j=0 s t t+1 + q t are vectors of length equal to the number of assets (c t+1 ) q t +1 + d i t+1 (1) (c t+1 ) R t+1 1= E t β u0 (c t+1 ) R u 0 (c t )

Size: px
Start display at page:

Download "j=0 s t t+1 + q t are vectors of length equal to the number of assets (c t+1 ) q t +1 + d i t+1 (1) (c t+1 ) R t+1 1= E t β u0 (c t+1 ) R u 0 (c t )"

Transcription

1 1 Aet Prce: overvew Euler equaton C-CAPM equty premum puzzle and rk free rate puzzle Law of One Prce / No Arbtrage Hanen-Jagannathan bound reoluton of equty premum puzzle Euler equaton agent problem X X max β t u c t t Pr t j=0 t c t t + q a t a t+1 t W t t t W t+1 t+1 = y t+1 t+1 + q t a +1 t+1 + d t+1 t+1 a t+1 t a comment: a t and qt Euler equaton are vector of length equal to the number of aet u 0 a (c t ) q t = βe t u 0 a (c t+1 ) q t +1 + d t+1 (1) u 0 (c t )= βe t u 0 (c t+1 ) R t+1 1= E t β u0 (c t+1 ) R () u 0 t+1 (c t ) tranveralty condton a lm β j E 0 u 0 (c t+j ) q t+j a t+j =0 j prcng formula repeated ubttuton of (1) X u 0 a (c t+j ) q t = β j E t d t+j (3) u 0 (c t ) j=1 no bubble 1

2 tranveralty and t =1 complete market contency check revew A-D prce wth complete market (3) q t+j, = β c ( t, j ) Pr u 0 (c t ( t )) t t j u 0 t+1 j t 3 CCAPM (Conumpton Captal Aet Prcng Model) make ()and (3)operatonal: CCAPM ue aggregate conumpton n above equaton jutfcaton: equlbrum of repreentatve agent economy (Luca / Breeden) equlbrum wth complete market (Contantnde) complete market Pareto Optma repreentatve conumer (weghted utlty) back to Euler equaton u 0 (c t+1 ) R 1=E t β t+1 u 0 (c t ) Abence of arbtrage mple that there ext ome m t+1 uch that 1=E t mt+1 R t+1 THE emprcally tetable condton (agan) ntutve decompoton t the covarance that matter! 1=βE t µ u 0 (c t+1 ) E t Rt +1 + βcovt µ u 0 (c t+1 ),Rt +1 u 0 (c t ) u 0 (c t ) 4 Equty Premum Puzzle Euler equaton wth data on R tock market and R bond

3 mple log-normal calculaton preference and conumpton u 0 (c) = c γ ½ ¾ c t+1 1 = c exp ε c σ c c t ε c v N μc,σ µ c E c t+1 c t = μ c return ½ ¾ 1 R = 1+ r exp ε σ ε v N μ c,σ c E R = R =1+ r Euler takng log... tock and bond: " µ # γ R c t+1 1 = βe c t µ = β 1+ r ( c ) γ E t exp ε σ γε c + γ σ c µ 1 = β 1+ r ( c ) γ E t exp (1 + γ) γ 1 σ c γσ c 1 log 1+ r = log β + γ log c (1 + γ) γ σ c + γσ c 1 r f log 1+ r f = log β + γ log c (1 + γ) γ σ (4) c 1 r log (1 + r )= log β + γ log c (1 + γ) γ σ c + γσ c (5) premum: r r f log (1 + r ) log 1+ r f = γσ c (6) 3

4 Table removed due to copyrght retrcton. Kocherlakota, Narayana R. "The Equty Premum Puzzle: It' Stll a Puzzle." Journal of Economc Lterature 34, no. 1 (1996): 47 (Table 1). US data (from Mehra and Precott): r = 7% r f = 1% σ rc =.19% Kocherlakota need γ = 7 to match (6) equty premum puzzle to match (4) we need γ very hgh or very low rk free rate puzzle 4

5 Table removed due to copyrght retrcton. Kocherlakota, Narayana R. "The Equty Premum Puzzle: It' Stll a Puzzle." Journal of Economc Lterature 34, no. 1 (1996): (Table and 3). 5 Dcount Factor: LOP and NA I follow Cochrane and Hanen (199) cloely great paper to read two perod "now" and "then" (t and t +1f you prefer) J fundamental aet: x j payoff then q j now prce tack nto x and q (column) vector payoff pace for "then" P {p : p = c x for ome c R} prcng functon π (p) :P R then π (x) =q 5

6 defnton: Law of One Prce (LOP) hold f the prcng functon lnear c x = c 0 x then c q = c 0 q 1 π (c x) =c π (x) =c q defnton: dcount factor y P π (p) =E (yp) Rez repreentaton Theorem LOP (tochatc) dcount factor y P Let Y be the et of all dcount factor note: y may be negatve example: y = x 0 (Exx 0 ) 1 q note: f Exx 0 non-ngular then remove aet from x untl t! a non-ngular Exx 0 mean that (a) there a rk-free aet (b) there are two way of gettng the ame payoff Defnton: No Arbtrage (NA) hold p 0 π (p) 0 p > 0 (wth potve prob.) π (p) > 0 reult NA trctly potve dcount factor y > 0 Let Y ++ be the et of all dcount factor that are potve example 1 proof: m = β t u 0 (c then ) u 0 (c now ) π (c x) = π (c 0 x) cπ (x) = c 0 π (x) cq = c 0 q 6

