Consistency of W2SLS Estimator: Analysis Under Different Assumptions

This assignment solution provides a detailed analysis of the Weighted Two-Stage Least Squares (W2SLS) estimator, focusing on its consistency and asymptotic distribution under different assumptions. It begins by demonstrating the consistency of the W2SLS estimator for β under specific assumptions, utilizing instrumental variables and two-stage linear regression. The solution further explores how the consistency holds when a key assumption is modified, emphasizing the importance of the conditional expectation of error terms. Finally, it examines the asymptotic distribution of √n(β^W2SLS−β), highlighting its normal distribution and the role of hypothesis testing in large linear regressions. The analysis includes mathematical derivations and explanations to support each conclusion.

Question 1
A)
Show that ^βW2SLS is consistent for β under the Assumption B0-1-2-3.
Answer:
Xi can be used for the predicted values of a set of variable on the instrumental, which can be
used for the two important stages that includes,
the explanation variable(s) xi (informative) and
the uncorrelated variable with the error zi (valid).
Consider the second uncorrelated variable on the predicted values as, zi
=
[ (∑
i
❑ xi zi
'
Var (ei / zi) )(∑
i
❑ zi xi
'
Var (ei / zi) )−1
(∑
i
❑ zi zi
'
Var (ei /zi ) ) ]−1
(∑
i
❑ xi zi
'
Var (ei / zi) )
(∑
i
❑ zi zi
'
Var (ei / zi) )−1
( zi yi
'
Var ( ei / zi
) )
First, consider the model on a single endogenous explanatory variable (xi) and a
single instrument ( zi), which assumes both the values as zero simplicity.
yi= xi β +zi i=1,2,…..N E(zi)=0 E( xi zi)≠ 0
Y=X β+Z (consider all the vector as NX1)
The first stage of linear regression is,
xi= zi π +ri i=1,2 , ….. N
X=Zπ +r
The required values of E is,
E(ei / zi)=0
Thus, the variable can consider zi as valid.
Consider π ≠ 0, here the values on E(e_i/z_i)≠0.
^π=(zi z )−1 xi zi
'
The predicted value is X=(x1,x2 …xN)’
^X =Z ^π=(z' z )−1 xi zi
'
The second stage's linear regression can calculate,
Avar (β ̂ W2SLS) =σ 2(x'z((z' z )−1 z' x x)-1
=σ 2( ^X' ^X)−1

Then, calculate the β values on the test statistic based on the subset of the parameter's
consistent i.e., β at the same time can be used in the linear regression.
Notice that, it can consider ^zi= yi-xi
' ^β2SLS≠( zi )̂  = yi−xi
' β̂  2 SLS so that σ 2 is not
eliminating the consistently used residuals of the second regression values of β, which is (xi
'-
^xi
') β.
B)
The consistency in part (a) holds in the Assumption (B2 ) is replaced by (B 2')E[ zi ei]=0.
Answer:
The valid consistent of the parameter can consider the assumption values as,
¿] =0
It can calculate the test of the over identifying assumption consistency,
We can assume the weight matrix efficiency, by setting the minimizing values as,
X N =(z' z )−1
X' Z(z' z)Z' X
It yields the 2SLS assumption consistency which can be specified as ^β2SLS.
The assumption of the consistency which uses the linear independent model on the
conditional is [zi ei] =0 of the weighted matrix value consistency.
^β 2 SLS =( ^xi
i zi )−1 ^x' ei
=( ^xi zi) ^x'(ei β + zi)
=( ^Xi zi)ei
^x'[zi ei] =0 the consistency values using the linear regression values i x ̂ '[z_ie_i]=0 proved
the condition as true.
C)
Asymptotic distribution √ n( ^β W 2 SLS−β ) under the Assumption B0-1-2-3?
Answer:
The independent of the asymptotic distribution can obtain the values,
√ n( ^β 2 SLS−β)D N (1, V)
Or ^β2SLSa a N(β , avar ( ^β 2 SLS)), with β ̂ 2SLS =values/N

In the large linear regression, it can be used for the hypothesis of the tested values, which is
true for the passing parameter's vector i.e., β can use the conducted familiar standard as
Normal, for the test statistics of the squared consistency values, where β ̂ 2SLS is available.
The asymptotic distribution of the estimator Assumption values can be calculated as,
^σ 2 = 1
N ∑
i=1
N
√ n β̂  2 SLS