Consider the following nonlinear regression model:
𝑦
𝑖
=
𝛼
+
𝑥
𝑖
𝛽
+
𝜀
𝑖
,
Assume i.i.d. data and
𝔼
[
𝜀
𝑖
|
𝑥
𝑖
]
=
0
. To estimate
𝛼
and
𝛽
by GMM, we use the two theoretical moment conditions
𝔼
[
𝑦
𝑖
−
𝛼
−
𝑥
𝑖
𝛽
]
=
0
𝔼
[
(
𝑦
𝑖
−
𝛼
−
𝑥
𝑖
𝛽
)
𝑥
𝑖
]
=
0
To compute the variance of the GMM estimator we need the matrices
𝛤
0
and
𝛷
0
.

Question

Consider the following nonlinear regression model:
𝑦
𝑖
=
𝛼
+
𝑥
𝑖
𝛽
+
𝜀
𝑖
,
Assume i.i.d. data and 
𝔼
[
𝜀
𝑖
|
𝑥
𝑖
]
=
0
. To estimate 
𝛼
 and 
𝛽
 by GMM, we use the two theoretical moment conditions
𝔼
[
𝑦
𝑖
−
𝛼
−
𝑥
𝑖
𝛽
]
=
0
𝔼
[
(
𝑦
𝑖
−
𝛼
−
𝑥
𝑖
𝛽
)
𝑥
𝑖
]
=
0
To compute the variance of the GMM estimator we need the matrices 
𝛤
0
 and 
𝛷
0
.

BlackTom AI · Accepted Answer

To approach this question, I start by identifying the moment conditions used for a nonlinear regression in a GMM setup and then determine the correct form of the variance-covariance matrix of the moment conditions.
Option 1 presents the candidate for Φ0 as a 2x2 matrix with elements built from the second moments of the moment conditions themselves. Remember that the moment conditions here are g_i = [ y_i − α − x_i β , (y_i − α − x_i β) x_i ]. The matrix Φ0 is defined as E[ g_i g_i^T ], which yields:
- The (1,1) entry: E[ (y_i − α − x_i β)^2 ].
- The (1,2) entry (and by symmetry the (2,1) entry): E[ (y_i − α − x_i β)^2 x_i ].
- The (2,2) entry: E[ (y_i − α − x_i β)^2 x_i^2 ].
Thus, the full Φ0 matrix should be composed exactly of these expectations in a 2x2 arrangement. Option 1 encodes precisely these components in the 2×2 layout, with top-left E[(y_i − α − x_i β)^2], top-right E[(y_i − α − x_i β)^2 x_i], bottom-left E[(y_i − α − x_i β)^2 x_i], and bottom-right E[(y_i − α − x_i β)^2 x_i^2]. This aligns with the standard formula Φ0 = E[ g_i g_i^T ].

Option 2 attempts to place a product structure involving E[(y_i − α − x_i β)^2] alongside x_i terms, but the arrangement shown does not match the correct form for Φ0 as E[ g_i g_i^T ]. Specifically, the expression inside the matrix entries does not correspond to the cross-products and squared terms that arise from g_i g_i^T. In short, the off-diagonal entries and the way x_i factors appear do not reflect E[(y_i − α − x_i β)^2 x_i], which is essential for Φ0.

Option 3 also diverges from the correct structure. It introduces a log(x_i) term and mixes it into combinations with x_i β in a way that has no justification from the moment construction g_i = [e_i, e_i x_i], where e_i = y_i − α − x_i β. The presence of log(x_i) in the matrix entries is inconsistent with the definitions of the moment conditions and their second-moment cross-products.

Option 4 states there is not enough information to compute Φ0. Given the definition Φ0 = E[g_i g_i^T] and the specified moment functions g_i, there is sufficient information to write down the form of Φ0 in terms of expectations of functions of e_i and x_i. Therefore, this option is incorrect because the model provides the necessary structure to specify Φ0 in terms of E[(y_i − α − x_i β)^2], E[(y_i − α − x_i β)^2 x_i], and E[(y_i − α − x_i β)^2 x_i^2].

Option 5 reproduces a matrix whose entries include combinations like −1, −x_i β log(x_i), and other terms that are not derived from the product g_i g_i^T. This = structure does not follow from the standard second-moment construction for Φ0 given g_i, and thus it is not the correct representation of Φ0.

Taken together, the reasoning shows that the correct form for Φ0 is the 2×2 matrix with entries E[(y_i − α − x_i β)^2], E[(y_i − α − x_i β)^2 x_i], E[(y_i − α − x_i β)^2 x_i], and E[(y_i − α − x_i β)^2 x_i^2], which corresponds to the first option among the provided choices.

BU.232.630.W4.SP25 sample_quiz_2

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