HEC Lausanne - Advanced Econometrics
|
|
- Eileen Aleesha Richards
- 6 years ago
- Views:
Transcription
1 HEC Lausanne - Advanced Econometrics Christohe HURLI Correction Final Exam. January 4. C. Hurlin Exercise : MLE, arametric tests and the trilogy (3 oints) Part I: Maximum Likelihood Estimation (MLE) Question ( oints): since the random variales X ; ::; X are i:i:d: (.5 oint), the loglikelihood of the samle x ; ::; x is de ned as to e: with (.5 oint) ln f X (x i ; ) ` (; x) ln () X ln f X (x i ; ) () ln (c) + ln (x i ) () So, we have ( oint): ` (; x) ln () ln (c) + X ln (x i ) (3) Question ( oints): the ML estimator of is de ned as to e (.5 oint): The log-likelihood equation (FOC) is the following: g ; x arg max ` (; x) (4) R ln ` (; (5) with (.5 ln ` (; By solving this system, we get: + ln (c) X ln (x i ) (6) ln (c) X ln (x i ) (7)
2 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin The SOC is ased on the Hessian numer: H ; ln ` (; 3 ln (c) X ln (x i ) (8) Given the FOC, the Hessian numer is equal to (.5 oint): H ; x 3 ln (c) X ln (x i ) (9) 3 < () This numer is negative, so we have a maximum. The maximum likelihood estimators of the arameter is de ned as to e (.5 oint): ln (c) X ln (X i ) () Question 3 ( oints): The sequence of i:i:d: (.5 oint) random variales ln (X ) ; ::; ln (X ) satisfy E (ln (X i )) ln (c) : Given the WLL, we have (.5 oint): X ln (X i ) E (ln (X i )) ln (c) () By using the CMP theorem for a function g (z) ln (c) g z (.5 oint), we have: X ln (X i ) g (ln (c) ) (3) or equivalently ln (c) As a consequence (.5 oint): X ln (X i ) ln (c) ln (c) + (4) The estimator is (weakly) consistent. Question 4 ( oints): Since the rolem is regular (.5 oint), we have: d ; I ( ) (5) where denotes the true value of the arameter and I( ) the (average) Fisher information matrix for one oservation. The Fisher information matrix associated to the samle
3 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin 3 is equal to: I ( ) E ( H ( ; X)) (6) E + X 3 ln (c) ln (X i ) (7) ln (c) X E (ln (X i )) (8) (ln (c) ln (c) + ) (9) () () Since the samle is i:i:d:, we have I ( ) I () () As a consequence (.5 oint): d ; (3) or equivalently asy ; (4) Question 5 ( oints). The Fisher information matrix for one oservation is equal to: I ( ) (5) A rst natural consistent estimator is given y (.5 oint): I (6) A second ossile estimator is the BHHH estimator ased on the cross-roduct of the gradients: I X g i x i ; (7) with g i x i ; + ln (c) ln (x i) (8) So, a second consistent estimator is given y (.5 oint): I X + ln (c) ln (x i) (9)
4 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin 4 The asymtotic variance of is equal to (.5 oint): V asy I ( ) So, two alternative estimator of the asymtotic variance of are given y (.5 oint for each estimator): V asy (3) X V asy + ln (c) ln (x i) (3) Part II: Parametric tests Question 6 ( oints). Given the eyman Pearson lemma the rejection region of the UMP test of size is given y (.5 oint): (3) L ( ; x) L ( ; x) < K (33) where K is a constant determined y the size or equivalently It gives (.5 oint): ln ( ) ` ( ; x) ` ( ; x) < ln (K) (34) X ln (c)+ ln (x i )+ ln ( )+ ln (c) X ln (x i ) < ln (K) X ln (x i ) < K (35) where K ln (K) + (ln ( ) ln ( )) + ln (c) is a constant term. Since < ; we have ( oint): X ln (x i ) < K (36) with K 3 K ( X ln (x i ) > K 3 (37) ) : This inequality can e re-exressed as: with K 4 ln (c) form ( oint): ln (c) X ln (x i ) < K 4 (38) K 3 : The rejection region of the UMP test of size has the general W n x : o (x) < A where A is a constant determined y the size : Remark: the exressions of the constant terms K ; K 3 and K 4 are useless (no oint for these exressions). (39)
5 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin 5 Question 7 ( oints). Given the de nition of the error of tye I (.5 oint): Pr (Wj H ) (4) So, here we have (.5 oint): Pr asy < A ; (4) Then, we have: Pr < A asy A (; ) (4) where (:) denotes the cdf of the standard normal distriution. From this exression, we can deduce the critical value of the UMP test of size ( oint): A + () (43) Question 8 ( oints). Consider the test H : H : with < : The rejection region of the UMP test of size is ( oint): W x : (x) < + () (44) W does not deend on (.5 oint). It is also the rejection region of the UMP one-sided test (.5 oint): H : H : < Question 9 ( oints). Consider the one-sided tests: Test A: H : against H : < Test B: H : against H : > The non rejection regions of the UMP tests of size are (.5 oint): W A x : (x) > + (45) W B x : (x) < + So, non rejection region of the two-sided test of size is ( oint): W x : + < (x) < + (46) (47)
