Weak Instruments

A simple example of Staiger and Stock (1997)

Consider the following model

Let



Assume that

1. are iid.

2. which is a positive definite matrix.

3. 

4. 

Let where

1. Prove that So that the rank condition for identification is satisfied.

   
   
   

2. Prove that
   
   because the WLLN and CMT. Next we assume (or where c is a constant). In empirical work using instrumental variables (IV) regression, often the partial correlation between the instruments and the included endogenous variable is low, that is, the instruments are weak.

3. We now want to consider a case in which the rank condition is just barely satisfied, that is, is very close to zero. One way to do this is to replace Assumption (4) with
   
   Note
   
   where is the sample mean of

   Show that

   

4. Show that
   
   where is distributed and is the (2,2) element of .

   
   
   by a CLT.
   
   where by the CMT.

5. Show that
   
   
   
   where by the CMT. The above result says that can be express asymptotically the ratio of two normal random variables.

6. Is consistent?

   It is clear that
   
   Of course, is inconsistent.