# 8 - MIT OpenCourseWare http/ocw.mit.edu 14.30 Introduction...

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MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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Problem Set #8 - Solutions 14.30 - Intro. to Statistical Methods in Economics Tnst,ri~ctor:Konrwd Menzel Due: Tuesday, April 28, 2009 Question One: Law of Large Numbers and Central Limit Theorem Probably the two most important concepts that you will take away from this course are the Law of Large Numbers and the Central Limit Theorem and how they allow us to use averages to learn about the world around us. I. State the Law of Large Numbers (please, just copy it down from the lecture notes). Solution to 1: Suppose XI,. . . , Xn is a sequence of i.i.d. draws with IEIXi] = p and Var(Xi) = a2 < oo for all i. Then for any E > 0 (typically a small value), the sample mean satisfies lim P(IXn - pI > E) = 0 n We say that X, converges in probability to p. 2. Explain what the Law of Large Numbers tells us about the average of an i.i.d. (inde- pendent, identically distributed) sample of the random variable X with mean p and variance a2. Solution to 2: The law of large numbers tells us that the density of the average of an i.i.d. sample of a random variable X will be concentrated in an "epsilon ball" of radius E. Or, more rigorously, for any E > 0, if we take an infinite sample of a random variable, the density of its mean will be concentrated at p. Suppose you wanted to know the unemployment rate for residents of Cambridge during April 2009. The "unemployed" are defined as "Persons 16 years and over who had no employment during the reference week, were available for work, except for temporary illness, and had made specific efforts to find employment sometime during the 4-week period ending with the reference week. Persons who were waiting to be recalled to a job from which they had been laid off need not have been looking for work to be classified as unemployed." (Source: http://www.econmodel.com/classic/terms/ur.htm .) Suppose you utilize a phone survey to sample the random variable X = l(Emp1oyed) where I(-) is the indicator function for whether someone is employed.
1. Write down an estimator, &, of the unemployment rate, a, which is the fraction of the labor force that is unemployed. Is your estimator a Method of Moments estimator for a Bernoulli random variable? Solution to 1: The estimator of choice would be & = & c%, Xi. This estimator is a Method of Moments estimator which uses the mean of the distribution of Xi, which is a Bernoulli random variable since it only takes on the values 0 and 1. 2. Describe how the Law of Large Numbers applies to the estimator by stating what (at least three) conditions are required to hold about X in order for your estimator to be consistent (by the Law of Large Numbers you copied down from the lecture notes above). Solution to 2: The law of large numbers applies to this estimator since it is an

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## This note was uploaded on 01/30/2010 for the course STAT 430 taught by Professor Jones during the Fall '10 term at Napa Valley College.

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8 - MIT OpenCourseWare http/ocw.mit.edu 14.30 Introduction...

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