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

The question concerns selecting two moment conditions to identify the intercept α and slope β in the nonlinear regression y_i = α + β x_i + ε_i, with E[ε_i | x_i] = 0 and i.i.d. data, using GMM.

Option 1: E[y_i − α − e β x_i] = 0 and E[(y_i − α − β x_i) log(x_i)] = 0.
- Interpreting the first moment: y_i − α − β x_i is the composite error ε_i. Setting E[ε_i] = 0 is a valid moment condition, because E[ε_i] = E[E[ε_i | x_i]] = E[0] = 0 under the given assumption. This corresponds to the standard orthogonality with a constant instrument (a valid moment condition for identifying α and β).
- Interpreting the second moment: E[(ε_i) log(x_i)] = 0 follows from E[ε_i | x_i] = 0, since E[ε_i log(x_i)] = E[E[ε_i | x_i] log(x_i)] = E[0 · log(x_i)] = 0. Here, log(x_i) serves as an instrument that is a function of x_i. As long as log(x_i) is not perfectly correlated with the first instrument (the constant), this provides a second valid moment to achieve two-equation/two-parameter identification.
- Practical note: Using log(x_i) as an instrument is mathematically admissible under the exogeneity condition, but in practice one typically uses E[ε_i] = 0 and E[x_i ε_i] = 0, which corresponds to the standard moments with instruments 1 and x_i. The second moment with log(x_i) is a valid alternative instrument if log(x_i) is well-defined and exogenous, but it is not the conventional pair of moments for this simple linear-in-parameters setup.

Option 2: The two moments are E[y_i − α − β x_i] = 0 and E[(y_i − α − β x_i) log(x_i)] = 0.
- The first moment is the same as in Option 1, representing E[ε_i] = 0, which is valid given E[ε_i | x_i] = 0.
- The second moment uses log(x_i) as the instrument, which again relies on E[ε_i | x_i] = 0 to imply E[ε_i log(x_i)] = 0. This is a legitimate moment condition provided log(x_i) is a valid (non-degenerate) instrument that is uncorrelated with ε_i.
- However, there is a caveat: log(x_i) is only defined for x_i > 0. If the data include non-positive x_i values, this choice would be problematic. Even when defined, using log(x_i) changes the weighting of observations in the moment conditions and may affect efficiency relative to the conventional pair of moments. Conceptually, it remains an exogenous moment because it is a function of x_i and E[ε_i | x_i] = 0.

Additional considerations across options:
- Why not E[x_i (y_i − α − β x_i)] = 0? This is the traditional second moment, arising from E[ε_i x_i] = 0 when E[ε_i | x_i] = 0. It is a standard, straightforward choice that yields the same identification as using E[x_i ε_i] = 0. It is typically preferred for interpretability and numerical stability.
- Why might the log(x_i) moment be used? If you suspect a nonlinear relationship between x_i and ε_i or you want to exploit a nonlinear instrument to improve efficiency under certain distributions, a function of x_i like log(x_i) can still be exogenous and provide a valid moment. Yet its validity hinges on E[ε_i | x_i] = 0 and the positivity of x_i.

In summary, the first moment E[y_i − α − β x_i] = 0 is a straightforward and standard moment condition for this model. The second moment E[(y_i − α − β x_i) log(x_i)] = 0 is also valid under the exogeneity condition but relies on log(x_i) being a well-defined, exogenous instrument. The conventional two-moment pair is E[y_i − α − β x_i] = 0 and E[x_i(y_i − α − β x_i)] = 0, which corresponds to using the constant and x_i as instruments. The given options present two related moment specifications, with the first aligning with the standard choice and the second offering a legitimate but less conventional alternative depending on data support and modeling preferences.

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: Single choice

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