# Econ7630_Exam1b

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 Author: mattstam ID: 66061 Filename: Econ7630_Exam1b Updated: 2011-02-14 11:09:52 Tags: econometrics \"parameter estimation\" Folders: Description: Exam 1 (part b) for graduate level Econometrics 1 at LSU during spring 2011. Notation: Use @ for theta. ^ or ` before parameter indicates hat and bar (ex: `@ is theta-bar). Show Answers:

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1. Parameter Estimation Framework
• 1. Population Y
• 2. E[Y] = @
• 3. Var(Y) = sig2
• 4. Assume Y1, Y2, . . ., Yn is a random sample from population
• 5. Assume Yi are independtly and identically distributed (iid)
• 6. ^ indicates estimate
2. Estimation of @ Method 1: Method of Moments
• 1. E[Yi] = @ "population moment"
• 2. `Y = ^@ = sum1-n(Yi/n) "sample moment"
• "sum..." is the Estimator, which is an expression
3. Estimation of @ Method 2: Minimize
• 1. s = sum1-n(yi - @)2 given yi(data) = s(@)
• 2. Goal is to find @ that minimizes expression
• 3. @ in 1 known as "Least Squares Estimator of @"
4. Properties of Estimators (particular to some, not given for all)
• 1. Linearity:
• 2. Unbiasedness:
• 3. Minimum Variance (of estimator):
5. Best Linear Unbiased Estimator(BLUE)
6. Proof that `Y is BLUE
7. If Y~N(@, sig2) => `Y~
8. Matrix Notation for Estimation
9. Estimating sig2
10. Chi-squared Distribution
11. Wald Statistic
12. Matrix Chi-Square: X2(m)
13. Ybar =
^mu = (1/n)sum(Yi), where Yi~N(mu,sig2)
14. s2 =
^sig = [sum(Yi - Ybar)2]/(n - 1), where Yi~N(mu,sig2)
15. Show E(s2) = sig2, that is show that s2 is an unbiased estimator of population variance.
16. T-distribution
• t = (Ybar - mu)/[s/sqrt(n)]
• = (Ybar - mu)/se(Ybar)~tn-1
• Derivation:
17. Properties of t distribution
18. Confidence Interval
19. Ingredients of Hypothesis Test
20. Var(Ybar) =
^sig2/n = s2/n
21. se(Ybar) =
^sig/sqrt(n) = s/sqrt(n)
22. Elements of T distribution
• Let Y1...Yn be a random sample from a population Yi~N(mu,sig2)
23. Interval Estimation
24. Probability of rejecting Ho when it is true
25. Rejection Rules
Ho: mu = muo
H1: mu > muo
• Reject Ho and accept H1:
• If p <= alpha and t >= tc
• Fail to reject Ho:
• If p > alpha and t < tc
26. Rejection Rules
Ho: mu = muo
H1: mu < muo
• Reject Ho and accept H1:
• If p <= alpha and t <= -tc
• Fail to reject Ho:
• If p > alpha and t > -tc
27. Rejection Rules
Ho: mu = muo
H1: mu not equal to muo
"Two-tailed test"

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