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 problem in our own words to ensure clarity: we have the nonlinear regression y_i = α + e β x_i + ε_i with i.i.d. data and E[ε_i | x_i] = 0. To estimate α and β by GMM, we need two population moment conditions that involve the error term and the regressors, reflecting the exogeneity assumption.

Option 1: E[y_i − α − e β x_i] = 0 and E[(y_i − α − e β x_i) x_i^3] = 0. Here, the first moment is a plausible average residual condition, but the second uses x_i^3 rather than the regressor x_i (or some valid instrument) as the multiplier. In standard GMM for a single regressor, the second moment should involve (y_i − α − β x_i) multiplied by x_i (or by an appropriate instrument Z_i that is uncorrelated with ε_i). The appearance of x_i^3 is extraneous and not justified by the usual exogeneity/expression for the moments, so this option is not correct.

Option 2: There is not enough information to write two moment conditions. In fact, given E[ε_i | x_i] = 0, we can write at least two valid moment conditions, typically E[y_i − α − β x_i] = 0 and E[(y_i − α − β x_i) x_i] = 0 (or equivalents using valid instruments). Therefore, this statement is not accurate—they are providing an uninformative blanket claim instead of identifying the proper moments.

Option 3: The two moments are E[y_i − α − e β x_i] = 0 and E[y_i − e β x_i] = 0. The second moment omits the α and β x_i terms in a way that makes it equivalent to E[ε_i] = 0 only if the model is specified with α absorbed appropriately. It fails to incorporate the regression error in relation to the regressor x_i in a way that yields two independent, useful moment conditions for GMM estimation of α and β.

Option 4: The two moments are E[y_i − α − e β x_i] = 0 and E[(y_i − α − e β x_i)] = E(ε_i). The left-hand expression is a standard residual moment, but the right-hand expression is not a moment condition set to zero; instead, it states that the expectation equals the (unobserved) error’s expectation. While E(ε_i) = 0 under the usual assumption, listing it as a separate second moment condition is awkward and does not provide a proper, distinct moment equation that equals zero. Thus this option is not appropriate as a two-moment specification for GMM.

Option 5: The two moments are E[y_i − α − e β x_i] = 0 and E[(y_i − α − e β x_i) x_i^3] = 0. Similar to Option 1, while the first moment is plausible, the second uses x_i^3 instead of the regressor x_i (or a valid instrument). The multiplier should be a function of x_i that yields a valid orthogonality condition with ε_i; using x_i^3 does not correspond to the standard exogeneity condition for this model, so this option is incorrect.

In summary, each of the provided options either mis-specifies the second moment by using x_i^3 (instead of the usual x_i or a valid instrument), or includes a second moment that does not form a proper, zero-expectation condition aligned with E[ε_i | x_i] = 0. Consequently, none of the listed choices correctly identifies the two moment conditions you would typically use for GMM estimation of α and β in this model (namely E[y_i − α − β x_i] = 0 and E[(y_i − α − β x_i) x_i] = 0, or their equivalents with valid instruments).

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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