7 6 Hanen-Jagannathan bound all theore: q = E (mp) m = f (data, parameter) (ee Cochrane book) note p /q rate of return H-J bound: dagnotc tool for model of m pecal cae: data on a ngle exce return relatve to ome baelne aet then π (r) =0o that ntuton: need volatle σ m r = p/q p 0 /q 0 0 = Emr = EmEr + cov (m, r) = EmEr + σ m σ r corr (m, r) EmEr 1 σm σ r = corr (m, r) 1 EmEr 1 σ m σ r σ r σ m Er Em note: Em = 1/R f f therea rk free rate R f let generalze: for any random vector x we can conder the populaton regreon: m = a + x 0 b + e whch jut defne e unquely a havng Ee = 0and cov(x, e) =0 by defnton cov (e, x) =0 var (m) var (x 0 b) 7

8 dea compute x 0 b and var (x 0 b) to get lower bound check whether theore for y pa th tet b = [cov (x, x)] 1 cov (x, y) a = Ey Ex 0 b How to compute b? dea: f x = p then theory help... aume x = p note that o: cov (x, y) = q E (y) E (x) b = [cov (x, x)] 1 [q E (y) E (x)] var x 0 [cov (x, x)] 1 [q E (y) E (x)] = var (x)[var (x)] E (y) E (x) f we knew E (y) we have a lower bound otherwe feable regon for par (E (y),var(y)) convenent no need to recompute lower bound for each theory help ee where the theory fal 3 cae: rk-le return E (y) pnned down and rky return one exce-return q = 0 Sharpe rato and market prce of rk (what we dd before!) general cae very flexble, ee CH paper fgure.1: exce.,.3,.4 from CH paper 8

9 7 Reoluton (?) 7.1 Exotc Preference Rk Averon v. IES (Wel / Epten-Zn) frt-order rk averon (Epten-Zn) habt pertence e.g. u(c t αc t t ) (Abel / Campbell-Cochrane) lo-averon 7. Heterogenou Agent Incomplete Market unnured doyncratc rk (Mankw / Contantnde-Duffe) borrowng contrant (Euler wth nequalty) (Luttmer / Heaton-Luca) contraned optma wth lmted commtment (Alvarez-Jermann) 7.3 Knghtan Uncertanty rk v. uncertanty fear of not undertandng return / uncertanty over probablty dtrbuton / dere for robut decon (Hanen and Sargent) 7.4 No rk premum! no rk premum to explan... htorcal returnon tockwereunexpected (McGratten-Precott) bond are money low return tock more rky than ample (low probablty of a crah) (ee Retz, Cochrane, Wetzman and Barro) 9

10 8 Concluon Rk premum puzzle great example of nterplay between theory and data no trong conenu on reoluton yet many new dea new model hould explore revt the welfare cot of BC (Alvarez and Jermann) 10

Specification -- Assumptions of the Simple Classical Linear Regression Model (CLRM) 1. Introduction

Specification -- Assumptions of the Simple Classical Linear Regression Model (CLRM) 1. Introduction ECONOMICS 35* -- NOTE ECON 35* -- NOTE Specfcaton -- Aumpton of the Smple Clacal Lnear Regreon Model (CLRM). Introducton CLRM tand for the Clacal Lnear Regreon Model. The CLRM alo known a the tandard lnear

More information

Equilibrium with Complete Markets. Instructor: Dmytro Hryshko

Equilibrium with Complete Markets. Instructor: Dmytro Hryshko Equlbrum wth Complete Markets Instructor: Dmytro Hryshko 1 / 33 Readngs Ljungqvst and Sargent. Recursve Macroeconomc Theory. MIT Press. Chapter 8. 2 / 33 Equlbrum n pure exchange, nfnte horzon economes,

More information

Team. Outline. Statistics and Art: Sampling, Response Error, Mixed Models, Missing Data, and Inference

Team. Outline. Statistics and Art: Sampling, Response Error, Mixed Models, Missing Data, and Inference Team Stattc and Art: Samplng, Repone Error, Mxed Model, Mng Data, and nference Ed Stanek Unverty of Maachuett- Amhert, USA 9/5/8 9/5/8 Outlne. Example: Doe-repone Model n Toxcology. ow to Predct Realzed

More information

Additional File 1 - Detailed explanation of the expression level CPD

Additional File 1 - Detailed explanation of the expression level CPD Addtonal Fle - Detaled explanaton of the expreon level CPD A mentoned n the man text, the man CPD for the uterng model cont of two ndvdual factor: P( level gen P( level gen P ( level gen 2 (.).. CPD factor

More information

Statistical Properties of the OLS Coefficient Estimators. 1. Introduction

Statistical Properties of the OLS Coefficient Estimators. 1. Introduction ECOOMICS 35* -- OTE 4 ECO 35* -- OTE 4 Stattcal Properte of the OLS Coeffcent Etmator Introducton We derved n ote the OLS (Ordnary Leat Square etmator ˆβ j (j, of the regreon coeffcent βj (j, n the mple

More information

1 The Sidrauski model

1 The Sidrauski model The Sdrausk model There are many ways to brng money nto the macroeconomc debate. Among the fundamental ssues n economcs the treatment of money s probably the LESS satsfactory and there s very lttle agreement

