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MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 200 9 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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14.30 Introduction to Statistical Methods in Economics Lecture Notes 1 Konrad Menzel February 3, 2009 1 Introduction and Overview This class will give you an introduction to Probability Theory and the main tools of Statistics. Probability is a mathematical formalism to describe and analyze situations in which we do not have perfect knowledge of all relevant factors. In modern life, we are all routine consumers of statistical studies in fields ranging from medicine to sociology, and probabilistic reasoning is crucial to follow most of the recent debates in economics and finance. In the first half of this class, we’ll talk about probabilities as a way of describing genuine risk - or our subjective lack of information - over events. Example 1 In subprime lending, banks offered mortgages to borrowers who were much less likely to repay than their usual clientele. In order to manage the risks involved in lending to prospective home-owners who do not own much that could serve as collateral, thousands of these loans were bundled and resold as ”mortgage backed securities,” i.e. the bank which made the original loans promised to pay the holder of that paper whatever repayment it received on the loans. Eventually, there were more complicated financing schemes under which the pool was divided into several ”tranches”, where a first tranche was served first, i.e. if the tranche had a nominal value of, say, 10 million dollars, anyone holding a corresponding claim got repaid whenever the total of repayments in the pool surpassed 10 million dollars. The lower tranches were paid out according to whatever money was left after serving the high-priority claims. How could it be that the first ”tranche” from a pool with many very risky loans was considered to be ”safe” when each of the underlying mortgages was not? The low-priority tranches were considered riskier - why? And why did in the end even the ”safe” securities turn out to be much riskier in retrospect than what everyone in the market anticipated? We’ll get back to this when we talk about the Law of Large Numbers, and under which conditions it works, and when it doesn’t. Usually in order to answer this type of question, you’ll have to know a lot about the distribution (i.e. the relative likelihood) of outcomes, but in some cases you’ll actually get by with much less: in some cases you are only concerned with ”typical” values of the outcome, like expectations or other moments of a distribution. In other cases you may only be interested in an average over many repetitions of a random experiment, and in this situation the law of large numbers and the central limit theorem can sometimes give you good approximations without having to know much about the likelihood of different outcomes in each individual experiment.
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