If limn probx 0 then plim x x to in words the

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If lim(n )Prob[|x |  >   ] 0 then, plim x . x    to  . In words, the probability that the difference betwe converges in probability θ ε ε θ n n en x  and   is larger than   for any    goes to zero.  x  becomes arbitrarily close to  . Mean square convergence is sufficient (not necessary)  for convergence in probability. (We will not require other, broader definitions of convergence, such as  "almost sure convergence.")
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Part 11: Asymptotic Distribution  Theory Mean Square Convergence ™    8/42
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Part 11: Asymptotic Distribution  Theory Probability Limits and Expecations What is the difference between E[xn] and plim xn? ™    9/42 P n n A notation plim x      x = θ →θ
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Part 11: Asymptotic Distribution  Theory Consistency of an Estimator If the random variable in question, xn is an estimator (such as the mean), and if plim xn = θ Then xn is a consistent estimator of θ. Estimators can be inconsistent for two reasons: (1) They are consistent for something other than the thing that interests us. (2) They do not converge to constants. They are not consistent estimators of anything. We will study examples of both. ™    10/42
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Part 11: Asymptotic Distribution  Theory The Slutsky Theorem Assumptions: If xn is a random variable such that plim xn = θ. For now, we assume θ is a constant. g(.) is a continuous function with continuous derivatives. g(.) is not a function of n. Conclusion: Then plim[g(xn)] = g[plim(xn)] assuming g[plim(xn)] exists. (VVIR!) Works for probability limits. Does not work for expectations. ™    11/42 μ = μ μ n n n n E[x ]= ; plim(x ) ,  E[1/x ]= ?; plim(1/x )=1/
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Part 11: Asymptotic Distribution  Theory Slutsky Corollaries ™    12/42 θ μ ± = θ ± μ × = θ × μ = θ μ μ = θ μ n n n n n n n n n n x  and y  are two sequences of random variables with probability limits   and  .   Plim (x  y )      (sum) Plim (x  y )      (product) Plim (x / y )  /   (product, if     0) Plim[g(x ,y )]  g(  ,  ) assuming it exists and g(.) is continuous with continuous partials, etc.
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Part 11: Asymptotic Distribution  Theory Slutsky Results for Matrices Functions of matrices are continuous functions of the elements of the matrices. Therefore, If plim A n = A and plim B n = B (element by element), then plim( A n-1) = [plim A n]-1 = A -1 and plim( A n B n) = plim A nplim B n = AB ™    13/42
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Part 11: Asymptotic Distribution  Theory Limiting Distributions Convergence to a kind of random variable instead of to a constant xn is a random sequence with cdf Fn(xn). If plim xn = θ (a constant), then Fn(xn) becomes a point. But, Fn may converge to a specific random variable. The distribution of that random variable is the limiting distribution of xn.
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