Consider the following nonlinear regression model:
𝑦
𝑖
=
𝛼
+
𝑒
𝛽
𝑥
𝑖
+
𝜀
𝑖
,
Assume i.i.d. data and
𝔼
[
𝜀
𝑖
|
𝑥
𝑖
]
=
0
. To estimate
𝛼
and
𝛽
by GMM, we need two moment conditions. Choose the best answer below:

Question

Consider the following nonlinear regression model:
𝑦
𝑖
=
𝛼
+
𝑒
𝛽
𝑥
𝑖
+
𝜀
𝑖
,
Assume i.i.d. data and 
𝔼
[
𝜀
𝑖
|
𝑥
𝑖
]
=
0
. To estimate 
𝛼
 and 
𝛽
 by GMM, we need two moment conditions. Choose the best answer below:

BlackTom AI · Accepted Answer

We start by restating the setup: a nonlinear regression model y_i = α + β x_i + ε_i with i.i.d. observations and E[ε_i | x_i] = 0. To estimate α and β via GMM, we pick moment conditions that are consistent with the model’s structure and the exogeneity of x_i (via E[ε_i | x_i] = 0). In this context, the natural moments use the residual (the difference between observed and predicted values) and, optionally, its interaction with the regressor x_i.

Option 1: The two moments are E[y_i − α − eβ x_i] = 0 and E[(y_i − α − eβ x_i) x_i] = 0.
- The first moment E[y_i − α − eβ x_i] = 0 corresponds to the average residual being zero, which is a standard univariate moment implying the model fits on average. However, the residual term should be β x_i, not eβ x_i, if we intend to match the typical specification y_i = α + β x_i + ε_i. If e represents the error term ε_i, then writing eβ x_i mixes the notation unpredictably, because the residual used in the moment condition should be a function of the parameters and data, not the random error separately. The second moment E[(y_i − α − eβ x_i) x_i] = 0 similarly uses the same residual form multiplied by x_i. If the intended meaning is E[(y_i − α − β x_i) x_i] = 0, then this option is close to the standard second moment, but the appearance of eβ x_i rather than β x_i introduces ambiguity and could be misleading.

Option 2: There is not enough information to write two moment conditions.
- In this setting, under E[ε_i | x_i] = 0 and i.i.d. data, we typically can write two valid moment conditions using the residual and its product with x_i. Therefore, asserting there isn’t enough information is not correct: the standard GMM setup indeed provides two (valid) moment conditions for identification with one regressor and an intercept.

Option 3: The two moments are E[y_i − α − eβ x_i] = 0 and E[(y_i − α − eβ x_i) x_i] = 0.
- This option mirrors option 1 and is essentially the same construction, though again the use of eβ x_i instead of β x_i introduces the same notational confusion. If interpreted as the usual residual  (y_i − α − β x_i) used in the moment conditions, then these two moments are the common choice for GMM with a linear regression structure: the unconditional mean of the residual is zero, and the residual is orthogonal to x_i.

Option 4: The two moments are E[y_i − α − eβ x_i] = 0 and E[x_i eβ x_i] = 0.
- The second moment here is problematic. E[x_i eβ x_i] = 0 would imply the expectation of x_i times some function eβ x_i is zero, which does not align with the usual orthogonality condition derived from E[ε_i | x_i] = 0. The left-hand side uses the residual, but the right-hand side uses a product that does not include the residual, breaking the standard construction of moment conditions for GMM in a regression context.

Option 5: The two moments are E[y_i − α − eβ x_i] = 0 and E[(y_i − α − eβ x_i)] = 0 = E(ε_i).
- The second moment collapses to the same as the first, or to a statement about the expectation of ε_i alone, which is not a valid separate moment condition given the exogeneity assumption E[ε_i | x_i] = 0. This would be redundant or incorrect in the standard GMM framework for this model.

Putting this together, the clean, standard pair of moment conditions (ignoring the notational quirk e vs β in the residual itself) is that the mean of the residual is zero and the residual is orthogonal to the regressor x_i, i.e., E[y_i − α − β x_i] = 0 and E[(y_i − α − β x_i) x_i] = 0. All options that attempt to express the residual-based moments with a clear residual term are aiming at this structure, but the presence of eβ x_i rather than β x_i introduces interpretational ambiguity in the residual definition. The options that present a second moment involving the product of the residual with x_i are the ones that align with the usual GMM identification strategy for α and β, since they enforce E[(y_i − α − β x_i) x_i] = 0 in addition to the zero-mean residual condition.

In summary, the correct conceptual family among the choices are those that express a pair of moments with the residual y_i − α − β x_i (or its equivalent) equal to zero, and the residual multiplied by x_i equal to zero. The presence of eβ x_i instead of β x_i is a notational issue; what matters is whether the moments capture E[y_i − α − β x_i] = 0 and E[(y_i − α − β x_i) x_i] = 0. The option that presents both a zero-mean residual and a cross-moment with x_i in this residual form is the structurally correct approach, while options that place the product outside the residual or that split the moments into inconsistent expressions fail to reflect the standard GMM identification conditions.

Consider the following nonlinear regression model: 𝑦 𝑖 = 𝛼 + 𝑒 𝛽 𝑥 𝑖 + 𝜀 𝑖 , Assume i.i.d. data and 𝔼 [ 𝜀 𝑖 | 𝑥 𝑖 ] = 0 . To estimate 𝛼 and 𝛽 by GMM, we need two moment conditions. Choose the best answer below:

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类似问题

Consider the following nonlinear regression model: 𝑦 𝑡 = 𝛼 𝑥 𝑡 𝛽 + 𝜀 𝑡 Assume i.i.d. data and 𝔼 [ 𝜀 𝑡 | 𝑥 𝑡 ] = 0 . To estimate 𝛼 and 𝛽 by GMM, we need two moment conditions. Choose the best answer below.

Consider the following nonlinear regression model: yi=α+eβxi+εi, Assume i.i.d. data and 𝔼[εi|xi]=0. To estimate α and β by GMM, we need two moment conditions. Choose the best answer below:

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