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Course: ECON 203c, Spring 2008
School: UCLA
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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
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 + &quot;iifor i = 1; : : : ; nwhere0=&quot;i j xi[1; 2 ; : :
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
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
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
UCLA - ECON - 203c
UCLA - ECON - 203c
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 ) &gt; 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
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
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
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
Korea University - ECON - 205
6 Integration (A.4)6.1 Indenite Integral Consider a continuous function f (x), where f (x) &gt; 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;
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
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
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
Korea University - ECON - 205
Theorem 41 (Thm 30.5, p.828). Let f : R1 R1 be a C2 function. For any point a &lt;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
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
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
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 &gt; j. A is called a lower-triangular matrix if ai j = 0 for i &lt; j. A is calleda diagonal matrix if ai j = 0 for i = j.Theorem 56 (Fact 26.11, p.731). Th
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #9This assignment is due at the beginning of class on Friday, November 30, 2007. You must submitsolutions to all problems. As indicated on the course outline, solutions will be graded for bothcontent and clarity of exposit
University of Regina - STAT - 351
Statistics 351 (Fall 2007)Covariance Zero Does Not Imply IndependenceThe following exercise illustrates that two random variables can have covariance zero yet need notbe independent.Exercise. Consider the random variable X dened by P (X = 1) = 1/4, P
University of Regina - STAT - 351
Statistics 351 (Fall 2007)Fishers F -distribution and Statistical Hypothesis TestingFor Problem #4 on page 27 you are asked to manipulate Fishers F -distribution. The F -distributionarises in the following context. (See Section 10.9 in the Stat 251 tex
University of Regina - STAT - 351
Statistics 351.001 Fall 2008Final Exam InformationThe date and location of the nal exam are Friday, December 12, 2008, 9:00 a.m. 12:00 p.m. Education Building, room 230 (ED 230)Please ensure that you know where the location of the test room is and th
University of Regina - STAT - 351
Statistics 351 (Fall 2007)The Gamma FunctionSuppose that p &gt; 0, and deneup1 eu du.(p) :=0We call (p) the Gamma function and it appears in many of the formul of density functionsfor continuous random variables such as the Gamma distribution, Beta di
University of Regina - STAT - 351
Statistics 351 Fall 2007Independence of X and S 2Theorem. Suppose that X1 , . . . , Xn are independent N (0, 1) random variables. If1X=nnXii=11and S =n1n2(Xi X )2i=1denote the sample mean and sample variance, respectively, then X and S 2 a
University of Regina - STAT - 351
Statistics 351 (Fall 2007)The Jacobian for Polar CoordinatesExample. Determine the Jacobian for the change-of-variables from cartesian coordinatesto polar coordinates.Solution. The traditional letters to use arex = r cos and y = r sin .However, to a
University of Regina - STAT - 351
Statistics 351 (Fall 2007)Review of Linear AlgebraSuppose that A is the symmetric matrix1 1 0A = 1 2 1 .013Determine the eigenvalues and eigenvectors of A.Recall that a real number is an eigenvalue of A if Av = v for some vector v = 0. We callv a
University of Regina - STAT - 351
Statistics 351 Midterm #1 October 10, 2007This exam has 4 problems and 6 numbered pages.You have 50 minutes to complete this exam. Please read all instructions carefully, and checkyour answers. Show all work neatly and in order, and clearly indicate yo
University of Regina - STAT - 351
Statistics 351 Midterm #1 October 6, 2008This exam has 4 problems and 6 numbered pages.You have 50 minutes to complete this exam. Please read all instructions carefully, and checkyour answers. Show all work neatly and in order, and clearly indicate you
University of Regina - STAT - 351
Statistics 351 Midterm #1 October 6, 2008This exam has 4 problems and 6 numbered pages.You have 50 minutes to complete this exam. Please read all instructions carefully, and checkyour answers. Show all work neatly and in order, and clearly indicate you
University of Regina - STAT - 351
Statistics 351 Midterm #2 November 16, 2007This exam is worth 50 points.There are 5 problems on 5 numbered pages.You have 50 minutes to complete this exam. Please read all instructions carefully, and checkyour answers. Show all work neatly and in orde
University of Regina - STAT - 351
Statistics 351 (Fall 2007)The Density and Characteristic Function Denitions of Multivariate NormalitySuppose that the random vector X = (X, Y ) has a multivariate normal distribution with meanvector and covariance matrix given by=xyand =2xx y.
