UCLA : ECON 203c : final

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final_2008s

UCLA - ECON - 203c

Economics 203CIntroduction to EconometricsSystem ModelsSpring, 2008Moshe BuchinskyDepartment of EconomicsUCLAEconomics 203Introduction to Econometrics: System ModelsFinal ExaminationJune 11, 2008This is a 3 hour open-book exam. You are allowed t

final_2008s_answers

UCLA - ECON - 203c

ECON 203C: System ModelsFinal 2008 Suggested AnswersHisayuki YoshimotoLast Modied: June 14, 20081Question 4 (40 points)Consider the binary model given by the following latent variableyi = x0 + "iifor i = 1; : : : ; nwhere0="i j xi[1; 2 ; : :

Hypothesis Tests of ML and GMM Estimation

UCLA - ECON - 203c

ECON 203C: System ModelsTA Note 6: Version 1Hypothesis Tests of ML and GMM EstimationsHisayuki YoshimotoLast Modied: May 20, 2008Abstract:1Wald, Lagrange Multiplier (LM), and Likelihood Ratio (LR) Statistics1.1Review of ML and GMM estimatorNNA

Binary Choice (Probit and Logit) Models

UCLA - ECON - 203c

ECON 203C: System ModelsTA Note 4: Version 3Binary Choice (Probit and Logit) ModelsHisayuki YoshimotoLast Modied: May 7, 2008Abstract: In section 1, we review Bernoulli random variable that is basis of binary choice model. In section 2, wesolve Fina

final_2003s_answers

UCLA - ECON - 203c

Econ 203c, Spring 2003, Final ExamQuestion 1y i x i z u iiz 1i z v 1iiz 2i z v 2iiAssume: Eu i |x i , z covu i , v 1i covu i , v 2i 0 and covv 1i , v 2i 0iNote: There was a typo that was not corrected during the exam: covv 1i , v 2i 0. Iwill be

final_2006s

UCLA - ECON - 203c

final_2006s_answers

UCLA - ECON - 203c

math-2011-note13

Korea University - ECON - 205

DF (x )(y x ) = 0. However,F (1 t )x + t y) F (x )t 0t(1 t )F (x ) + tF (y) F (x ) limt 0t= F (y) F (x ) > 0,g (0) = lima contrdiction.17.3 Calculus Criteria for Concavity (21.1)Theorem 63 (Thm 21.3(p.511) and Thm 21.5(p.513). Let F : U R1 be

math-2011-note1

Korea University - ECON - 205

Mathematics for Economists1 Introduction1.1 Motivation Why do we need to know mathematics in order to learn economics? What is economics? In economics we learn how the economy works in various situations. An economy consists of various people (consu

math-2011-note2

Korea University - ECON - 205

2.7 Differentiability and Continuity A function f is differentiable at x0 if the following limit existslimh0f (x0 + h) f (x0 ).h A function f is continuous at x0 if limh0 f (x0 + h) exists and limh0 f (x0 +h) = f (x0 ).Example 1 (Non-differentiab

math-2011-note3

Korea University - ECON - 205

4.2 Inverse FunctionsDenition 4 (p.76). For a given function f : E1 R1 , E1 R1 , we say a functiong : E2 R1 , E2 R1 , is an inverse of f ifg( f (x) = x for all x E1 andf (g(y) = y for all y E2 .1 Example: f (x) = 3 2x, g(y) = 2 (3 y) If f has an in

math-2011-note4

Korea University - ECON - 205

6 Integration (A.4)6.1 Indenite Integral Consider a continuous function f (x), where f (x) > 0 for all x. Consider the area under the graph of y = f (x) from a certain point a to anotherpoint x and denote it by A(x; a). What is the derivative of A(x;

math-2011-note5

Korea University - ECON - 205

9 Limits and Open Sets (ch.12)9.1 Sequences in Real NumbersA sequence of real numbers cfw_x1 , x2 , , xn , is an assignment of a real number xnto each natural number n.Denition 7 (p.255). Let cfw_x1 , x2 , , xn , be a sequence of real numbers. A rea

math-2011-note7

Korea University - ECON - 205

Theorem 34 (Thm 13.3, p.291). The general quadratic form Q(x1 , , xk ) = i j ai j xi x jcan be written as1a11 2 a12 1 a1kx12 1 a12 a22 1 a2k x2 22x1 x2 xk ... . ,.... . .....11xkakk2 a1 k2 a2 k or xT Ax, where A is a symmetric m

math-2011-note8

Korea University - ECON - 205

12 Calculus of Several Variables (ch.14)12.1 Partial DerivativeDenition 25 (p.300). Let F : Rk R. The partial derivative of f with respect to xi atx0 = (x0 , , x0 ) is dened as1kF (x0 , , x0 + h, , x0 ) F (x0 , , x0 , , x0 )F 0ii11kk(x ) = l

math-2011-note9

Korea University - ECON - 205

Theorem 41 (Thm 30.5, p.828). Let f : R1 R1 be a C2 function. For any point a <b R1 , there is a point c (a, b) such that f (b) = f (a) + f (a)(b a) + 1 f (c)(b 2a)2 .Proof. Dene2f (b) f (a) f (a)(b a) .(b a)2Suppose that f (x) = M for all x (a, b

math-2011-note10

Korea University - ECON - 205

Example 6. Suppose that we have a production function Q = kxa yb . Then,Q= akxa1 yb ,x 2Q= abkxa1 yb1 , x yQ= bkxa yb1y 2Q= abkxa1 yb1 y x14 Some Linear Algebra14.1 Deniteness of Quadratic Forms (16.2)Denition 27. Let A be an m m symmetric

math-2011-note11

Korea University - ECON - 205

Theorem 53 (Thm 11.2, p.243). Let A = (a1 , a2 , , am). If a1 , a2 , , am are linearlyindependent if and only if det A = 0.Proof. a) only if: If a1 , a2 , , am are linearly independent, then A is one-to-one andonto, hence has an inverse function. Call

math-2011-note12

Korea University - ECON - 205

Denition 36 (p.161). An m m matrix A = (ai j ) is called an upper-triangular matrixif ai j = 0 for i > j. A is called a lower-triangular matrix if ai j = 0 for i < j. A is calleda diagonal matrix if ai j = 0 for i = j.Theorem 56 (Fact 26.11, p.731). Th

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