6 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin 6 Since () ( ) ; this region can e rewritten as: W x : (x) < (48) The region rejection of the two-sided test of size is (.5 oint): W x : (x) > (49) Question ( oints). By de nition of the ower function, we have (.5 oint): Under the alternative: P () Pr (Wj H ) 8 6 (5) asy H ; with 6 : The ower function can e exressed as: P () Pr W H Pr + Pr < + + The ower function is (.5 oint): < < + + Pr < (5) (5) P () (53) When tends to in nity, two cases have to e considered. If >, then we have (.5 oint): lim P () ( ) + ( ) (54) If <, then we have (.5 oint): lim P () (+) + (+) + (55) Whatever the value of ; the ower function tends to one: The test is consistent. Part III: The trilogy lim P () (56) Question (3 oints). The LR test statistic is de ned as to e (.5 oint): LR H ` ; x H ` ; x (57)
7 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin 7 where H and H resectively denote the ML estimator of the arameter under the null and the alternative. We know that under the alternative H (x) ln (c) As a consequence, we have (.5 oint): X ln (x i ) ln (c) X ln (x i ) (58) H (x) (4 ) (59) The log-likelihood of the samle under the alternative is equal to (.5 oint): H ` ; x ln H ln (c) + H X ln (x H i ) H ln () :3 (6) Under the null, the arameter is known ( :5) and the log-likelihood of the samle is equal to (.5 oint) H ` ; x ln ( ) ln (c) + X ln (x i ) :5 ln (:5) :5 4 + :5 373:87 (6) The LR test statistic (realisation) is equal to (.5 oint): LR (x) H ` ; x H ` ; x ( 373: :3) 9:3 (6) Under some regularity conditions and under the null, we have: LR d () (63) since there is only one restriction imosed. The critical region for a 5% signi cance level is: W x : LR (x) > :95 () 3:845 (64) Conclusion: for a 5% signi cance level, we reject the null H : :5 (.5 oint). Question (3 oints). In this case, the Wald test statistic (realisation) is equal to (.5 oint): H :5 H :5 ( :5) Wald (x) V H asy H 6:5 (65) Under some regularity conditions and under the null, we have (.5 oint): Wald d () (66)
8 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin 8 since there is only one restriction imosed. The critical region for a 5% signi cance level is (.5 oint): W x : Wald (x) > :95 () 3:845 (67) Conclusion: for a 5% signi cance level, we reject the null H : :5 (.5 oint). Question 3 (4 oints). The LM test statistic is de ned as to e (.5 oint): LM s ( c ; x i) I ( c ) (68) where s ( c ; x) is the score of the unconstrained model evaluated at c : In our case, the score is de ned y (.5 oint): s (; ln ` (; + ln (c) The realisation of the score evaluated at c is equal to ( oint) X ln (X i ) (69) s ( c ; x) + X c ln (c) c ln (x i ) c :5 + :5 4 :5 : (7) The estimate of the Fisher information numer associated to the samle I ( c ) is equal to (.5 oint): I ( c ) 44:4444 (7) c :5 So, the realisation of the LM test statistic is (.5 oint): LM (x) s ( c; x) I ( c ) : 44:4444 : (7) Under some regularity conditions and under the null, we have (.5 oint): LM d () (73) since there is only one restriction imosed. The critical region for a 5% signi cance level is: W x : LM (x) > :95 () 3:845 (74) Conclusion: for a 5% signi cance level, we reject the null H : :5 (.5 oint).
9 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin 9 Exercise : Logit model and MLE (5 oints) Question ( oints). the conditional distriution of the deendent variale Y i is a Bernoulli distriution with a (conditional) success roaility equal to i Pr (y i j x i ) : (.5 oint) y i j x i Bernoulli ( i ) Since the variales fy i ; x i g are i:i:d: (.5 oint), the conditional log-likelihood of the samle fy i ; x i g corresonds to the sum of the (log) roaility mass functions associated to the conditional distriutions of y i given x i for i ; ::; (.5 oint): with ` (; yj x) X ln f yjx (y i j x i ; ) (75) f yjx (y i j x i ; ) yi i ( i ) yi (76) As a consequence, the (conditional) log-likelihood of the samle fy i ; x i g is equal to (.5 oint): ` (; yj x) X y i ln (x i ) + X ( y i ) ln ( (x i )) (77) or equivalently ` (; yj x) X ex (x i y i ln ) + ex (x i ) + X ( y i ) ln ex (x i ) + ex (x i ) (78) Question ( oints). By de nition of the score vector, we have ( oint): s (; yj (; yj X (x i y i (x i i X y i (x i ) (x i )x i X (x i ( y i ) (x i i X (x i ( y i ) ) (x i )x i (79) Since (x) (x) ( (x)) ; this exression can e simli ed as follows (.5 oint): s (; yj x) X (x i ) ( y (x i )) i (x i ) X y i ( (x i )) x i x i X X ( y i ) (x i ) x i X (y i y i (x i ) (x i ) + y i (x i )) x i ( y i ) (x i ) ( (x i )) (x i ) x i X (y i (x i )) x i (8)
10 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin So, the score vector of the logit model is simly de ned y (.5 oint): s (; yj x) X x i (y i (x i )) (8) Question 3 ( oints). The Hessian matrix is a K K matrix given y (.5 oint): So, we have (.5 oint): H (; yj (; (; yj (8) H (; yj x) X x i (x i (83) The Hessian matrix is de ned as to e (oint) : H (; yj x) X x i (x i ) x i (84) or equivalently (since (x i ) is a scalar) H (; yj x) X (x i ) x ix i (85) Question 4 (4 oints). The rst ste consists in writing a function Matla to de ne the oosite of the log-likelihood. The following syntax ( oints) is roosed, cf. Figure : Figure : Log-Likelihood function The second ste consists in using the numerical otimisation algorithm of Matla y using the following syntax, cf. Figure ( oints).