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #1 Solutions2. (a) If X Unif[0, 2], then FX (x) = x for 0 x 2, and if Y Exp(3), then FY (y ) = 1 ey/32for y &gt; 0. Since X and Y are independent, we conclude thatFX,Y (x, y ) = FX (x) FY (y ) =x1 ey/32for 0 x 2 and y &gt;
University of Regina - STAT - 351
Statistics 351 Fall 2007 Midterm #1 Solutions1. (a) By denition,x yfX (x) =2e exdy = 2ey= 2e2x , x &gt; 0,exandxyyx yfY (y ) =2e eydx = 2ex= 2ey 1 ey ,e0y &gt; 0.01. (b) Since fX,Y (x, y ) = fX (x) fY (y ), we immediately conclude that
University of Regina - STAT - 351
Statistics 351 Fall 2008 Midterm #1 Solutions1. (a) By denition,yefX (x) == ex ,ydy = exx &gt; 0.xNote that X Exp(1) so that E(X ) = 1.1. (b) By denition,yey dx = yey ,fY (y ) =y &gt; 0.0Note that Y (2, 1).1. (c) By denition,fY |X =x (y ) =
University of Regina - STAT - 351
Statistics 351 Fall 2008 Midterm #1 Solutions1. (a) By denition,xefY (y ) == ey ,xdx = eyy &gt; 0.yNote that Y Exp(1) so that E(Y ) = 1.1. (b) By denition,xex dy = xex ,fX (x) =x &gt; 0.0Note that X (2, 1).1. (c) By denition,fY |X =x (y ) =
University of Regina - STAT - 351
Statistics 351 Fall 2007 Midterm #2 Solutions1. (a) Let1 111B=and b =20so thatBX + b =1 111X1X2+20=X1 X2 2X1 + X2Y1Y2== Y.By Theorem V.3.1, we conclude that Y N (B + b, B B ) where1 111E(Y) = B + b =12+10=00andcov(Y) =
University of Regina - STAT - 351
Stat 351 Fall 2007Solutions to Assignment #4Problem #3, page 55: Suppose that X + Y = 2. By denition of conditional density,fX,X +Y (x, 2).fX +Y (2)fX |X +Y =2 (x) =We now nd the joint density fX,X +Y (x, 2). Let U = X and V = X + Y so that X = U a
University of Regina - STAT - 351
Stat 351 Fall 2007Solutions to Assignment #51. (a) We begin by calculating E(Y1 ). That is,E(Y1 ) = 1 P (Y = 1) + (1) P (Y = 1) = p (1 p) = 2p 1.We now notice that Sn+1 = Sn + Yn+1 . Therefore,E(Sn+1 |X1 , . . . , Xn ) = E(Sn + Yn+1 |X1 , . . . , Xn
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #7 Solutions1. If X1 , X2 are independent N (0, 1) random variables, then by Denition I, Y1 = X1 3X2 + 2 isnormal with mean E (Y1 ) = E (X1 ) 3E (X2 ) + 2 = 2 and variance var(Y1 ) = var(X1 3X2 + 2) =var(X1 ) + 9 var(X2 )
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #8 SolutionsExercise 7.1, page 134: By Denition I, we know that X and Y X are normally distributed.Therefore, by Theorem 7.1, X and Y X are independent if and only if cov(X, Y X ) = 0. Wecomputecov(X, Y X ) = cov(X, Y ) c
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #9 Solutions1. (a) By Denition I, we see that X1 X2 is normally distributed with meanE (X1 X2 ) = E (X1 ) E (X2 ) = 0and variancevar(X1 X2 ) = var(X1 ) + 2 var(X2 ) 2 cov(X1 , X2 ) = 1 + 2 22 = 1 2 .That is, X1 X2 = Y wh
University of Regina - STAT - 351
Statistics 351Intermediate ProbabilityFall 2008 (200830)Final Exam SolutionsInstructor: Michael Kozdron1. (a) We see that fX,Y (x, y ) 0 for all 0 &lt; x, y &lt; 1, and that1425y dy = y10xy dx dy =fX,Y (x, y ) dx dy =011y500= 1.0Thus, fX,Y i
University of Regina - STAT - 351
Statistics 351 (Fall 2007)Simple Random WalkSuppose that Y1 , Y2 , . . . are i.i.d. random variables with P cfw_Y1 = 1 = P cfw_Y1 = 1 = 1/2,and dene the discrete time stochastic process cfw_Sn , n = 0, 1, . . . by setting S0 = 0 andnSn =Yi .i=1We
University of Regina - STAT - 351
Misprints and Corrections toAn Intermediate Course in ProbabilityMisprintsPage25272749516265656997124128140141141141141151156156183183183183184184184Line155111491391813314846945891212141481013Text
University of Regina - STAT - 351
Make sure that this examination has 13 numbered pagesUniversity of ReginaDepartment of Mathematics &amp; StatisticsFinal Examination200830(December 12, 2008)Statistics 351Intermediate ProbabilityName:Student Number:Instructor: Michael KozdronTime:
