You are interested in purchasing a home in Santa Barbara. To determine if it is a good time to buy, you collect the selling prices of all homes in the city over the past three years, giving you 497 observations. Let us denote observation 't' as Pt , t=1,...,497 We assume that [Pt]t=1497 is a sequence of independent and identically distributed random variables.
1. Construct an estimator of the mean and median of Pt.
2.Are your estimators unbiased? What is the variance of each estimator?
3. Suppose that you grouped the sales according to years, so you could estimate the mean and median sale price of a home in each of the three years. Suppose further, that our assumption that [Pt]t=1497 is an i.i.d. sequence is erroneous. In fact, there are two types of homes, call them nice with mean price un and average with mean price ua , where un > ua.
4. If the composition of sales over time has shifted toward nice homes, which estimator would shift more over time? Carefully explain your logic.