11 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin Figure : Main rogram Question 5 ( oints). Since the rolem is regular, the asymtotic variance covariance matrix of the MLE estimator is (.5 oint): V asy I I A consistent estimator for average Fisher information matrix, ased on the Hessian matrix, is (.5 oint): I X H i ; yi j x i H ; yj x (87) So, an estimator of the asymtotic variance covariance matrix of the MLE estimator is given y (.5 oint): In this case, we have (.5 oint): V asy I H ; yj x (86) (88) V asy X x i x i x i (89) Question 6 (3 oints). The following syntax is roosed, cf. Figure 3:
12 Mid-term 3. Advanced Econometrics. HEC Lausanne. C. Hurlin Figure 3: Asymtotic standard errors Exercise 3: OLS and heteroscedasticity (5 oints) The aim of this code is to comute the OLS estimate of the arameter of a linear model, denoted eta in the code (.5 oint) and the corresonding White consistent standard errors ( oint). The matrix M denotes the estimator de ned y (.5 oint): M X " i x i x i (9) where " i denotes the OLS residual for the i th unit and x i the corresonding vector of exlicative variales. The code uses a nite samle correction (.5 oint) for M (Davidson and MacKinnon, 993). The matrix V corresonds to the White consistent estimate of the asymtotic variance covariance matrix of the OLS estimator ( oint). The vector std corresonds to the roust asymtotic standard errors (.5 oint).
Exercises Chapter 4 Statistical Hypothesis Testing
Exercises Chapter 4 Statistical Hypothesis Testing Advanced Econometrics - HEC Lausanne Christophe Hurlin University of Orléans December 5, 013 Christophe Hurlin (University of Orléans) Advanced Econometrics
More informationECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Spring 2013 Instructor: Victor Aguirregabiria
ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Spring 2013 Instructor: Victor Aguirregabiria SOLUTION TO FINAL EXAM Friday, April 12, 2013. From 9:00-12:00 (3 hours) INSTRUCTIONS:
More informationEcon 583 Homework 7 Suggested Solutions: Wald, LM and LR based on GMM and MLE
Econ 583 Homework 7 Suggested Solutions: Wald, LM and LR based on GMM and MLE Eric Zivot Winter 013 1 Wald, LR and LM statistics based on generalized method of moments estimation Let 1 be an iid sample
More informationECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2016 Instructor: Victor Aguirregabiria
ECOOMETRICS II (ECO 24S) University of Toronto. Department of Economics. Winter 26 Instructor: Victor Aguirregabiria FIAL EAM. Thursday, April 4, 26. From 9:am-2:pm (3 hours) ISTRUCTIOS: - This is a closed-book
More informationA Very Brief Summary of Statistical Inference, and Examples
A Very Brief Summary of Statistical Inference, and Examples Trinity Term 2009 Prof. Gesine Reinert Our standard situation is that we have data x = x 1, x 2,..., x n, which we view as realisations of random
More informationElements of Asymptotic Theory. James L. Powell Department of Economics University of California, Berkeley
Elements of Asymtotic Theory James L. Powell Deartment of Economics University of California, Berkeley Objectives of Asymtotic Theory While exact results are available for, say, the distribution of the
More informationChapter 6: Endogeneity and Instrumental Variables (IV) estimator
Chapter 6: Endogeneity and Instrumental Variables (IV) estimator Advanced Econometrics - HEC Lausanne Christophe Hurlin University of Orléans December 15, 2013 Christophe Hurlin (University of Orléans)
More informationChapter 1. GMM: Basic Concepts
Chapter 1. GMM: Basic Concepts Contents 1 Motivating Examples 1 1.1 Instrumental variable estimator....................... 1 1.2 Estimating parameters in monetary policy rules.............. 2 1.3 Estimating
More informationTesting Linear Restrictions: cont.
