Lecture 3 Flashcards
Prove that in a multiple regression model with deterministic regressors, beta hat converges to beta in second mean and list the assumptions.
Prove that in a multiple regression model with stochastic regressors, beta hat converges to beta in second mean and list the assumptions.
List the 4 more simplistic conditions for the consistency of Beta hat.
State and prove the simplistic condition regarding the convergence of M_n hat to M.
z_i ^ 2 uncorrelated with z_j ^2 which is satisfied when z_i iid with finite 4th moments.
from first to second, we can move expectation inside because the only covariances that are different than 0 is when i = j.
The bounding M here is not the usual M but any bounding M since we assumed that the 4th moments are finite.
using chebyshev: mean 0 sequence that is uncorrelated converges in second mean when …
State the fact regarding 1-F(x) concerning positive RVs with finite mean.
State and prove the fact regarding 1-F(x) concerning positive RVs with finite mean.
Give the definition of Uniform Integrability.
Give the 3 sufficient conditions for Uniform Integrability.
State formally and prove that having just a little bit more than first moment is a sufficient condition for UI.
State formally and prove that being identically distributed and having bounded first moment is a sufficient condition for UI.
State formally and prove that the fact that we can find a bounding RV with finite first moment is a sufficient condition for UI.
State the necessary condition for Uniform Integrability.
State and prove the necessary condition for Uniform Integrability.
State a condition that is not sufficient for UI and prove that it is not sufficient.
