lecture_note_090416

lecture_note_090416 - Econ 5280 Applied Econometrics...

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Econ 5280: Applied Econometrics Instructor: Professor Jin Seo Cho ©Jin Seo Cho, 2016, All Rights Reserved.
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Chapter 1 Introduction 1.1 Classification of Data We can classify data based on the following matrix: Independent Non-independent Identically distributed I.I.D. Stationary Non-identically distributed I.N.I.D. Non-stationary In this course, we will mainly treat I.I.D. case, where I.I.D. means Independently and Identically Distributed. 1.2 Classification of Model To handle data, we use a specific model. In econometrics, there are so many models. However, we can classify these models into following four categories: Reduced-Form Structural-Form Linear OLS IV, GMM Non-linear NLS, ML GMM, ML Here, reduced-form focuses to probability structure of data and structure-form focuses to econometric model implied by economic theory. In this course, we will mainly discuss reduced form case. 1
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Chapter 2 Review of Probability Theory 2.1 Single Random Variables Economic data are understood as realizations of random numbers. There are two sorts of random numbers, implying that there are two sorts of economic data sets. The first is the case in which each realization has a probability mass, and the second is the case in which each observation does not have a probability mass. The first and second are called discrete and continuous random variables respectively. We consider specific examples belonging to these random variables. 2.1.1 Discrete Distributions Discrete random variables are defined by their probability density functions. Suppose that X is a random variable, and its realization has to be one of k possible events, { x 1 , ,x k } . In the language of probability, each event has probability to come true, and probability cannot be less than zero and greater than one, so that 0 P ( X = x i ) 1 . We don’t know in advance what X would be. But, we know for sure that X will be one of { x 1 , ,x k } , so that P ( X { x 1 , ,x k }) = 1 . 2
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In other words, sum of the probability of each event is one. That is, k i = 1 P ( X = x i ) = 1 . We may abbreviate P ( X = x i ) as p i for each i = 1 , ,k . Given this, probability density function ( PDF ) is defined as a functional relationship between ( x 1 , ,x k ) and ( p 1 , ,p k ) . 1 Discrete random variables are characterized by the probability density function. If two random variables X and Y have the same probability density function, then we say that they are identically distributed. We will see popular discrete random variables below. Before this, we need to see relevant definitions to the PDF needed for future reference. For a function of X , say f ( X ) , we define the expected value of f ( X ) (or population mean of f ( X ) ) by the following formula: E [ f ( X )] = k i = 1 f ( x i ) P ( X = x i ) . As an example, suppose that f ( X ) = X . Then the population mean of X is defined as E [ X ] = k i = 1 x i P ( X = x i ) .
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