Testing Linear Restrictions: cont. The F-statistic is closely connected with the R of the regression. In fact, if we are testing q linear restriction, can write the F-stastic as F = (R u R r)=q ( R u)=(n
More informationECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2014 Instructor: Victor Aguirregabiria
ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2014 Instructor: Victor guirregabiria SOLUTION TO FINL EXM Monday, pril 14, 2014. From 9:00am-12:00pm (3 hours) INSTRUCTIONS:
More informationChapter 4: Constrained estimators and tests in the multiple linear regression model (Part III)
Chapter 4: Constrained estimators and tests in the multiple linear regression model (Part III) Florian Pelgrin HEC September-December 2010 Florian Pelgrin (HEC) Constrained estimators September-December
More informationSTK4900/ Lecture 7. Program
STK4900/9900 - Lecture 7 Program 1. Logistic regression with one redictor 2. Maximum likelihood estimation 3. Logistic regression with several redictors 4. Deviance and likelihood ratio tests 5. A comment
More informationEstimating Time-Series Models
Estimating ime-series Models he Box-Jenkins methodology for tting a model to a scalar time series fx t g consists of ve stes:. Decide on the order of di erencing d that is needed to roduce a stationary
More informationMax. Likelihood Estimation. Outline. Econometrics II. Ricardo Mora. Notes. Notes
Maximum Likelihood Estimation Econometrics II Department of Economics Universidad Carlos III de Madrid Máster Universitario en Desarrollo y Crecimiento Económico Outline 1 3 4 General Approaches to Parameter
More informationsimple if it completely specifies the density of x
3. Hypothesis Testing Pure significance tests Data x = (x 1,..., x n ) from f(x, θ) Hypothesis H 0 : restricts f(x, θ) Are the data consistent with H 0? H 0 is called the null hypothesis simple if it completely
More informationChapter 3. GMM: Selected Topics
Chater 3. GMM: Selected oics Contents Otimal Instruments. he issue of interest..............................2 Otimal Instruments under the i:i:d: assumtion..............2. he basic result............................2.2
More information1 Extremum Estimators
FINC 9311-21 Financial Econometrics Handout Jialin Yu 1 Extremum Estimators Let θ 0 be a vector of k 1 unknown arameters. Extremum estimators: estimators obtained by maximizing or minimizing some objective
More informationMaximum Likelihood (ML) Estimation
Econometrics 2 Fall 2004 Maximum Likelihood (ML) Estimation Heino Bohn Nielsen 1of32 Outline of the Lecture (1) Introduction. (2) ML estimation defined. (3) ExampleI:Binomialtrials. (4) Example II: Linear
More informationECON 4130 Supplementary Exercises 1-4
HG Set. 0 ECON 430 Sulementary Exercises - 4 Exercise Quantiles (ercentiles). Let X be a continuous random variable (rv.) with df f( x ) and cdf F( x ). For 0< < we define -th quantile (or 00-th ercentile),
More informationFinal Exam. 1. (6 points) True/False. Please read the statements carefully, as no partial credit will be given.
1. (6 points) True/False. Please read the statements carefully, as no partial credit will be given. (a) If X and Y are independent, Corr(X, Y ) = 0. (b) (c) (d) (e) A consistent estimator must be asymptotically
More informationExercises Econometric Models
Exercises Econometric Models. Let u t be a scalar random variable such that E(u t j I t ) =, t = ; ; ::::, where I t is the (stochastic) information set available at time t. Show that under the hyothesis
More informationElements of Asymptotic Theory. James L. Powell Department of Economics University of California, Berkeley
Elements of Asymtotic Theory James L. Powell Deartment of Economics University of California, Berkeley Objectives of Asymtotic Theory While exact results are available for, say, the distribution of the
More informationIntroduction Large Sample Testing Composite Hypotheses. Hypothesis Testing. Daniel Schmierer Econ 312. March 30, 2007
Hypothesis Testing Daniel Schmierer Econ 312 March 30, 2007 Basics Parameter of interest: θ Θ Structure of the test: H 0 : θ Θ 0 H 1 : θ Θ 1 for some sets Θ 0, Θ 1 Θ where Θ 0 Θ 1 = (often Θ 1 = Θ Θ 0
More informationLECTURE 7 NOTES. x n. d x if. E [g(x n )] E [g(x)]
LECTURE 7 NOTES 1. Convergence of random variables. Before delving into the large samle roerties of the MLE, we review some concets from large samle theory. 1. Convergence in robability: x n x if, for
More informationMicroeconomics Fall 2017 Problem set 1: Possible answers
Microeconomics Fall 07 Problem set Possible answers Each answer resents only one way of solving the roblem. Other right answers are ossible and welcome. Exercise For each of the following roerties, draw
More informationi) the probability of type I error; ii) the 95% con dence interval; iii) the p value; iv) the probability of type II error; v) the power of a test.
Problem Set 5. Questions:. Exlain what is: i) the robability of tye I error; ii) the 95% con dence interval; iii) the value; iv) the robability of tye II error; v) the ower of a test.. Solve exercise 3.
