Notes2 - Lecture Notes 2 Econ 410 Introduction to...

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Lecture Notes 2 Econ 410 – Introduction to Econometrics 1 Review of Statistics Population vs. Sample Many times, we want to answer quantitative questions that are related to the distribution of a random variable in a large group of people, the population. However, it is often impossible (or too expensive) to gather information about all the members of the population. This problem is solved by selecting a sample at random from the population, and by using this sample to infer about the characteristics of the population from which the sample was drawn. Statistical methods used to study the population distribution of a random variable Estimation : computing a “best guess” numerical value for an unknown characteristic of a population distribution from a sample of data. Estimator : a function of a sample of data, used to estimate an unknown population characteristic. An estimator is a random variable, because it is function of a randomly selected sample. What are desirable characteristics of the sampling distribution of an estimator? Unbiasedness - if Y µ ˆ is a generic estimator of the unknown population characteristic Y µ , the condition for unbiasedness is: ( ) Y Y E µ µ = ˆ Consistency - an estimator is consistent if, as the sample size increases, it converges in probability to the population characteristic it estimates: Y p Y µ µ → ˆ Efficiency - an estimator Y µ ˆ is said to be more efficient than another estimator Y µ ~ if ( ) ( ) Y Y Var Var µ µ ~ ˆ < Therefore, the more efficient estimator is the one with the tightest sampling distribution. Estimating the population mean The natural estimator of the population mean Y µ is the sample average Y . ( ) Y Y E µ = so Y is an unbiased estimator for Y µ Y p Y µ → by the Law of large numbers, so Y is a consistent estimator for Y µ . Y is the most efficient of all the linear unbiased estimators ( Y is the Best Linear Unbiased Estimator). This means that Y is most efficient among all the estimators that are unbiased and are linear functions of the observations in the sample.
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Lecture Notes 2 Econ 410 – Introduction to Econometrics 2 Another property of Y : it is the estimator that provides the best fit to data The sample average is the estimator that solves the following problem: ( ) = n i i m m Y 1 2 min In other words, the sum of the squared differences between the observations and Y is the smallest of all possible estimators. For this reason, Y is called the least squares estimator of Y µ . Estimating the population variance and standard deviation The sample variance (standard deviation) can be used as an estimator of the population variance (standard deviation). The sample variance is defined as: ( ) = = n i i Y Y Y n s 1 2 2 1 1 The sample standard deviation is defined as: 2 Y Y s s = Unbiasedness The sample variance is an unbiased estimator for 2 Y σ , that is: ( ) 2 2 Y Y s E σ = Consistency of the sample variance The sample variance is a consistent estimator of the population variance, that is: 2 2 Y p Y s σ →
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