More information

Notes on Kehoe Perri, Econometrica 2002

Notes on Kehoe Perri, Econometrica 2002 Notes on Kehoe Perr, Econometrca 2002 Jonathan Heathcote November 2nd 2005 There s nothng n these notes that s not n Kehoe Perr NBER Workng Paper 7820 or Kehoe and Perr Econometrca 2002. However, I have

More information

Interest rates parity and no arbitrage as equivalent equilibrium conditions in the international financial assets and goods markets

Interest rates parity and no arbitrage as equivalent equilibrium conditions in the international financial assets and goods markets Interet rate party and no arbtrage a equvalent equlbrum condton n the nternatonal fnancal aet and good market Stefano Bo, Patrce Fontane, Cuong Le Van To cte th veron: Stefano Bo, Patrce Fontane, Cuong

More information

j) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1

j) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1 Random varables Measure of central tendences and varablty (means and varances) Jont densty functons and ndependence Measures of assocaton (covarance and correlaton) Interestng result Condtonal dstrbutons

More information

Copyright (C) 2008 David K. Levine This document is an open textbook; you can redistribute it and/or modify it under the terms of the Creative

Copyright (C) 2008 David K. Levine This document is an open textbook; you can redistribute it and/or modify it under the terms of the Creative Copyrght (C) 008 Davd K. Levne Ths document s an open textbook; you can redstrbute t and/or modfy t under the terms of the Creatve Commons Attrbuton Lcense. Compettve Equlbrum wth Pure Exchange n traders

More information

This appendix presents the derivations and proofs omitted from the main text.

This appendix presents the derivations and proofs omitted from the main text. Onlne Appendx A Appendx: Omtted Dervaton and Proof Th appendx preent the dervaton and proof omtted from the man text A Omtted dervaton n Secton Mot of the analy provded n the man text Here, we formally

More information

The multivariate Gaussian probability density function for random vector X (X 1,,X ) T. diagonal term of, denoted

The multivariate Gaussian probability density function for random vector X (X 1,,X ) T. diagonal term of, denoted Appendx Proof of heorem he multvarate Gauan probablty denty functon for random vector X (X,,X ) px exp / / x x mean and varance equal to the th dagonal term of, denoted he margnal dtrbuton of X Gauan wth

More information

Improvements on Waring s Problem

Improvements on Waring s Problem Improvement on Warng Problem L An-Png Bejng, PR Chna apl@nacom Abtract By a new recurve algorthm for the auxlary equaton, n th paper, we wll gve ome mprovement for Warng problem Keyword: Warng Problem,

More information

Idiosyncratic Investment (or Entrepreneurial) Risk in a Neoclassical Growth Model. George-Marios Angeletos. MIT and NBER

Idiosyncratic Investment (or Entrepreneurial) Risk in a Neoclassical Growth Model. George-Marios Angeletos. MIT and NBER Idosyncratc Investment (or Entrepreneural) Rsk n a Neoclasscal Growth Model George-Maros Angeletos MIT and NBER Motvaton emprcal mportance of entrepreneural or captal-ncome rsk ˆ prvate busnesses account

More information

Economics 2450A: Public Economics Section 10: Education Policies and Simpler Theory of Capital Taxation

Economics 2450A: Public Economics Section 10: Education Policies and Simpler Theory of Capital Taxation Economcs 2450A: Publc Economcs Secton 10: Educaton Polces and Smpler Theory of Captal Taxaton Matteo Parads November 14, 2016 In ths secton we study educaton polces n a smplfed verson of framework analyzed

More information

PROBLEM SET 7 GENERAL EQUILIBRIUM

PROBLEM SET 7 GENERAL EQUILIBRIUM PROBLEM SET 7 GENERAL EQUILIBRIUM Queston a Defnton: An Arrow-Debreu Compettve Equlbrum s a vector of prces {p t } and allocatons {c t, c 2 t } whch satsfes ( Gven {p t }, c t maxmzes βt ln c t subject

More information

Supporting Materials for: Two Monetary Models with Alternating Markets

Supporting Materials for: Two Monetary Models with Alternating Markets Supportng Materals for: Two Monetary Models wth Alternatng Markets Gabrele Camera Chapman Unversty Unversty of Basel YL Chen Federal Reserve Bank of St. Lous 1 Optmal choces n the CIA model On date t,

More information

Economics 8105 Macroeconomic Theory Recitation 1

Economics 8105 Macroeconomic Theory Recitation 1 Economcs 8105 Macroeconomc Theory Rectaton 1 Outlne: Conor Ryan September 6th, 2016 Adapted From Anh Thu (Monca) Tran Xuan s Notes Last Updated September 20th, 2016 Dynamc Economc Envronment Arrow-Debreu

More information

Welfare Properties of General Equilibrium. What can be said about optimality properties of resource allocation implied by general equilibrium?

Welfare Properties of General Equilibrium. What can be said about optimality properties of resource allocation implied by general equilibrium? APPLIED WELFARE ECONOMICS AND POLICY ANALYSIS Welfare Propertes of General Equlbrum What can be sad about optmalty propertes of resource allocaton mpled by general equlbrum? Any crteron used to compare

More information

Asset Pricing. Chapter IX. The Consumption Capital Asset Pricing Model. June 20, 2006

Asset Pricing. Chapter IX. The Consumption Capital Asset Pricing Model. June 20, 2006 Chapter IX. The Consumption Capital Model June 20, 2006 The Representative Agent Hypothesis and its Notion of Equilibrium 9.2.1 An infinitely lived Representative Agent Avoid terminal period problem Equivalence

More information

Supporting Information for: Two Monetary Models with Alternating Markets

Supporting Information for: Two Monetary Models with Alternating Markets Supportng Informaton for: Two Monetary Models wth Alternatng Markets Gabrele Camera Chapman Unversty & Unversty of Basel YL Chen St. Lous Fed November 2015 1 Optmal choces n the CIA model On date t, gven

More information

Pythagorean triples. Leen Noordzij.

