). d) [5pts] Graph the steady state of the model (kt+1 = kt).
Akash M.
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