More informationGraduate Econometrics I: Maximum Likelihood I
Graduate Econometrics I: Maximum Likelihood I Yves Dominicy Université libre de Bruxelles Solvay Brussels School of Economics and Management ECARES Yves Dominicy Graduate Econometrics I: Maximum Likelihood
More informationGreene, Econometric Analysis (6th ed, 2008)
EC771: Econometrics, Spring 2010 Greene, Econometric Analysis (6th ed, 2008) Chapter 17: Maximum Likelihood Estimation The preferred estimator in a wide variety of econometric settings is that derived
More informationBackground. GLM with clustered data. The problem. Solutions. A fixed effects approach
Background GLM with clustered data A fixed effects aroach Göran Broström Poisson or Binomial data with the following roerties A large data set, artitioned into many relatively small grous, and where members
More informationAsymptotic F Test in a GMM Framework with Cross Sectional Dependence
Asymtotic F Test in a GMM Framework with Cross Sectional Deendence Yixiao Sun Deartment of Economics University of California, San Diego Min Seong Kim y Deartment of Economics Ryerson University First
More informationOn the asymptotic sizes of subset Anderson-Rubin and Lagrange multiplier tests in linear instrumental variables regression
On the asymtotic sizes of subset Anderson-Rubin and Lagrange multilier tests in linear instrumental variables regression Patrik Guggenberger Frank Kleibergeny Sohocles Mavroeidisz Linchun Chen\ June 22
More informationDiscrete Dependent Variable Models
Discrete Dependent Variable Models James J. Heckman University of Chicago This draft, April 10, 2006 Here s the general approach of this lecture: Economic model Decision rule (e.g. utility maximization)
More informationProblem set 1 - Solutions
EMPIRICAL FINANCE AND FINANCIAL ECONOMETRICS - MODULE (8448) Problem set 1 - Solutions Exercise 1 -Solutions 1. The correct answer is (a). In fact, the process generating daily prices is usually assumed
More informationSuggested Solution for PS #5
Cornell University Department of Economics Econ 62 Spring 28 TA: Jae Ho Yun Suggested Solution for S #5. (Measurement Error, IV) (a) This is a measurement error problem. y i x i + t i + " i t i t i + i
More informationMarkov-Switching Models with Endogenous Explanatory Variables. Chang-Jin Kim 1
Markov-Switching Models with Endogenous Explanatory Variables by Chang-Jin Kim 1 Dept. of Economics, Korea University and Dept. of Economics, University of Washington First draft: August, 2002 This version:
More information(a) (3 points) Construct a 95% confidence interval for β 2 in Equation 1.
Problem 1 (21 points) An economist runs the regression y i = β 0 + x 1i β 1 + x 2i β 2 + x 3i β 3 + ε i (1) The results are summarized in the following table: Equation 1. Variable Coefficient Std. Error
More informationThe following document is intended for online publication only (authors webpage).
The following document is intended for online ublication only (authors webage). Sulement to Identi cation and stimation of Distributional Imacts of Interventions Using Changes in Inequality Measures, Part
More informationTesting and Model Selection
Testing and Model Selection This is another digression on general statistics: see PE App C.8.4. The EViews output for least squares, probit and logit includes some statistics relevant to testing hypotheses
More informationJohan Lyhagen Department of Information Science, Uppsala University. Abstract
Why not use standard anel unit root test for testing PPP Johan Lyhagen Deartment of Information Science, Usala University Abstract In this aer we show the consequences of alying a anel unit root test that
More informationLecture 21. Hypothesis Testing II
Lecture 21. Hypothesis Testing II December 7, 2011 In the previous lecture, we dened a few key concepts of hypothesis testing and introduced the framework for parametric hypothesis testing. In the parametric
More informationChapter 2. Dynamic panel data models
Chapter 2. Dynamic panel data models School of Economics and Management - University of Geneva Christophe Hurlin, Université of Orléans University of Orléans April 2018 C. Hurlin (University of Orléans)
More information1. The Multivariate Classical Linear Regression Model
Business School, Brunel University MSc. EC550/5509 Modelling Financial Decisions and Markets/Introduction to Quantitative Methods Prof. Menelaos Karanasos (Room SS69, Tel. 08956584) Lecture Notes 5. The
More informationMaximum Likelihood Asymptotic Theory. Eduardo Rossi University of Pavia
Maximum Likelihood Asymtotic Theory Eduardo Rossi University of Pavia Slutsky s Theorem, Cramer s Theorem Slutsky s Theorem Let {X N } be a random sequence converging in robability to a constant a, and
More informationChapter 2. Review of basic Statistical methods 1 Distribution, conditional distribution and moments
Chapter 2. Review of basic Statistical methods 1 Distribution, conditional distribution and moments We consider two kinds of random variables: discrete and continuous random variables. For discrete random
More informationBOOTSTRAP FOR PANEL DATA MODELS
BOOSRAP FOR PAEL DAA MODELS Bertrand HOUKAOUO Université de Montréal, CIREQ July 3, 2008 Preliminary, do not quote without ermission. Astract his aer considers ootstra methods for anel data models with
More information11. Bootstrap Methods
11. Bootstrap Methods c A. Colin Cameron & Pravin K. Trivedi 2006 These transparencies were prepared in 20043. They can be used as an adjunct to Chapter 11 of our subsequent book Microeconometrics: Methods
More informationStatistical Treatment Choice Based on. Asymmetric Minimax Regret Criteria
Statistical Treatment Coice Based on Asymmetric Minimax Regret Criteria Aleksey Tetenov y Deartment of Economics ortwestern University ovember 5, 007 (JOB MARKET PAPER) Abstract Tis aer studies te roblem
More informationMaximum Likelihood Tests and Quasi-Maximum-Likelihood
Maximum Likelihood Tests and Quasi-Maximum-Likelihood Wendelin Schnedler Department of Economics University of Heidelberg 10. Dezember 2007 Wendelin Schnedler (AWI) Maximum Likelihood Tests and Quasi-Maximum-Likelihood10.
