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 chose among the following moment conditions:
𝔼
[
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
]
=
0

𝔼
[
(
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
)
𝑥
𝑡
]
=
0

𝔼
[
(
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
)
1
𝑥
𝑡
]
=
0
Choose the most appropriate answer below:

Question

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 chose among the following moment conditions:
𝔼
[
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
]
=
0

𝔼
[
(
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
)
𝑥
𝑡
]
=
0

𝔼
[
(
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
)
1
𝑥
𝑡
]
=
0
Choose the most appropriate answer below:

BlackTom AI · Accepted Answer

Question restatement: We are given a nonlinear regression model y_t = α x_t^β (interpreting the notation as α times x_t raised to the power β) with E[ε_t | x_t] = 0 and i.i.d. data. The proposed GMM moment conditions are:
1) E[y_t − α x_t^β] = 0
2) E[(y_t − α x_t^β) x_t] = 0
3) E[(y_t − α x_t^β) 1/x_t] = 0
We are to assess which of the listed options about the validity of these moment conditions is correct.

Option 1: Only the first and second equations are valid moment conditions to estimate α and β by GMM.
- Why the first condition is valid: Since y_t − α x_t^β = ε_t and E[ε_t | x_t] = 0, taking expectation over the joint distribution yields E[y_t − α x_t^β] = E[ε_t] = 0. This aligns with the standard conditional mean zero assumption.
- Why the second condition is valid: E[(y_t − α x_t^β) x_t] = E[ε_t x_t]. Using the law of iterated expectations and E[ε_t | x_t] = 0, we get E[ε_t x_t] = E[x_t E(ε_t | x_t)] = E[x_t · 0] = 0. So this is a valid moment condition.
- Why the third condition is not considered here: The option asserts only first and second are valid, implying the third is invalid. However, given E[ε_t | x_t] = 0, E[(y_t − α x_t^β) 1/x_t] = E[ε_t / x_t] = E[ (1/x_t) E(ε_t | x_t) ] = E[ (1/x_t) · 0 ] = 0. Thus the third condition is also valid. Therefore this option is incorrect.

Option 2: Only the second and third equations are valid moment conditions to estimate α and β by GMM.
- The reasoning above shows the first condition is also valid because E[y_t − α x_t^β] = 0. Since the first condition holds, asserting that only the second and third are valid is incorrect. This option is therefore false.

Option 3: None of the equations above are valid moment conditions to estimate α and β by GMM.
- This claim contradicts the derivation: with E[ε_t | x_t] = 0, each of the moments 1) E[y_t − α x_t^β], 2) E[(y_t − α x_t^β) x_t], and 3) E[(y_t − α x_t^β) 1/x_t] equals zero. Hence at least some of them are valid, so this option is incorrect.

Option 4: Only the first and third equations are valid moment conditions to estimate α and β by GMM.
- As shown, the second condition is also valid since E[(y_t − α x_t^β) x_t] = 0 under E[ε_t | x_t] = 0. Therefore excluding the second condition makes this option false. This reasoning indicates option 4 is incorrect.

Option 5: All equations are valid moment conditions to estimate α and β by GMM.
- This matches the standard result under E[ε_t | x_t] = 0. The first moment is zero because ε_t has mean zero, the second is zero because the product ε_t x_t has zero mean given the conditional mean independence, and the third is zero because ε_t/x_t has zero mean conditional on x_t. Putting these together, each of the three moments provides a valid moment condition for GMM estimation of α and β. Thus this option is consistent with the assumed model and moments and is the correct characterization among the choices provided.

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Similar Questions

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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