Pythagorean triples. Leen Noordzij. Pythagorean trple. Leen Noordz Dr.l.noordz@leennoordz.nl www.leennoordz.me Content A Roadmap for generatng Pythagorean Trple.... Pythagorean Trple.... 3 Dcuon Concluon.... 5 A Roadmap for generatng Pythagorean

More information

Information Acquisition in Global Games of Regime Change (Online Appendix)

Information Acquisition in Global Games of Regime Change (Online Appendix) Informaton Acquton n Global Game of Regme Change (Onlne Appendx) Mchal Szkup and Iabel Trevno Augut 4, 05 Introducton Th appendx contan the proof of all the ntermedate reult that have been omtted from

More information

Improvements on Waring s Problem

Improvements on Waring s Problem Imrovement on Warng Problem L An-Png Bejng 85, PR Chna al@nacom Abtract By a new recurve algorthm for the auxlary equaton, n th aer, we wll gve ome mrovement for Warng roblem Keyword: Warng Problem, Hardy-Lttlewood

More information

Scattering of two identical particles in the center-of. of-mass frame. (b)

Scattering of two identical particles in the center-of. of-mass frame. (b) Lecture # November 5 Scatterng of two dentcal partcle Relatvtc Quantum Mechanc: The Klen-Gordon equaton Interpretaton of the Klen-Gordon equaton The Drac equaton Drac repreentaton for the matrce α and

More information

In the figure below, the point d indicates the location of the consumer that is under competition. Transportation costs are given by td.

In the figure below, the point d indicates the location of the consumer that is under competition. Transportation costs are given by td. UC Berkeley Economcs 11 Sprng 006 Prof. Joseph Farrell / GSI: Jenny Shanefelter Problem Set # - Suggested Solutons. 1.. In ths problem, we are extendng the usual Hotellng lne so that now t runs from [-a,

More information

Expected Value and Variance

Expected Value and Variance MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or

More information

k t+1 + c t A t k t, t=0

k t+1 + c t A t k t, t=0 Macro II (UC3M, MA/PhD Econ) Professor: Matthas Kredler Fnal Exam 6 May 208 You have 50 mnutes to complete the exam There are 80 ponts n total The exam has 4 pages If somethng n the queston s unclear,

More information

Mixed Taxation and Production Efficiency

Mixed Taxation and Production Efficiency Floran Scheuer 2/23/2016 Mxed Taxaton and Producton Effcency 1 Overvew 1. Unform commodty taxaton under non-lnear ncome taxaton Atknson-Stgltz (JPubE 1976) Theorem Applcaton to captal taxaton 2. Unform

More information

Binomial Distribution: Tossing a coin m times. p = probability of having head from a trial. y = # of having heads from n trials (y = 0, 1,..., m).

Binomial Distribution: Tossing a coin m times. p = probability of having head from a trial. y = # of having heads from n trials (y = 0, 1,..., m). [7] Count Data Models () Some Dscrete Probablty Densty Functons Bnomal Dstrbuton: ossng a con m tmes p probablty of havng head from a tral y # of havng heads from n trals (y 0,,, m) m m! fb( y n) p ( p)

More information

Chapter 6 The Effect of the GPS Systematic Errors on Deformation Parameters

Chapter 6 The Effect of the GPS Systematic Errors on Deformation Parameters Chapter 6 The Effect of the GPS Sytematc Error on Deformaton Parameter 6.. General Beutler et al., (988) dd the frt comprehenve tudy on the GPS ytematc error. Baed on a geometrc approach and aumng a unform

More information

Small signal analysis

Small signal analysis Small gnal analy. ntroducton Let u conder the crcut hown n Fg., where the nonlnear retor decrbed by the equaton g v havng graphcal repreentaton hown n Fg.. ( G (t G v(t v Fg. Fg. a D current ource wherea

More information

Lotteries, Sunspots, and Incentive Constraints 1

Lotteries, Sunspots, and Incentive Constraints 1 Lottere, Sunpot, and Incentve Contrant 1 Tmothy J. Kehoe Department of Economc, Unverty of Mnneota, Mnneapol, Mnneota 55455 Reearch Department, Federal Reerve Bank of Mnneapol, Mnneapol, Mnneota 55480

More information

GMM Method (Single-equation) Pongsa Pornchaiwiseskul Faculty of Economics Chulalongkorn University

GMM Method (Single-equation) Pongsa Pornchaiwiseskul Faculty of Economics Chulalongkorn University GMM Method (Sngle-equaton Pongsa Pornchawsesul Faculty of Economcs Chulalongorn Unversty Stochastc ( Gven that, for some, s random COV(, ε E(( µ ε E( ε µ E( ε E( ε (c Pongsa Pornchawsesul, Faculty of Economcs,

More information

A A Non-Constructible Equilibrium 1

A A Non-Constructible Equilibrium 1 A A Non-Contructbe Equbrum 1 The eampe depct a eparabe contet wth three payer and one prze of common vaue 1 (o v ( ) =1 c ( )). I contruct an equbrum (C, G, G) of the contet, n whch payer 1 bet-repone

