A Primer on Asymptotics
Eric Zivot Department of Economics University of Washington September 30, 2003 Revised: October 7, 2009
1
Introduction
Consistency Asymptotic Normality
The two main concepts in asymptotic theory covered in these notes are
Intuitio
Econ 583 Fall 2009 HW #2
Eric Zivot Due: Wednesday, October 14
1. Recall Chebychevs inequality: Let by any random variable with [ ] = and var( ) = 2 Then for every 0 Pr(| | ) var( ) 2 =2 2
Suppose ( 2 ) Using Chebychevs inequality, determine the upper bo
TESTING THE CAPM REVISITED
Surajit Raya*, N. E. Savinb and Ashish Tiwaric
July 14, 2009
a
Morgan Stanley, IM-Global Risk & Analysis, 522 5th Avenue, New York, NY 10036* Department of Economics, Tippie College of Business, University of Iowa, 108 John Papp
Introduction to Dynamic Panel Data: Autoregressive Models with Fixed Eects
Eric Zivot
December 2, 2009
Dynamic Panel Data
yim = yi,m1 + i + im i = 1, . . . , n (individuals) m = 1, . . . , M (time periods) Typical assumptions 1. Stationarity: | < 1 2. E [
Introduction to GMM with Weak Instruments
Eric Zivot
December 9, 2009
Monte Carlo Experiment to Illustrate Problem with 2SLS with Weak Instruments Reference: Zivot, Startz and Nelson (1998). Valid Condence Intervals and Inference in the Presence of Weak I
Multiple Equation Linear GMM
Eric Zivot
November 23, 2009
Multiple Equation Linear GMM
Notation yiM , i = individual; M = equation There are M linear equations, yim = Remarks: 1. No a priori assumptions about cross equation error correlation 2. No cross e
Multiple Equation GMM with Common Coecients: Panel Data
Eric Zivot
November 25, 2009
Multi-equation GMM with common coecients Example (panel wage equation) LW 69i = + S 69i + IQi + EXP R69i + i1 LW 80i = + S 80i + IQi + EXP R80i + i2 Note: common coecient
Hypothesis Testing in a Likelihood Framework
Eric Zivot
November 18, 2009
Wald Statistic The Wald statistic is based directly on the asymptotic normal distribution of mle : mle N , I (mle
A
An implication of the asymptotic normality result is that the us
Maximum Likelihood Estimation
Eric Zivot May 14, 2001 This version: November 15, 2009
1
1.1
Maximum Likelihood Estimation
The Likelihood Function
The joint density is an n dimensional function of the data x1 , . . . , xn given the parameter vector . The j
Maximum Likelihood Estimation
Eric Zivot
November 16, 2009
The Likelihood Function Let X1, . . . , Xn be an iid sample with probability density function (pdf) f (xi; ), where is a (k 1) vector of parameters that characterize f (xi; ). Example: Let XiN (,
Nonlinear GMM
Eric Zivot
November 2, 2009
Nonlinear GMM estimation occurs when the K GMM moment conditions g(wt, ) are nonlinear functions of the p model parameters .
The moment conditions g(wt, ) may be K p nonlinear functions satisfying E [g(wt, 0)] =
Single Equation Linear GMM with Serially Correlated Moment Conditions
Eric Zivot
November 2, 2009
Univariate Time Series Let cfw_yt be an ergodic-stationary time series with E [yt] = and var(yt) < . A fundamental decomposition result is the following: Wol
The Stata Journal (2003) 3, Number 1, pp. 131
Instrumental variables and GMM: Estimation and testing
Mark E. Schaer Christopher F. Baum HeriotWatt University Boston College Steven Stillman New Zealand Department of Labour
Abstract. We discuss instrumental
Example: stylized consumption function (Campbell and Mankiw (1990) ct = 0 + 1yt + 2rt + t, t = 1, . . . , T = 0zt + t L=3 where ct = the log of real per capita consumption (excluding durables), yt = the log of real disposable income, and rt = the ex post
Hypothesis Testing for Linear Models The main types of hypothesis tests are
Overidentication restrictions Coecient restrictions (linear and nonlinear) Subsets of orthogonality restrictions Instrument relevance.
Remark: One should always rst test the over
Single Equation Linear GMM Consider the linear regression model yt zt 0 t Engodeneity The model allows for the possibility that some or all of the elements of zt may be correlated with the error term t (i.e., E [ztk t] 6= 0 for some k). If E [ztk i] 6= 0,
Time Series Concepts A stochastic process cfw_Yt is a sequence of random variables indexed by t=1 time t : cfw_. . . , Y1, Y2, . . . , Yt, Yt+1, . . . A realization of a stochastic process is the sequence of observed data cfw_yt : t=1 cfw_. . . , Y1 = y1,
Economics 583: Econometric Theory I A Primer on Asymptotics
Eric Zivot
October 7, 2009
The two main concepts in asymptotic theory that we will use are
Consistency Asymptotic Normality Intuition
consistency: as we get more and more data, we eventually kn
Supplemental materials for this article are available through the TAS web page at http:/www.amstat.org/publications.
Teachers Corner
Understanding Convergence Concepts: A Visual-Minded and Graphical Simulation-Based Approach
Pierre L AFAYE DE M ICHEAUX an