More informationLECTURE 13: TIME SERIES I
1 LECTURE 13: TIME SERIES I AUTOCORRELATION: Consider y = X + u where y is T 1, X is T K, is K 1 and u is T 1. We are using T and not N for sample size to emphasize that this is a time series. The natural
More informationLECTURE 10: NEYMAN-PEARSON LEMMA AND ASYMPTOTIC TESTING. The last equality is provided so this can look like a more familiar parametric test.
Economics 52 Econometrics Professor N.M. Kiefer LECTURE 1: NEYMAN-PEARSON LEMMA AND ASYMPTOTIC TESTING NEYMAN-PEARSON LEMMA: Lesson: Good tests are based on the likelihood ratio. The proof is easy in the
More information1.5 Testing and Model Selection
1.5 Testing and Model Selection The EViews output for least squares, probit and logit includes some statistics relevant to testing hypotheses (e.g. Likelihood Ratio statistic) and to choosing between specifications
More informationLecture 3 Consistency of Extremum Estimators 1
Lecture 3 Consistency of Extremum Estimators 1 This lecture shows how one can obtain consistency of extremum estimators. It also shows how one can find the robability limit of extremum estimators in cases
More informationLecture 32: Asymptotic confidence sets and likelihoods
Lecture 32: Asymptotic confidence sets and likelihoods Asymptotic criterion In some problems, especially in nonparametric problems, it is difficult to find a reasonable confidence set with a given confidence
More informationASYMPTOTIC RESULTS OF A HIGH DIMENSIONAL MANOVA TEST AND POWER COMPARISON WHEN THE DIMENSION IS LARGE COMPARED TO THE SAMPLE SIZE
J Jaan Statist Soc Vol 34 No 2004 9 26 ASYMPTOTIC RESULTS OF A HIGH DIMENSIONAL MANOVA TEST AND POWER COMPARISON WHEN THE DIMENSION IS LARGE COMPARED TO THE SAMPLE SIZE Yasunori Fujikoshi*, Tetsuto Himeno
More informationNotes on Instrumental Variables Methods
Notes on Instrumental Variables Methods Michele Pellizzari IGIER-Bocconi, IZA and frdb 1 The Instrumental Variable Estimator Instrumental variable estimation is the classical solution to the roblem of
More informationIntroduction to Estimation Methods for Time Series models Lecture 2
Introduction to Estimation Methods for Time Series models Lecture 2 Fulvio Corsi SNS Pisa Fulvio Corsi Introduction to Estimation () Methods for Time Series models Lecture 2 SNS Pisa 1 / 21 Estimators:
More informationEstimation and Detection
Estimation and Detection Lecture : Detection Theory Unknown Parameters Dr. ir. Richard C. Hendriks //05 Previous Lecture H 0 : T (x) < H : T (x) > Using detection theory, rules can be derived on how to
More informationECONOMET RICS P RELIM EXAM August 24, 2010 Department of Economics, Michigan State University
ECONOMET RICS P RELIM EXAM August 24, 2010 Department of Economics, Michigan State University Instructions: Answer all four (4) questions. Be sure to show your work or provide su cient justi cation for
More information2014 Preliminary Examination
014 reliminary Examination 1) Standard error consistency and test statistic asymptotic normality in linear models Consider the model for the observable data y t ; x T t n Y = X + U; (1) where is a k 1
More informationEconometrics Midterm Examination Answers
Econometrics Midterm Examination Answers March 4, 204. Question (35 points) Answer the following short questions. (i) De ne what is an unbiased estimator. Show that X is an unbiased estimator for E(X i
More informationLikelihood Ratio Test in High-Dimensional Logistic Regression Is Asymptotically a Rescaled Chi-Square
Likelihood Ratio Test in High-Dimensional Logistic Regression Is Asymptotically a Rescaled Chi-Square Yuxin Chen Electrical Engineering, Princeton University Coauthors Pragya Sur Stanford Statistics Emmanuel
More informationAdvanced Quantitative Methods: maximum likelihood
Advanced Quantitative Methods: Maximum Likelihood University College Dublin 4 March 2014 1 2 3 4 5 6 Outline 1 2 3 4 5 6 of straight lines y = 1 2 x + 2 dy dx = 1 2 of curves y = x 2 4x + 5 of curves y
More informationGMM estimation of spatial panels
MRA Munich ersonal ReEc Archive GMM estimation of spatial panels Francesco Moscone and Elisa Tosetti Brunel University 7. April 009 Online at http://mpra.ub.uni-muenchen.de/637/ MRA aper No. 637, posted
More informationSome General Types of Tests
Some General Types of Tests We may not be able to find a UMP or UMPU test in a given situation. In that case, we may use test of some general class of tests that often have good asymptotic properties.
