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# hws01 - Economics 140A Answers to Exercise 1 1(From Fall...

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Economics 140A Answers to Exercise 1 1. (From Fall 1992 °nal examination.) a) To assess the quality of estimates obtained from the actual sales prices of homes, we must determine the statistical properties of the estimators from which the estimates are constructed. To determine the statistical properties of the estimators, we must construct the probability model that underlies the data we gather. To form the probability model, we begin by assuming that the prices form a sequence of independent and identically distributed random variables. We further assume that the distribution of P i , denoted F P ( ° ) , is continuous so that f P ( ° ) is the density function for P i (the sub-subscript i is omitted from the distribution and density functions because the random variables are identically distributed). Two measures of the location of the distribution of P i , which provides a measure of the typical value of the price of a home in Santa Barbara, are the mean and median. A natural estimator of the mean of P i is the sample mean ° P n , where in this case the sample size is n = 497 , de°ned as ° P 497 = 1 497 497 X i =1 P i : A natural estimator of the median of P i is the sample median P ( n +1 2 ) . To obtain the formula for the estimator, begin by ordering the data from the smallest selling price to the largest selling price. The elements of the resulting sequence f P (1) ; : : : ; P ( n ) g are de°ned as: P (1) is the smallest selling price, which is not generally the °rst selling price recorded in the sample, so P (1) does not generally equal P 1 . The sample median is the element of f P (1) ; : : : ; P ( n ) g for which half of the elements are smaller and half are larger. Because there are an odd number of elements in the sequence, the sample median is P ( n +1 2 ) = P (249) : (Because there are an odd number of observations, the sample median is unique.) b) To determine if the sample mean and sample median are unbiased, we must be clear about the quantity that is being estimated. Let us begin with the case in which f P ( ° ) is symmetric. Let ° = E [ P i ] . If f P ( ° ) is symmetric about ° , then the mean and median both equal ° . In this case, the sample mean and sample median are estimating the same quantity. Is the sample mean unbiased? The bias of the sample mean equals E [ ° P n ] ± ° . We have E [ ° P n ] = E [ 1 n n X i =1 P i ] = 1 n n X i =1 E [ P i ] = °: The second equality follows from the facts that the expectations operator is linear and the sample size is not a random variable, while the third equality 1

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follows from the fact that P i is identically distributed. The sample mean is an unbiased estimator of the mean. Is the sample median unbiased? The bias of the sample median equals E [ P ( n +1 2 ) ] ± ° . Wilks (1962, Mathematical Statistics , page 236, result 8.7.2) shows that F P f P ( n +1 2 ) g has a beta distribution with parameters ( n +1 2 ; n ± ( n +1 2 ) + 1) = ( n +1 2 ; n +1 2 ) . Mood et al. (1974, Introduction to the Theory of Statistics , page 116, theorem 19) give formulas for the mean and variance of F P f P ( n ° 1 2 +1) g . From Mood et al.
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hws01 - Economics 140A Answers to Exercise 1 1(From Fall...

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