Lecture17-2010 - Denition of Estimator and Choosing among...

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Defnition oF Estimator and Choosing among Estimators: Lecture XVII Charles B. Moss October 9, 2010 I. What is an Estimator? A. In the next several lectures we will be discussing statistical esti- mators and estimation. The book divides this discussion into the estimation of a single number such as a mean or standard devia- tion or the estimation of a range such as a conFdence interval. B. At the most basic level, the deFnition of an estimator involves the distinction between a sample and a population. 1. In general we assume that we have a random variable ( X ) with some distribution function. 2. Next, we assume that we want to estimate something about that population, for example we may be interested in esti- mating the mean of the population or probability that the outcome will lie between two numbers. a) ±or example, in a farm-planning model we may be inter- ested in estimating the expected return for a particular crop. b) In a regression context, we may be interested in estimating the average e²ect of price or income on the quantity of goods consumed. 3. This estimation is typically based on a sample of outcomes drawn from the population instead of the population itself. C. Common point estimators are the sample moments 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XVII Fall 2010 1. Sample Mean ¯ X = 1 n n X i =1 X i (1) 2. Sample Variance S 2 X = 1 n n X i =1 ± X t ¯ X ² 2 = 1 n n X i =1 X 2 t ± ¯ X ² 2 (2) 3. The k th sample moment around zero M k = 1 n n X i =1 X k i (3) 4.Th e k th moment around the mean ˜ M k = 1 n n X i =1 ± X i ¯ X ² k (4) 5. Sample Covariance Cov ( X,Y )= S xy = 1 n n X i =1 ± X i ¯ X ²± Y i ¯ Y ² (5) 6. Sample Correlation ρ xy = S xy S xx S
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Lecture17-2010 - Denition of Estimator and Choosing among...

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