Applications of Generalized Method of Moments Estimation
نویسنده
چکیده
T he method of moments approach to parameter estimation dates back more than 100 years (Stigler, 1986). The notion of a moment is fundamental for describing features of a population. For example, the population mean (or population average), usually denoted m, is the moment that measures central tendency. If y is a random variable describing the population of interest, we also write the population mean as E( y), the expected value or mean of y. (The mean of y is also called the first moment of y.) The population variance, usually denoted s or Var( y), is defined as the second moment of y centered about its mean: Var( y) 5 E[( y 2 m)]. The variance, also called the second central moment, is widely used as a measure of spread in a distribution. Since we can rarely obtain information on an entire population, we use a sample from the population to estimate population moments. If { yi: i 5 1, . . . , n} is a sample from a population with mean m, the method of moments estimator of m is just the sample average: y# 5 ( y1 1 y2 1 . . . 1 yn)/n. Under random sampling, y# is unbiased and consistent for m regardless of other features of the underlying population. Further, as long as the population variance is finite, y# is the best linear unbiased estimator of m. An unbiased and consistent estimator of s also exists and is called the sample variance, usually denoted s. Method of moments estimation applies in more complicated situations. For example, suppose that in a population with m . 0, we know that the variance is three times the mean: s 5 3m. The sample average, y#, is unbiased and consistent
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