Consistencyof an EstimatorIf the random variable in question, Xn is an estimator (suchas the mean), and ifplim Xp = 0Then X is a consistent estimator of 0.Estimators can be inconsistent for two reasons:(1) They are consistent for something other than thething that interests us.(2) They do not converge to constants. They are notconsistent estimators of anything.We will study examples of both
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
TheSlutskyTheoremAssumptions: IfXn is a random variable such that plim X = .For now, we assume θ is a constantg() is a continuous function with continuousderivatives. 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 forexpectations.E[区,]=μ; plim(×,) = μ, E[1/区,]=?; plim(1/区,)=1/μ
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. E[x ]= ; plim(x ) , E[1/x ]=?; plim(1/x )=1/ n n n n =
SlutskyCorollariesX, and y, are two sequences of random variables withprobability limits and μ.Plim (x,± y.)=θ ± μ (sum)Plim (x, × y,)= × μ (product)Plim (x,/ y,)= / μ (product, if μ ± O)Plim[g(xp,y,)) = g(0 , μ) assuming it exists and g(.) iscontinuous with continuous partials, etc
Slutsky Corollaries = = = = 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