Central Limit Theorem: Empirical Examples and Applications, Study notes of Introduction to Macroeconomics

Download Central Limit Theorem: Empirical Examples and Applications and more Study notes Introduction to Macroeconomics in PDF only on Docsity! Empirical Examples of the Central Limit Theorem: Lecture XVI I. Back to Asymptotic Normality A. The characteristic function of a random variable X is defined as           cos sin cos sin itX X t E e E tX i tX E tX iE tX                   Note that this definition parallels the definition of the moment-generating function   tXXM t E e    1. Like the moment-generating function there is a one-to-one correspondence between the characteristic function and the distribution of random variable. Two random variables with the same characteristic function are distributed the same. 2. The characteristic function of the uniform distribution function of the uniform distribution function is   1itX t e   The characteristic function of the Normal distribution function is   2 2 2 tit X t e     The Gamma distribution function       1 0, r r Xx e f X X r        which implies the characteristic function     1 1 X r t it     B. Taking a Taylor series expansion of around the point 0t  yields          2 2 1 1 0 0 0 1! 2! : Z Z Z Zt t t o t X Z n               To work with this expression we note that  0 1X  For any random variable X , and      0k k kX i E X  AEB 6933 – Mathematical Statistics for Food and Resource Economics Lecture XVI Professor Charles B. Moss Fall 2007 2 Putting these two results into the second-order Taylor series expansion             2 2 2 2 2 2 2 1 1 2 2 : 0, 1 E ZE Z t t t o t o t i i E Z E Z          Thus,                       2 2 2 2 2 2 2 2 2 1 1 0 0 0 1! 2! 1 2 1 1 : ~ 0,1 2 2 Z Z Z Z iy t t t o t E ZE Z t t o t i i t t o t E e y N                    II. Wrapping Up Loose Ends. A. Most of these examples can be found in Casella and Berger Chapter 4. B. Application of Holder’s Inequality. 1. Holder’s Inequality       qqpp YEXEXYEXYE 11  2. Example 4.7.1: If X and Y have means X , Y and variances 2 X , 2 Y , respectively, we can apply the Cauchy-Schwartz Inequality (Holders inequality with 1 2 p q  ) to get            2 1 22 1 2 YXYX YEXEYXE   Squaring both sides and substituting for variances and covariances yields    222, YXYXCov  Which implies that the absolute value of the correlation coefficient is less than one. C. Application of Chebychev’s Inequality: 1. Chebychev’s Inequality: Let X be a random variable and let  g x be a nonnegative function. Then, for any 0r        r XgE rXgP  2. Example 4.7.3: The most widespread use of Chebychev’s Inequality involves means and variances. Let     2 2 x g x     , where  E X  and  2 V X  . Let 2r t .     22 2 2 2 2 2 11 t X E t t X P                      