Consider the following nonlinear regression model:
𝑦
𝑖
=
𝛼
+
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 at least two moment conditions. Chose the correct answer below.

Question

Consider the following nonlinear regression model:
𝑦
𝑖
=
𝛼
+
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 at least two moment conditions. Chose the correct answer below.

BlackTom AI · Accepted Answer

Examining the problem, we are dealing with a simple linear regression y_i = α + β x_i + ε_i, with E[ε_i | x_i] = 0 and i.i.d. data. In a GMM framework, valid moment conditions come from the orthogonality between the structural error and instruments (here, the regressor and a constant).
Option 1: E[y_i − α − β x_i] = 0 and E[(y_i − α − β x_i) x_i] = 0. These are the standard mean and slope-related moment conditions one can derive from E[ε_i | x_i] = 0 by multiplying ε_i with 1 (the constant) and with x_i, respectively. They are valid because the expectation of the error conditional on x is zero, which implies these residuals are uncorrelated with the instruments. This pair provides at least two independent moments needed for GMM estimation of α and β.
Option 2: E[(y_i − α − β x_i) x_i β x_i − 1] = 0. This mixes the residual with a nonlinear combination of instruments (x_i and β x_i) and includes a constant shift of −1 inside the moment. There is no theoretical justification for including β x_i inside the moment like this, and the −1 term is arbitrary; such a moment would not vanish in expectation under the model, making it invalid as a moment condition for identification and estimation.
Option 3: E[y_i − α − β x_i] = 0. While this is a valid single moment (the mean of the residual), using only this moment is insufficient to identify both α and β in a single-equation system. GMM requires as many independent moments as parameters to identify all parameters; here we have two parameters but only one moment, so this alone cannot estimate both α and β.
Option 4: E[(y_i − α − β x_i) x_i β x_i − 1] = 0. Similar to Option 2, this crafts a nonlinear and unjustified moment that combines the residual with β x_i in a way that would not be satisfied in general under the assumed model. It would not serve as a valid, reliable moment condition for GMM estimation of α and β.
Option 5: Any of the answers provides appropriate moment conditions for estimation. This is incorrect because not all listed moments are valid or sufficient. Some are invalid due to inappropriate constructions or insufficient number of moments for identification.

Consider the following nonlinear regression model: 𝑦 𝑖 = 𝛼 + 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 at least two moment conditions. Chose the correct answer below.

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Similar Questions

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: yt=αxβt+εt Assume i.i.d. data and 𝔼[εt|xt]=0. To estimate α and β by GMM, we need two moment conditions. Choose the best answer below:

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