Consider the following nonlinear regression model:
yt=αx
β
t
+εt
Assume i.i.d. data and 𝔼[εt|xt]=0. To estimate α and β by GMM, we chose among the following moment conditions:
𝔼[yt−αx
β
t
]=0
𝔼[(yt−αx
β
t
)xt]=0
𝔼[(yt−αx
β
t
)
1

xt
]=0
Choose the most appropriate answer below:

Question

Consider the following nonlinear regression model:
yt=αx
β
t
+εt
Assume i.i.d. data and 𝔼[εt|xt]=0. To estimate α and β by GMM, we chose among the following moment conditions:
𝔼[yt−αx
β
t
]=0
𝔼[(yt−αx
β
t
)xt]=0
𝔼[(yt−αx
β
t
)
1

xt
]=0
Choose the most appropriate answer below:

BlackTom AI · Accepted Answer

In this nonlinear regression setup, y_t = α x_t^β + ε_t with E[ε_t | x_t] = 0. A core property of exogeneity is that any moment of the form E[ ε_t g(x_t) ] = 0 holds for any function g with finite variance, because E[ ε_t g(x_t) ] = E[ E[ ε_t | x_t] g(x_t) ] = 0.

Option 1: E[ y_t − α x_t^β ] = 0. Here the residual is ε_t, and multiplying by the constant 1 yields E[ ε_t ] = 0. This is valid since the conditional mean of ε_t given x_t is zero, which implies its unconditional mean is also zero. Therefore this is a valid moment condition.

Option 2: E[ (y_t − α x_t^β) x_t ] = 0. This uses the residual ε_t multiplied by x_t. By the law above, E[ ε_t x_t ] = E[ E[ ε_t | x_t] x_t ] = 0, so this is also a valid moment condition. The presence of x_t does not introduce bias under the exogeneity assumption as long as moments are finite.

Option 3: E[ (y_t − α x_t^β) (1/x_t) ] = 0. This is of the same spirit: multiply the residual by a function of x_t, namely 1/x_t. Since E[ ε_t | x_t ] = 0, we have E[ ε_t (1/x_t) ] = E[ E[ ε_t | x_t] (1/x_t) ] = 0, provided E[ |ε_t| / |x_t| ] < ∞ and x_t ≠ 0 with probability 1. Thus this is also a valid moment condition, though one must be careful about the support of x_t and moments’ finiteness in practice.

Option 4: The claim that only the first and second equations are valid would be incorrect because, given exogeneity E[ε_t | x_t] = 0, any function of x_t can be used to construct valid moments, including 1/x_t as in Option 3. There is no theoretical restriction forcing only first and second to be valid; the structure E[ ε_t g(x_t) ] = 0 supports a wide class of g, subject to regularity.

Option 5: The claim that none of the equations are valid is inconsistent with the exogeneity implication. Since E[ε_t | x_t] = 0, at least the moment with g(x_t) ≡ 1 is valid, so this option is incorrect.

Putting these together, each of the three moment conditions uses a function g(x_t) (namely, 1, x_t, and 1/x_t) that satisfies E[ ε_t g(x_t) ] = 0 under the stated exogeneity, with caveats about support and finite moments for the third case. The reasoning shows why the first two are clearly valid and the third is valid under standard regularity conditions; the other options misstate the exogeneity implications or rely on invalid conclusions about which moments are acceptable.

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