Econ7630_Exam1b

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Author:
mattstam
ID:
66061
Filename:
Econ7630_Exam1b
Updated:
2011-02-14 11:09:52
Tags:
econometrics \"parameter estimation\"
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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).
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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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