Lecture+3+Statistics

# Lecture+3+Statistics - ECON 123A, Fall 2011, Lecture 3 3-1...

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ECON 123A, Fall 2011, Lecture 3 Dale J. Poirier 3-1 Lecture 3 REVIEW OF STATISTICS The last lecture discussed a probability framework in which all quantities were known . We now permit some of these quantities ( parameters ) to be unknown and not directly observable. We will review three activities involving parameters: C point estimation C interval estimation C hypothesis testing

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ECON 123A, Fall 2011, Lecture 3 Dale J. Poirier 3-2 Point Estimation Suppose we are interested in some unknown property 2 of the distribution of some observable quantity Y of interest. C Examples for 2 : mean, variance, mode, median, q th percentile, etc. C A key insight of statistical theory is that we can learn about 2 by selecting a sample Y i (i = 1, 2, . .., n) from the distribution/population. C We focus on the case 2 = : .
ECON 123A, Fall 2011, Lecture 3 Dale J. Poirier 3-3 C Functions of the data Y i (i = 1, . .., n) that do not involve unknown quantities like 2 are called statistics . Examples of Statistics: sample mean, sample variance, sample median, and the q th sample percentile. C Statistics designed with a purpose in mind, namely, to provide numerical estimates of a particular unknown feature 2 of the underlying population from which we are sampling, are known as estimators . Example: Suppose 2 = : . The sample mean and the sample median are potentially interesting estimators of 2 .

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ECON 123A, Fall 2011, Lecture 3 Dale J. Poirier 3-4 Classical (frequentist) statistics is based on the ex ante sampling distribution of an estimator . C The distribution of the possible values an estimator can take over repeated samples of size n is called its sampling distribution . C An estimator with a “desirable” sampling distribution is preferred, and the realized estimate it produces is to be used.
ECON 123A, Fall 2011, Lecture 3 Dale J. Poirier 3-5 C But what constitutes a “desirable” sampling distribution? B In general, we would like an estimator that produces estimates as close as possible to the unknown value of : . B In other words, we would like the sampling distribution of an estimator to be tightly centered on : .

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ECON 123A, Fall 2011, Lecture 3 Dale J. Poirier 3-6 Note: An estimator is a rule or a procedure that processes the data. C Before the data are collected, we do not know what value this procedure will yield. C After we have the data and apply the procedure we obtain a realization, i.e., a numerical value (an estimate ).
ECON 123A, Fall 2011, Lecture 3 Dale J. Poirier 3-7 Since sampling distributions are fundamental in frequentist econometrics, their understanding is essential. C Suppose we wish to estimate the mean of some distribution. C As an estimator consider the sample mean based on a random (iid) sample of n = 10 observations:

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ECON 123A, Fall 2011, Lecture 3 Dale J. Poirier 3-8 C But why should we use rather than some other estimator?
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## This note was uploaded on 12/13/2011 for the course ECON 123a taught by Professor Staff during the Fall '08 term at UC Irvine.

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Lecture+3+Statistics - ECON 123A, Fall 2011, Lecture 3 3-1...

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