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

Problem #1. Known: All required parameters. Schematic: Find: Depth of freezing as function of time. Strategy:

Problem #1. Known: All required parameters. Schematic: Find: Depth of freezing as function of time. Strategy: BEE 3500 013 Prelm Soluton Problem #1 Known: All requred parameter. Schematc: Fnd: Depth of freezng a functon of tme. Strategy: In thee mplfed analy for freezng tme, a wa done n cla for a lab geometry,

More information

Two-Layered Model of Blood Flow through Composite Stenosed Artery

Two-Layered Model of Blood Flow through Composite Stenosed Artery Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 4, Iue (December 9), pp. 343 354 (Prevouly, Vol. 4, No.) Applcaton Appled Mathematc: An Internatonal Journal (AAM) Two-ayered Model

More information

A NOTE ON CES FUNCTIONS Drago Bergholt, BI Norwegian Business School 2011

A NOTE ON CES FUNCTIONS Drago Bergholt, BI Norwegian Business School 2011 A NOTE ON CES FUNCTIONS Drago Bergholt, BI Norwegan Busness School 2011 Functons featurng constant elastcty of substtuton CES are wdely used n appled economcs and fnance. In ths note, I do two thngs. Frst,

More information

Liquidity and Asset Pricing (Prof. Pedersen) Short Sale Constraints Due to Limited Commitment

Liquidity and Asset Pricing (Prof. Pedersen) Short Sale Constraints Due to Limited Commitment Lqudty and Asset Prcng (Prof. Pedersen) Short Sale Constrants Due to Lmted Commtment Stjn Van Neuwerburgh March 7, 2005 I. Introducton In the frst module of ths class we wll study asset prcng mplcatons

More information

Credit Card Pricing and Impact of Adverse Selection

Credit Card Pricing and Impact of Adverse Selection Credt Card Prcng and Impact of Adverse Selecton Bo Huang and Lyn C. Thomas Unversty of Southampton Contents Background Aucton model of credt card solctaton - Errors n probablty of beng Good - Errors n

More information

PROBABILITY PRIMER. Exercise Solutions

PROBABILITY PRIMER. Exercise Solutions PROBABILITY PRIMER Exercse Solutons 1 Probablty Prmer, Exercse Solutons, Prncples of Econometrcs, e EXERCISE P.1 (b) X s a random varable because attendance s not known pror to the outdoor concert. Before

More information

Chapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.

Chapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise. Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the

More information

e i is a random error

e i is a random error Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown

More information

Problem Set 3. 1 Offshoring as a Rybzcynski Effect. Economics 245 Fall 2011 International Trade

Problem Set 3. 1 Offshoring as a Rybzcynski Effect. Economics 245 Fall 2011 International Trade Due: Thu, December 1, 2011 Instructor: Marc-Andreas Muendler E-mal: muendler@ucsd.edu Economcs 245 Fall 2011 Internatonal Trade Problem Set 3 November 15, 2011 1 Offshorng as a Rybzcynsk Effect There are

More information

Optimal Expectations

Optimal Expectations Optmal Expectaton Marku K. Brunnermeer rnceton Unverty Jonathan A. arker rnceton Unverty and NBER June 2003 Frt Draft: Augut 2002 Abtract Th paper ntroduce a tractable, tructural model of ubjectve belef.

More information

JAB Chain. Long-tail claims development. ASTIN - September 2005 B.Verdier A. Klinger

JAB Chain. Long-tail claims development. ASTIN - September 2005 B.Verdier A. Klinger JAB Chan Long-tal clams development ASTIN - September 2005 B.Verder A. Klnger Outlne Chan Ladder : comments A frst soluton: Munch Chan Ladder JAB Chan Chan Ladder: Comments Black lne: average pad to ncurred

More information

Start Point and Trajectory Analysis for the Minimal Time System Design Algorithm

Start Point and Trajectory Analysis for the Minimal Time System Design Algorithm Start Pont and Trajectory Analy for the Mnmal Tme Sytem Degn Algorthm ALEXANDER ZEMLIAK, PEDRO MIRANDA Department of Phyc and Mathematc Puebla Autonomou Unverty Av San Claudo /n, Puebla, 757 MEXICO Abtract:

More information

However, since P is a symmetric idempotent matrix, of P are either 0 or 1 [Eigen-values

However, since P is a symmetric idempotent matrix, of P are either 0 or 1 [Eigen-values Fall 007 Soluton to Mdterm Examnaton STAT 7 Dr. Goel. [0 ponts] For the general lnear model = X + ε, wth uncorrelated errors havng mean zero and varance σ, suppose that the desgn matrx X s not necessarly

More information

APPLICATIONS: CHEMICAL AND PHASE EQUILIBRIA

APPLICATIONS: CHEMICAL AND PHASE EQUILIBRIA 5.60 Sprn 2007 Lecture #28 pae PPLICTIOS: CHMICL D PHS QUILIBRI pply tattcal mechanc to develop mcrocopc model for problem you ve treated o far wth macrocopc thermodynamc 0 Product Reactant Separated atom

More information

β0 + β1xi and want to estimate the unknown

β0 + β1xi and want to estimate the unknown SLR Models Estmaton Those OLS Estmates Estmators (e ante) v. estmates (e post) The Smple Lnear Regresson (SLR) Condtons -4 An Asde: The Populaton Regresson Functon B and B are Lnear Estmators (condtonal

