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

Let's break down what the question is asking and what each option is proposing about the matrix Φ0 in the context of GMM for a nonlinear regression with moment conditions. The setup uses two theoretical moment conditions: E[ y_i − α − x_i β ] = 0 and E[ (y_i − α − x_i β) x_i ] = 0, and we need the appropriate form of Φ0 to compute the variance of the GMM estimator.

Option 1: The matrix Φ0 is: Φ0 = E[ −1  − x_i β log(x_i)  − x_i  − x_i β + 1 log(x_i) ].
- This expression is trying to assemble Φ0 as a vector or matrix that contains expectations involving transformations like log(x_i) and products with x_i or β. However, Φ0 in the GMM context is typically the variance (or the expected outer product) of the moment conditions and their derivatives with respect to parameters, evaluated at the true parameter values. A single row vector with terms like −1, − x_i β log(x_i), etc., does not align with the standard construction of Φ0 for this model, which would involve E[ g_i g_i' ] or E[ ∂g_i/∂θ ... ] structures. This option appears to mix components in a way that does not correspond to the conventional Φ0 form, and the inclusion of log(x_i) in this mixed fashion is inappropriate unless motivated by a specific transformation of the score not indicated in the question. Overall, this option misidentifies the structural form of Φ0.

Option 2: The matrix Φ0 is: Φ0 = E[ 1  x_i β log(x_i)  x_i  x_i β + 1 log(x_i) ].
- Here, the proposed Φ0 contains a first element of 1 and cross-terms with x_i, β, and log(x_i) in a single expectation. Similar to Option 1, the stated structure does not reflect the standard construction of the Φ0 matrix for GMM with the given two moment conditions. The presence of mixed terms like β log(x_i) and 1 log(x_i) without a clear outer-product or derivative framework suggests a misunderstanding of how Φ0 is formed (usually as E[g_i g_i' ] or E[ ∂g_i/∂θ g_i' ] depending on the estimator). This option is unlikely to be correct because it does not correspond to the conventional variance or the information-type matrices used in GMM inference.

Option 3: There is not enough information to compute the matrix Φ0.
- This choice points to a missing piece: to form Φ0, one needs either the explicit form of the moment functions g_i(θ) = [ y_i − α − x_i β, (y_i − α − x_i β) x_i ]' and the corresponding derivatives, or at least a specification of which form of Φ0 (e.g., Φ0 = E[ g_i g_i' ] for a homoskedastic setting, or Φ0 = E[ ∂g_i/∂θ g_i' ] for a certain GMM weighting). Since the problem statement mentions computing the variance of the GMM estimator and asks for Φ0 and Γ0, and does not provide the full distributional details or higher-order moments of the errors, the claim that there is not enough information could be accurate in a general sense if we strictly require the exact forms. However, in many standard textbook treatments, Φ0 is defined in terms of the moment conditions and the data, and with the given moment functions, Φ0 can be derived up to the inclusion of E[ ε_i^2 ] or E[ ε_i^2 x_i^2 ] terms, depending on the chosen weighting. So, this option asserts a lack of information, which may be true only if one insists on a fully explicit numeric form, but conceptually Φ0 can be characterized from the moment conditions.

Option 4: The matrix Φ0 is: Φ0 = [ 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]  E[(y_i − α − x_i β)^2 x_i^2] ].
- This presentation resembles a 2x2 block-structured matrix built from second moments of the residuals times x_i powers, which is not the standard form for Φ0 in the GMM framework with the given two moments. Φ0 typically involves either E[ g_i g_i' ] or E[ ∂g_i/∂θ g_i' ], where g_i contains the two moment conditions. Using squared residuals multiplied by x_i or x_i^2 would correspond to higher-order moments or to a specific heteroskedasticity-robust form that is not implied by the baseline two moment conditions here. Therefore, this option does not align with the conventional Φ0 construction for the described model.

Option 5: The matrix Φ0 is: Φ0 = [ α  x_i β  x_i β  x_i β log(x_i) ].
- This option shows a matrix whose entries mix the parameters α and β with x_i and log(x_i) in a way that does not reflect a valid second-moment or derivative structure for Φ0. The appearance of α inside Φ0 without an expectation over the data, and the mixture of parameter and data terms in a non-matrix-outer-product form, indicates a misunderstanding of how Φ0 is formed. In short, this cannot be the correct specification for the Φ0 matrix in this GMM problem.

Across all these options, the common thread is that Φ0 should be formed from the second-moment structure of the moment conditions and, if needed, the cross-derivatives with respect to the parameters, evaluated at the true θ. The provided alternatives either misplace terms, mix components without the outer-product or derivative logic, or attempt to place parameter terms inside an expectation in a way that does not fit standard GMM theory. The claim that there is not enough information (Option 3) could be defensible if one insists on an explicit numeric expression requiring a specified form of g_i and its derivatives; otherwise, with the moment conditions supplied, a standard form for Φ0 can be discussed conceptually, though its exact numeric form would depend on which variant of Φ0 is used (e.g., homoskedastic vs. heteroskedastic GMM).

In summary, each option deviates from the conventional construction of Φ0 for the given two-moment GMM problem, either by proposing incorrect structural forms or by introducing terms that do not follow from the moment conditions. The most cautious stance, given only the problem statement, is to acknowledge that an explicit Φ0 requires specifying the exact GMM variant and the form of the residuals; without that, the question of a unique correct Φ0 form remains unsettled. Nevertheless, the provided options clearly do not align with the standard Φ0 derivation for these moment conditions.

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