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Unformatted text preview: Estimation Hypothesis Tests Con dence Intervals Comparing Means from Di erent Populations Scatterplots, Sample Covariance, Sample Correlation Introduction to Econometrics Chapter 3: Review of Statistics Geo rey Williams gwilliams@econ.rutgers.edu September 27, 2010 Geo rey Williams gwilliams@econ.rutgers.edu Introduction to Econometrics Chapter 3: Review of Statisti Estimation Hypothesis Tests Con dence Intervals Comparing Means from Di erent Populations Scatterplots, Sample Covariance, Sample Correlation Estimators and Estimation We've already talked about estimation  using a sample to get a sense of the mean of the population. Now let's give this idea a formal de nition. An estimator is a function that is performed on a sample of data  in theory, it can be any function you apply to the data. An estimate is a speci c value we get for the estimator on a speci c draw. So how do we talk about the fact that we'd like our estimate to measure something in the population? Geo rey Williams gwilliams@econ.rutgers.edu Introduction to Econometrics Chapter 3: Review of Statisti Estimation Hypothesis Tests Con dence Intervals Comparing Means from Di erent Populations Scatterplots, Sample Covariance, Sample Correlation Bias We usually use the hat symbol to show that a variable is an estimator. So, for example, ^ Y would be a standard symbol for an estimator of Y . Bias is, very simply, the di erence between the expected value of ^ Y and Y , E ( ^ Y ) Y This number is zero (and we say that ^ Y is unbiased ) if E ( ^ Y ) = Y Geo rey Williams gwilliams@econ.rutgers.edu Introduction to Econometrics Chapter 3: Review of Statisti Estimation Hypothesis Tests Con dence Intervals Comparing Means from Di erent Populations Scatterplots, Sample Covariance, Sample Correlation Consistency Consistency is the property that as the sample size increases, the estimator gets arbitrarily close to the true parameter. In notation, this is ^ Y p ! Y Geo rey Williams gwilliams@econ.rutgers.edu Introduction to Econometrics Chapter 3: Review of Statisti Estimation Hypothesis Tests Con dence Intervals Comparing Means from Di erent Populations Scatterplots, Sample Covariance, Sample Correlation E ciency While e ciency is technically a little more complicated than this, for our purposes we can simply say that e ciency is about variance. If we have two estimators, both of which are unbiased, the one with the lower variance is more e cient (it stays closer to the true parameter value). Geo rey Williams gwilliams@econ.rutgers.edu Introduction to Econometrics Chapter 3: Review of Statisti Estimation Hypothesis Tests Con dence Intervals Comparing Means from Di erent Populations Scatterplots, Sample Covariance, Sample Correlation Example: Y Bias: As we discussed in Chapter 2, E ( Y i ) = Y , so E ( Y ) = E " 1 n n X i = 1 Y i # = 1 n n X i = 1 E ( Y i ) = 1 n n Y = Y So bias is zero; the estimator Y is unbiased....
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 Fall '10
 Otusbo
 Econometrics

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