More informationCh. 5 Hypothesis Testing
Ch. 5 Hypothesis Testing The current framework of hypothesis testing is largely due to the work of Neyman and Pearson in the late 1920s, early 30s, complementing Fisher s work on estimation. As in estimation,
More information6. MAXIMUM LIKELIHOOD ESTIMATION
6 MAXIMUM LIKELIHOOD ESIMAION [1] Maximum Likelihood Estimator (1) Cases in which θ (unknown parameter) is scalar Notational Clarification: From now on, we denote the true value of θ as θ o hen, view θ
More informationHotelling s Two- Sample T 2
Chater 600 Hotelling s Two- Samle T Introduction This module calculates ower for the Hotelling s two-grou, T-squared (T) test statistic. Hotelling s T is an extension of the univariate two-samle t-test
More informationLet us first identify some classes of hypotheses. simple versus simple. H 0 : θ = θ 0 versus H 1 : θ = θ 1. (1) one-sided
Let us first identify some classes of hypotheses. simple versus simple H 0 : θ = θ 0 versus H 1 : θ = θ 1. (1) one-sided H 0 : θ θ 0 versus H 1 : θ > θ 0. (2) two-sided; null on extremes H 0 : θ θ 1 or
More informationIntroduction Model secication tests are a central theme in the econometric literature. The majority of the aroaches fall into two categories. In the r
Reversed Score and Likelihood Ratio Tests Geert Dhaene Universiteit Gent and ORE Olivier Scaillet Universite atholique de Louvain January 2 Abstract Two extensions of a model in the resence of an alternative
More informationA Very Brief Summary of Statistical Inference, and Examples
A Very Brief Summary of Statistical Inference, and Examples Trinity Term 2008 Prof. Gesine Reinert 1 Data x = x 1, x 2,..., x n, realisations of random variables X 1, X 2,..., X n with distribution (model)
More informationTesting Weak Cross-Sectional Dependence in Large Panels
esting Weak Cross-Sectional Deendence in Large Panels M. Hashem Pesaran University of Southern California, and rinity College, Cambridge January, 3 Abstract his aer considers testing the hyothesis that
More informationEconomics 620, Lecture 13: Time Series I
Economics 620, Lecture 13: Time Series I Nicholas M. Kiefer Cornell University Professor N. M. Kiefer (Cornell University) Lecture 13: Time Series I 1 / 19 AUTOCORRELATION Consider y = X + u where y is
More informationGeneralized linear models
Generalized linear models Douglas Bates November 01, 2010 Contents 1 Definition 1 2 Links 2 3 Estimating parameters 5 4 Example 6 5 Model building 8 6 Conclusions 8 7 Summary 9 1 Generalized Linear Models
More informationEconomics 583: Econometric Theory I A Primer on Asymptotics
Economics 583: Econometric Theory I A Primer on Asymptotics Eric Zivot January 14, 2013 The two main concepts in asymptotic theory that we will use are Consistency Asymptotic Normality Intuition consistency:
More informationBenoît MULKAY Université de Montpellier. January Preliminary, Do not quote!
Bivariate Probit Estimation for Panel Data: a two-ste Gauss-Hermite Quadrature Aroach with an alication to roduct and rocess innovations for France Benoît MULKAY Université de Montellier January 05 Preliminary,
More informationSpring 2012 Math 541B Exam 1
Spring 2012 Math 541B Exam 1 1. A sample of size n is drawn without replacement from an urn containing N balls, m of which are red and N m are black; the balls are otherwise indistinguishable. Let X denote
More informationEstimating function analysis for a class of Tweedie regression models
Title Estimating function analysis for a class of Tweedie regression models Author Wagner Hugo Bonat Deartamento de Estatística - DEST, Laboratório de Estatística e Geoinformação - LEG, Universidade Federal
More informationCollection of Formulae and Statistical Tables for the B2-Econometrics and B3-Time Series Analysis courses and exams
Collection of Formulae and Statistical Tables for the B2-Econometrics and B3-Time Series Analysis courses and exams Lars Forsberg Uppsala University Spring 2015 Abstract This collection of formulae is
More informationtests 17.1 Simple versus compound
PAS204: Lecture 17. tests UMP ad asymtotic I this lecture, we will idetify UMP tests, wherever they exist, for comarig a simle ull hyothesis with a comoud alterative. We also look at costructig tests based
More informationExtensions to the Basic Framework II
Topic 7 Extensions to the Basic Framework II ARE/ECN 240 A Graduate Econometrics Professor: Òscar Jordà Outline of this topic Nonlinear regression Limited Dependent Variable regression Applications of
More informationAdvanced Econometrics II (Part 1)
Advanced Econometrics II (Part 1) Dr. Mehdi Hosseinkouchack Goethe University Frankfurt Summer 2016 osseinkouchack (Goethe University Frankfurt) Review Slides Summer 2016 1 / 22 Distribution For simlicity,
More information(c) i) In ation (INFL) is regressed on the unemployment rate (UNR):
BRUNEL UNIVERSITY Master of Science Degree examination Test Exam Paper 005-006 EC500: Modelling Financial Decisions and Markets EC5030: Introduction to Quantitative methods Model Answers. COMPULSORY (a)
More informationEconomics 241B Review of Limit Theorems for Sequences of Random Variables