More information

Lecture 3 January 31, 2017

Lecture 3 January 31, 2017 CS 224: Advanced Algorthms Sprng 207 Prof. Jelan Nelson Lecture 3 January 3, 207 Scrbe: Saketh Rama Overvew In the last lecture we covered Y-fast tres and Fuson Trees. In ths lecture we start our dscusson

More information

A Dynamic Heterogeneous Beliefs CAPM

A Dynamic Heterogeneous Beliefs CAPM A Dynamc Heterogeneous Belefs CAPM Carl Charella School of Fnance and Economcs Unversty of Technology Sydney, Australa Roberto Dec Dpartmento d Matematca per le Scenze Economche e Socal Unversty of Bologna,

More information

Generalized Linear Methods

Generalized Linear Methods Generalzed Lnear Methods 1 Introducton In the Ensemble Methods the general dea s that usng a combnaton of several weak learner one could make a better learner. More formally, assume that we have a set

More information

Lecture 21: Numerical methods for pricing American type derivatives

Lecture 21: Numerical methods for pricing American type derivatives Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)

More information

Adverse Selection with individual- and joint-life annuities. Susanne Pech *) Working Paper No November 2004

Adverse Selection with individual- and joint-life annuities. Susanne Pech *) Working Paper No November 2004 DEPARTMENT OF ECONOMICS JOHANNES KEPLER UNIVERSITY OF LINZ Advere Selecton wth ndvdual- and ont-lfe annute by Suanne Pech *) Workng Paper No. 041 November 004 Johanne Kepler Unverty of Lnz Department of

More information

Batch RL Via Least Squares Policy Iteration

Batch RL Via Least Squares Policy Iteration Batch RL Va Leat Square Polcy Iteraton Alan Fern * Baed n part on lde by Ronald Parr Overvew Motvaton LSPI Dervaton from LSTD Expermental reult Onlne veru Batch RL Onlne RL: ntegrate data collecton and

More information

2.3 Least-Square regressions

2.3 Least-Square regressions .3 Leat-Square regreon Eample.10 How do chldren grow? The pattern of growth vare from chld to chld, o we can bet undertandng the general pattern b followng the average heght of a number of chldren. Here

More information

Online Appendix to Asset Pricing in Large Information Networks

Online Appendix to Asset Pricing in Large Information Networks Onlne Appendx to Asset Prcng n Large Informaton Networks Han N. Ozsoylev and Johan Walden The proofs of Propostons 1-12 have been omtted from the man text n the nterest of brevty. Ths Onlne Appendx contans

More information

Boning Yang. March 8, 2018

Boning Yang. March 8, 2018 Concentraton Inequaltes by concentraton nequalty Introducton to Basc Concentraton Inequaltes by Florda State Unversty March 8, 2018 Framework Concentraton Inequaltes by 1. concentraton nequalty concentraton

More information

Economics 101. Lecture 4 - Equilibrium and Efficiency

Economics 101. Lecture 4 - Equilibrium and Efficiency Economcs 0 Lecture 4 - Equlbrum and Effcency Intro As dscussed n the prevous lecture, we wll now move from an envronment where we looed at consumers mang decsons n solaton to analyzng economes full of

More information

Designing Service Competitions among Heterogeneous Suppliers

Designing Service Competitions among Heterogeneous Suppliers Degnng Servce Competton among Heterogeneou Suppler Ehan Elah and Saf Benaafar Graduate Program n Indutral & Sytem Engneerng Department of Mechancal Engneerng Unverty of Mnneota, Mnneapol, M 55455 elah@me.umn.edu

More information

Iterative Methods for Searching Optimal Classifier Combination Function

Iterative Methods for Searching Optimal Classifier Combination Function htt://www.cub.buffalo.edu Iteratve Method for Searchng Otmal Clafer Combnaton Functon Sergey Tulyakov Chaohong Wu Venu Govndaraju Unverty at Buffalo Identfcaton ytem: Alce Bob htt://www.cub.buffalo.edu

More information

Chapter 11. Supplemental Text Material. The method of steepest ascent can be derived as follows. Suppose that we have fit a firstorder

Chapter 11. Supplemental Text Material. The method of steepest ascent can be derived as follows. Suppose that we have fit a firstorder S-. The Method of Steepet cent Chapter. Supplemental Text Materal The method of teepet acent can be derved a follow. Suppoe that we have ft a frtorder model y = β + β x and we wh to ue th model to determne

More information

(1 ) (1 ) 0 (1 ) (1 ) 0

(1 ) (1 ) 0 (1 ) (1 ) 0 Appendx A Appendx A contans proofs for resubmsson "Contractng Informaton Securty n the Presence of Double oral Hazard" Proof of Lemma 1: Assume that, to the contrary, BS efforts are achevable under a blateral

More information

Variable Structure Control ~ Basics

Variable Structure Control ~ Basics Varable Structure Control ~ Bac Harry G. Kwatny Department of Mechancal Engneerng & Mechanc Drexel Unverty Outlne A prelmnary example VS ytem, ldng mode, reachng Bac of dcontnuou ytem Example: underea

More information

When Do Switching Costs Make Markets More or Less Competitive?