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #1This assignment is due at the beginning of class on Monday, September 10, 2007. You must submitall problems that are marked with an asterix (*).1.* Send me an email to say Hello. If I have never taught you before, tell
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #5This assignment is due at the beginning of class on Friday, October 5, 2007. You must submitsolutions to all problems. As indicated on the course outline, solutions will be graded for bothcontent and clarity of expositio
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #7This assignment is due at the beginning of class on Friday, November 9, 2007. You must submitsolutions to all problems. As indicated on the course outline, solutions will be graded for bothcontent and clarity of expositi
University of Regina - STAT - 351
Stat 351 Fall 2007Assignment #8This assignment is due at the beginning of class on Wednesday, November 14, 2007. You do notneed to submit any problems for grading.1. Exercise 7.1, page 1342. Problem #9, page 144 Problem #10, page 145 Problem #11,
University of Regina - STAT - 351
Statistics 351Probability IFall 2006 (200630)Final Exam SolutionsInstructor: Michael Kozdron1. (a) Solving for X and Y gives X = U V and Y = V U V , so that the Jacobian of thistransformation isxxuvvuJ=== v vu + vu = v.yyv 1 uuvBy Theo
University of Regina - STAT - 351
Statistics 351Probability IFall 2007 (200730)Final Exam SolutionsInstructor: Michael Kozdron1. (a) We see that fX,Y (x, y ) 0 for all x, y , and that1yfX,Y (x, y ) dx dy =0114y 3 dy = y 48xy dx dy =00= 1.0Thus, fX,Y is a legitimate densit
University of Regina - STAT - 351
Make sure that this examination has 11 numbered pagesUniversity of ReginaDepartment of Mathematics &amp; StatisticsFinal Examination200730(December 14, 2007)Statistics 351Probability IName:Student Number:Instructor: Michael KozdronTime: 3 hoursRea
University of Regina - STAT - 351
Make sure that this examination has 11 numbered pagesUniversity of ReginaDepartment of Mathematics &amp; StatisticsFinal Examination200630(December 13, 2006)Statistics 351Probability IName:Student Number:Instructor: Michael KozdronTime: 3 hoursRea
University of Regina - STAT - 351
Statistics 351 Midterm #1 October 18, 2006This exam has 4 problems and 5 numbered pages.You have 50 minutes to complete this exam. Please read all instructions carefully, and checkyour answers. Show all work neatly and in order, and clearly indicate yo
University of Regina - STAT - 351
Statistics 351 Midterm #2 November 17, 2006This exam is worth 40 points.There are 5 problems on 5 numbered pages. You may attempt all ve and yourfour highest scores will be taken as your mark. You might want to read all vequestions before you begin.Y
University of Regina - STAT - 351
Statistics 351 Fall 2006 (Kozdron) Midterm #1 Solutions1. (a) By denition, fX |Y =y (x) is given byfX |Y =y (x) =We being by calculating4 y 2fY (y ) =0fX,Y (x, y ).fY (y )11dx =4 y2for 0 &lt; y &lt; 2. Therefore,fX |Y =y (x) =114 y21=4 y2,
University of Regina - STAT - 351
Statistics 351 Fall 2006 (Kozdron) Midterm #2 Solutions1. (a) Recall that a square matrix is strictly positive denite if and only if the determinantsof all of its upper block diagonal matrices are strictly positive. Since2 22 3=we see that det( 1 )
Lone Star College - BIOL - 2402
Art-labeling Activity: Figure 21.15Part ADrag the appropriate labels to their respective targets.This content requires Adobe Flash Player 10.0.0.0 or newer.ANSWER:ViewCorrectIP: Class I and Class II MHC ProteinsClick on the link or the image below
St. Johns - PHI - 2240C