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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BlackTom AI · Accepted Answer

The question concerns choosing two moment conditions for GMM estimation of a nonlinear regression y_t = α x_t^β + ε_t under E[ε_t | x_t] = 0.

First, note a key principle: if E[ε | x] = 0, then the residual ε_t = y_t − α x_t^β is mean-independent of x_t, so any function of x_t can be used as a valid instrument to form moment conditions. In particular, E[ε_t] = 0 is a valid moment, and E[ε_t φ(x_t)] = 0 for any nonstochastic φ(x_t) is also valid. The choice of φ(x_t) is up to identification considerations, but it must be a function that yields a nondegenerate system when paired with the residual.

Option analysis:
- Option: "The two moments are E[y_t − α x_t^β] = 0 and E[(y_t − α x_t^β) log(...)]." This statement tries to specify a second moment using a log transformation, but it is incomplete as written: it does not specify the exact second function of x_t to use in the second moment (the log is shown, but the argument of the log is missing). Moreover, even if we include log x_t, the choice is unconventional and not guaranteed to identify both α and β uniquely without additional regularity or normalization. The core requirement is two independent moment conditions based on E[ε_t φ(x_t)] = 0 for some φ, not a generic “log” form that may fail if φ is not properly chosen or if x_t takes values that make log ill-defined. In short, this option is underspecified and potentially invalid as stated.

- General principle to create valid moments: Since E[ε_t | x_t] = 0, we have E[ε_t] = 0 and E[ε_t φ(x_t)] = 0 for any suitable φ. A natural two-moment setup would be:
  • m1(θ) = E[y_t − α x_t^β] = 0
  • m2(θ) = E[(y_t − α x_t^β) φ(x_t)] = 0 for some φ such that the resulting system identifies α and β. The choice of φ could be x_t^k, indicators, splines, or other functions of x_t, provided the model is identified and moments are finite. Using log(x_t) is only valid if x_t > 0 and the chosen φ leads to a nondegenerate system; otherwise it may fail to identify the parameters or be ill-posed.

- Why other potential readings of the option are problematic: If the second moment uses log(y_t) or a log-transformed residual without a proper rationale, it generally does not align with the exogeneity condition E[ε|x]=0 and can introduce bias or non-identification. The key is tying the second moment to a function of x_t, not to an arbitrary transformation of y_t.

In summary, the idea that the two moments come from E[y_t − α x_t^β] = 0 and E[(y_t − α x_t^β) log x_t] = 0 is only acceptable if log x_t is well-defined for all observations and the resulting moment system uniquely identifies α and β. More broadly, any valid two-moment GMM setup should specify two independent φ(x_t) functions to form E[(y_t − α x_t^β) φ_j(x_t)] = 0 for j = 1, 2, ensuring identification. The given answer fragment is incomplete because it does not clearly specify the second moment function φ(x_t) and may not guarantee identification or even validity in all cases.

BU.232.630.W6.SP25 Quiz 2 solutions

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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