2_Probability and Statistics

n and iid then the random variable n x z i2 i 1 2 has

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Unformatted text preview: ent, standard normal random variables in the following way: Suppose that Z i N (0,1) , i = 1,..., n and iid Then, the random variable n X º å Z i2 i =1 2 has a chi-square distribution with n degrees of freedom. We write this as X cn 2 If X cn , then E (X ) = n and Var (X ) = 2n (For you: Verify this.) VER. 9/11/2012. © P. KOLM 26 The t Distribution The t distribution is a workhorse in classical statistics and multiple regression analysis. We obtain it from (1) a standard normal, and (2) a chi-square random variable in the following way: 2 Suppose Z N (0,1) and X cn , and that Z and X are independent Then, the random variable T= Z X n has a t distribution with n degrees of freedom. We write this as T tn If T tn , then E (T ) = 0 (for n > 1) and Var (T ) = VER. 9/11/2012. © P. KOLM n (for n > 2 )4 n -2 27 The F Distribution The F distribution is used for testing hypotheses in the context of multiple regression analysis. We obtain it from two chi-square random variables in the following way: 2 2 Suppose X1 ck and X 2 ck , and that X 1 and X 2 are independent 1 2 Then, the random variable F= X1 / k1 X 2 / k2 has an F distribution with (k1, k2 ) degrees of freedom. We write thisas F Fk ,k 1 VER. 9/11/2012. © P. KOLM 28 2 Statistics VER. 9/11/2012. © P. KOLM 29 Statistical Inference The goal of statistical inference in econometrics5 is to construct an empirical model that describes the behavior of observed financial data There are two main frameworks: o Classical methods (our main focus in these lectures) Uses methods from classical probability theory to make statistical inferences from sampled data Parameters are treated as fixed but unknown numbers Two branches6: (1) parametric, and (2) nonparametric methods o Bayesian methods Combines prior knowledge with the given data, using Bayes’ theorem, to make statistical inferences Parameters are treated as random variables VER. 9/11/2012. © P. KOLM 30 Population, Sample and Measurement Population Sample VER. 9/11/2012. © P. KOLM Theory Measurement • Behaviors • Characteristics • Choices • Patterns 31 Classical Inference Population Sample Econometrics Measurement • Behaviors • Characteristics • Choices • Patterns Inference about the entire population VER. 9/11/2012. © P. KOLM 32 Random Sampling Classical statistical inference relies on using sampling and sample data effectively A sample of n observations on one or more variables (denoted Y = (Y1,Y2,...,Yn ) ) is a random sample if the n observations are drawn independently from the same population, or probability distribution, f (y; q) o f (y; q) is assumed to be known except for the value of the parameter q We say that Y = (Y1,Y2,...,Yn ) are independent and identically distributed (iid) VER. 9/11/2012. © P. KOLM 33 Statistic and Estimator A statistic is any function computed from the data in a sample, i.e. W = h(Y ) where Y = (Y1,Y2,...,Yn ) o Note: A statistic a random variable Main idea in statistical inference: If we can sample from the population, then we can infer (i.e. learn) something about q This process is called estimation ˆ A point estimate, q , is a statistic computed from a sample that gives a single estimated value for q ˆ The mathematical formu...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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