Economics 241B Review of Limit Theorems for Sequences of Random Variables Convergence in Distribution The previous de nitions of convergence focus on the outcome sequences of a random variable. Convergence
More informationChapter 6. Phillip Hall - Room 537, Huxley
Chater 6 6 Partial Derivatives.................................................... 72 6. Higher order artial derivatives...................................... 73 6.2 Matrix of artial derivatives.........................................74
More informationAdvanced Econometrics
Advanced Econometrics Dr. Andrea Beccarini Center for Quantitative Economics Winter 2013/2014 Andrea Beccarini (CQE) Econometrics Winter 2013/2014 1 / 156 General information Aims and prerequisites Objective:
More informationTopic 7: Using identity types
Toic 7: Using identity tyes June 10, 2014 Now we would like to learn how to use identity tyes and how to do some actual mathematics with them. By now we have essentially introduced all inference rules
More information1 Quantitative Techniques in Practice
1 Quantitative Techniques in Practice 1.1 Lecture 2: Stationarity, spurious regression, etc. 1.1.1 Overview In the rst part we shall look at some issues in time series economics. In the second part we
More informationModels, Testing, and Correction of Heteroskedasticity. James L. Powell Department of Economics University of California, Berkeley
Models, Testing, and Correction of Heteroskedasticity James L. Powell Department of Economics University of California, Berkeley Aitken s GLS and Weighted LS The Generalized Classical Regression Model
More informationEstimation of the large covariance matrix with two-step monotone missing data
Estimation of the large covariance matrix with two-ste monotone missing data Masashi Hyodo, Nobumichi Shutoh 2, Takashi Seo, and Tatjana Pavlenko 3 Deartment of Mathematical Information Science, Tokyo
More informationThe power performance of fixed-t panel unit root tests allowing for structural breaks in their deterministic components
ATHES UIVERSITY OF ECOOMICS AD BUSIESS DEPARTMET OF ECOOMICS WORKIG PAPER SERIES 23-203 The ower erformance of fixed-t anel unit root tests allowing for structural breaks in their deterministic comonents
More informationThe Logit Model: Estimation, Testing and Interpretation
The Logit Model: Estimation, Testing and Interpretation Herman J. Bierens October 25, 2008 1 Introduction to maximum likelihood estimation 1.1 The likelihood function Consider a random sample Y 1,...,
More informationMLE and GMM. Li Zhao, SJTU. Spring, Li Zhao MLE and GMM 1 / 22
MLE and GMM Li Zhao, SJTU Spring, 2017 Li Zhao MLE and GMM 1 / 22 Outline 1 MLE 2 GMM 3 Binary Choice Models Li Zhao MLE and GMM 2 / 22 Maximum Likelihood Estimation - Introduction For a linear model y
More informationDirection: This test is worth 250 points and each problem worth points. DO ANY SIX
Term Test 3 December 5, 2003 Name Math 52 Student Number Direction: This test is worth 250 points and each problem worth 4 points DO ANY SIX PROBLEMS You are required to complete this test within 50 minutes
More informationSystem Reliability Estimation and Confidence Regions from Subsystem and Full System Tests
009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 0-, 009 FrB4. System Reliability Estimation and Confidence Regions from Subsystem and Full System Tests James C. Sall Abstract
More informationRobustness of classifiers to uniform l p and Gaussian noise Supplementary material
Robustness of classifiers to uniform l and Gaussian noise Sulementary material Jean-Yves Franceschi Ecole Normale Suérieure de Lyon LIP UMR 5668 Omar Fawzi Ecole Normale Suérieure de Lyon LIP UMR 5668
More informationEconometrics I, Estimation
Econometrics I, Estimation Department of Economics Stanford University September, 2008 Part I Parameter, Estimator, Estimate A parametric is a feature of the population. An estimator is a function of the
More informationQuick Review on Linear Multiple Regression
Quick Review on Linear Multiple Regression Mei-Yuan Chen Department of Finance National Chung Hsing University March 6, 2007 Introduction for Conditional Mean Modeling Suppose random variables Y, X 1,
More informationIntroduction to Maximum Likelihood Estimation
Introduction to Maximum Likelihood Estimation Eric Zivot July 26, 2012 The Likelihood Function Let 1 be an iid sample with pdf ( ; ) where is a ( 1) vector of parameters that characterize ( ; ) Example:
More informationChapter 7: Special Distributions
This chater first resents some imortant distributions, and then develos the largesamle distribution theory which is crucial in estimation and statistical inference Discrete distributions The Bernoulli
More informationCombining Logistic Regression with Kriging for Mapping the Risk of Occurrence of Unexploded Ordnance (UXO)
Combining Logistic Regression with Kriging for Maing the Risk of Occurrence of Unexloded Ordnance (UXO) H. Saito (), P. Goovaerts (), S. A. McKenna (2) Environmental and Water Resources Engineering, Deartment
More information