When Do Switching Costs Make Markets More or Less Competitive? When Do Swtchng Cot Make Market More or Le Compettve? Francco Ruz-Aleda December, 013 Abtract In a two-perod duopoly ettng n whch wtchng cot are the only reaon why product may be perceved a d erentated,

More information

CHAPTER X PHASE-CHANGE PROBLEMS

CHAPTER X PHASE-CHANGE PROBLEMS Chapter X Phae-Change Problem December 3, 18 917 CHAPER X PHASE-CHANGE PROBLEMS X.1 Introducton Clacal Stefan Problem Geometry of Phae Change Problem Interface Condton X. Analytcal Soluton for Soldfcaton

More information

Confidence intervals for the difference and the ratio of Lognormal means with bounded parameters

Confidence intervals for the difference and the ratio of Lognormal means with bounded parameters Songklanakarn J. Sc. Technol. 37 () 3-40 Mar.-Apr. 05 http://www.jt.pu.ac.th Orgnal Artcle Confdence nterval for the dfference and the rato of Lognormal mean wth bounded parameter Sa-aat Nwtpong* Department

More information

Economics 130. Lecture 4 Simple Linear Regression Continued

Economics 130. Lecture 4 Simple Linear Regression Continued Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do

More information

Perfect Competition and the Nash Bargaining Solution

Perfect Competition and the Nash Bargaining Solution Perfect Competton and the Nash Barganng Soluton Renhard John Department of Economcs Unversty of Bonn Adenauerallee 24-42 53113 Bonn, Germany emal: rohn@un-bonn.de May 2005 Abstract For a lnear exchange

More information

The Second Anti-Mathima on Game Theory

The Second Anti-Mathima on Game Theory The Second Ant-Mathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2-player 2-acton zero-sum games 2. 2-player

More information

Estimation of Finite Population Total under PPS Sampling in Presence of Extra Auxiliary Information

Estimation of Finite Population Total under PPS Sampling in Presence of Extra Auxiliary Information Internatonal Journal of Stattc and Analy. ISSN 2248-9959 Volume 6, Number 1 (2016), pp. 9-16 Reearch Inda Publcaton http://www.rpublcaton.com Etmaton of Fnte Populaton Total under PPS Samplng n Preence

More information

Productivity and Reallocation

Productivity and Reallocation Productvty and Reallocaton Motvaton Recent studes hghlght role of reallocaton for productvty growth. Market economes exhbt: Large pace of output and nput reallocaton wth substantal role for entry/ext.

More information

APPENDIX A Some Linear Algebra

APPENDIX A Some Linear Algebra APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,

More information

SCENARIO SELECTION PROBLEM IN NORDIC POWER MARKETS. Juha Ojala. Systems Analysis Laboratory, Helsinki University of Technology

SCENARIO SELECTION PROBLEM IN NORDIC POWER MARKETS. Juha Ojala. Systems Analysis Laboratory, Helsinki University of Technology 2.2.2003 SCENARIO SELECTION ROBLEM IN NORDIC OER MARKETS MAT-2.08 INDEENDENT RESEARCH ROJECT IN ALIED MATHEMATICS Juha Oala oala@cc.hut.f Sytem Analy Laboratory, Heln Unverty of Technology ortfolo Management

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

More information

Lecture 6: Recursive Preferences

Lecture 6: Recursive Preferences Lecture 6: Recursive Preferences Simon Gilchrist Boston Univerity and NBER EC 745 Fall, 2013 Basics Epstein and Zin (1989 JPE, 1991 Ecta) following work by Kreps and Porteus introduced a class of preferences

More information

18.1 Introduction and Recap

18.1 Introduction and Recap CS787: Advanced Algorthms Scrbe: Pryananda Shenoy and Shjn Kong Lecturer: Shuch Chawla Topc: Streamng Algorthmscontnued) Date: 0/26/2007 We contnue talng about streamng algorthms n ths lecture, ncludng

More information

Statistics for Business and Economics

Statistics for Business and Economics Statstcs for Busness and Economcs Chapter 11 Smple Regresson Copyrght 010 Pearson Educaton, Inc. Publshng as Prentce Hall Ch. 11-1 11.1 Overvew of Lnear Models n An equaton can be ft to show the best lnear

More information

Computing Correlated Equilibria in Multi-Player Games

Computing Correlated Equilibria in Multi-Player Games Computng Correlated Equlbra n Mult-Player Games Chrstos H. Papadmtrou Presented by Zhanxang Huang December 7th, 2005 1 The Author Dr. Chrstos H. Papadmtrou CS professor at UC Berkley (taught at Harvard,

More information

Harmonic oscillator approximation

Harmonic oscillator approximation armonc ocllator approxmaton armonc ocllator approxmaton Euaton to be olved We are fndng a mnmum of the functon under the retrcton where W P, P,..., P, Q, Q,..., Q P, P,..., P, Q, Q,..., Q lnwgner functon

More information

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have

More information

ELG3336: Op Amp-based Active Filters

ELG3336: Op Amp-based Active Filters ELG6: Op Amp-baed Actve Flter Advantage: educed ze and weght, and thereore paratc. Increaed relablty and mproved perormance. Smpler degn than or pave lter and can realze a wder range o uncton a well a

More information

Microeconomics: Auctions

Microeconomics: Auctions Mcroeconomcs: Auctons Frédérc Robert-coud ovember 8, Abstract We rst characterze the PBE n a smple rst prce and second prce sealed bd aucton wth prvate values. The key result s that the expected revenue

More information

3.2. Cournot Model Cournot Model

3.2. Cournot Model Cournot Model Matlde Machado Assumptons: All frms produce an homogenous product The market prce s therefore the result of the total supply (same prce for all frms) Frms decde smultaneously how much to produce Quantty

More information

Question 1 carries a weight of 25%; question 2 carries 20%; question 3 carries 25%; and question 4 carries 30%.

