Type up or write your solution legibly. Submit your assignment before the class. Your work should demonstrate how your arrive at each conclusion. You may study with your classmates, but you must independently write up your own solution
economics
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Type up or write your solution legibly. Submit your assignment before the class. Your work should demonstrate how your arrive at each conclusion. You may study with your classmates, but you must independently
write up your own solution
1. Study of Estimators. Let Xi be a random variable in a sample randomly drawn from a population
where i = 1,...,n. Xi has an unknown distribution but the mean and variance exist. Each observation
is independent and identically distributed (i.i.d.). Let µ = E(Xi) and σ
2 = var(Xi).
(a) The average of a sample of size n, mathematically X¯
n =
1
n ∑
n
i=1 Xi
, is often used as an estimator
of µ. Is X¯
n an unbiased estimator of µ? If not, what would be an unbiased estimator?
(b) What is the variance of the estimator X¯
n? (Remember: covariance of Xi and Xj equals zero if
they are independent and i 6= j.)
(c) What would be an ideal estimator of µ?
2. OLS Estimators. Suppose you have a large sample of data (yi
, xi),i = 1,...,N. You would like to
estimate a linear regression model below:
(a) Suppose you decided to estimate the classical linear regression model with OLS. We derived the
expectation and variance of ˆβ1 in class, but what assumptions do we need at each step to arrive
at the results?
(b) How can you improve the precision of β1 estimate?
(c) One of your friends suggests that you randomly split the sample in two, run regression analysis
on the two independent samples, and have the estimators be the average of the two samples. Let
ˆβ11,
ˆβ21 be the two estimators from the two samples for β1. Likewise for β0. But we will just
focus on one, the estimator for β1. The new estimator is:
