Exam 1

Irrelevant Instruments

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1. Consider the following regression
   
   , where is an unknown parameter and are . It is proposed to estimate by instrumental variable (IV) methods using a vector of instruments generated by
   
   Let be the IV estimator of obtained with the instruments Analyze the asymptotic behavior of and compare your results to those obtained in Homework #3.

   1. Show that satisfies
      
      and
      
      and interpret them.

      Note is a martingale difference sequence (MDS) since
      
      Its variance is
      
      where
      
      Then we use a WLLS for MDS
      
      Next we show
      
      since
      
      If
      
      then
      
      But this implies that
      
      as claimed. From a CLT for MDS
      
      Note is also a MDS since
      
      It is easy to show
      
      and
      
      Finally, consider the joint distribution of two elements in the vector. Any linear combination of these elements takes the form
      
      Then is also a MDS with a positive variance given by
      
      satisfying
      
      for and
      
      Furthermore
      
      Thus any linear combination of the two elements in the vector is asymptotically normal, implying a limiting mutivariate normal distribution
      
      We consider the 2SLS:
      
      Thus
      
      
      
      and
      
      where
      
      and
      
      It is easy to show
      
      where
      
      and
      
      Therefore
      
      This implies that
      
      Assume What happens to the asymptotic distribution of when That is, let after you have passed

      We know that
      
      as Note
      
      since
      
      
      
      This means
      
      The implication of the result is that
      
      if and then sequentially.