Question 1 carries a weight of 25%; question 2 carries 20%; question 3 carries 25%; and question 4 carries 30%. UNIVERSITY OF EAST ANGLIA School of Economcs Man Seres PGT Examnaton 017-18 FINANCIAL ECONOMETRICS ECO-7009A Tme allowed: HOURS Answer ALL FOUR questons. Queston 1 carres a weght of 5%; queston carres

More information

Two-factor model. Statistical Models. Least Squares estimation in LM two-factor model. Rats

Two-factor model. Statistical Models. Least Squares estimation in LM two-factor model. Rats tatstcal Models Lecture nalyss of Varance wo-factor model Overall mean Man effect of factor at level Man effect of factor at level Y µ + α + β + γ + ε Eε f (, ( l, Cov( ε, ε ) lmr f (, nteracton effect

More information

Pitfalls in the use of systemic risk measures*

Pitfalls in the use of systemic risk measures* Ptfalls n the use of systemc rsk measures* Peter Raupach, Deutsche Bundesbank; jont work wth Gunter Löffler, Unversty of Ulm, Germany ESCB Research Cluster 3, 1st Workshop, Athens * To appear n the Journal

More information

SELECTED PROOFS. DeMorgan s formulas: The first one is clear from Venn diagram, or the following truth table:

SELECTED PROOFS. DeMorgan s formulas: The first one is clear from Venn diagram, or the following truth table: SELECTED PROOFS DeMorgan s formulas: The frst one s clear from Venn dagram, or the followng truth table: A B A B A B Ā B Ā B T T T F F F F T F T F F T F F T T F T F F F F F T T T T The second one can be

More information

Online Appendix for A Simpler Theory of Optimal Capital Taxation by Emmanuel Saez and Stefanie Stantcheva

Online Appendix for A Simpler Theory of Optimal Capital Taxation by Emmanuel Saez and Stefanie Stantcheva Onlne Appendx for A Smpler Theory of Optmal Captal Taxaton by Emmanuel Saez and Stefane Stantcheva A. Antcpated Reforms Addtonal Results Optmal tax wth antcpated reform and heterogeneous dscount rates

More information

Multiple Choice. Choose the one that best completes the statement or answers the question.

Multiple Choice. Choose the one that best completes the statement or answers the question. ECON 56 Homework Multple Choce Choose the one that best completes the statement or answers the queston ) The probablty of an event A or B (Pr(A or B)) to occur equals a Pr(A) Pr(B) b Pr(A) + Pr(B) f A

More information

Lecture 3: Probability Distributions

Lecture 3: Probability Distributions Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the

More information

CS-433: Simulation and Modeling Modeling and Probability Review

CS-433: Simulation and Modeling Modeling and Probability Review CS-433: Smulaton and Modelng Modelng and Probablty Revew Exercse 1. (Probablty of Smple Events) Exercse 1.1 The owner of a camera shop receves a shpment of fve cameras from a camera manufacturer. Unknown

More information

f(x,y) = (4(x 2 4)x,2y) = 0 H(x,y) =

f(x,y) = (4(x 2 4)x,2y) = 0 H(x,y) = Problem Set 3: Unconstraned mzaton n R N. () Fnd all crtcal ponts of f(x,y) (x 4) +y and show whch are ma and whch are mnma. () Fnd all crtcal ponts of f(x,y) (y x ) x and show whch are ma and whch are

More information

Online Appendix. t=1 (p t w)q t. Then the first order condition shows that

Online Appendix. t=1 (p t w)q t. Then the first order condition shows that Artcle forthcomng to ; manuscrpt no (Please, provde the manuscrpt number!) 1 Onlne Appendx Appendx E: Proofs Proof of Proposton 1 Frst we derve the equlbrum when the manufacturer does not vertcally ntegrate

More information

Stabilizing, Pareto Improving Policies 1.

Stabilizing, Pareto Improving Policies 1. Stablzng, Pareto Improvng Polce 1. Draft. Reved July, 2009 by Carten Krabbe Nelen Unverta Cattolca Ittuto d Economca Poltca Va Necch, 5 20123 Mlano, Italy Carten.Nelen@uncatt.t Abtract One of the objectve

More information

CS294 Topics in Algorithmic Game Theory October 11, Lecture 7

CS294 Topics in Algorithmic Game Theory October 11, Lecture 7 CS294 Topcs n Algorthmc Game Theory October 11, 2011 Lecture 7 Lecturer: Chrstos Papadmtrou Scrbe: Wald Krchene, Vjay Kamble 1 Exchange economy We consder an exchange market wth m agents and n goods. Agent

More information

Endogenous timing in a mixed oligopoly consisting of a single public firm and foreign competitors. Abstract

Endogenous timing in a mixed oligopoly consisting of a single public firm and foreign competitors. Abstract Endogenous tmng n a mxed olgopoly consstng o a sngle publc rm and oregn compettors Yuanzhu Lu Chna Economcs and Management Academy, Central Unversty o Fnance and Economcs Abstract We nvestgate endogenous

More information

Monetary Policy and the Redistribution Channel

Monetary Policy and the Redistribution Channel Monetary Polcy and the Redstrbuton Channel Adren Auclert MIT MFM Meetng, NYU Stern May 31, 2014 Adren Auclert (MIT) Redstrbuton Channel May 31, 2014 1 / 16 Introducton Polcymakers and the redstrbuton channel

More information