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 use the following moment conditions:
𝔼
[
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
]
=
0
𝔼
[
(
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
)
𝑥
𝑡
]
=
0
To compute the variance of the estimates, we need to estimate the matrices
𝛤
0
and
𝛷
0
.

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 use the following moment conditions:
𝔼
[
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
]
=
0
𝔼
[
(
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
)
𝑥
𝑡
]
=
0
To compute the variance of the estimates, we need to estimate the matrices 
𝛤
0
 and 
𝛷
0
.

BlackTom AI · Accepted Answer

We are given a nonlinear regression model y_t = α x_t^β + ε_t with i.i.d. data and E[ε_t | x_t] = 0. The GMM moment conditions are:
1) E[y_t − α x_t^β] = 0
2) E[(y_t − α x_t^β) x_t] = 0
To compute the asymptotic variance of the GMM estimates, we need the matrix Γ0, which is the expectation of the Jacobian of the moment conditions with respect to the parameter vector θ = (α, β). Denoting g1,t = y_t − α x_t^β and g2,t = (y_t − α x_t^β) x_t, the Jacobian ∂g_t/∂θ is a 2×2 matrix consisting of the derivatives of g1,t and g2,t with respect to α and β:
- For g1,t = y_t − α x_t^β:
  ∂g1,t/∂α = −x_t^β
  ∂g1,t/∂β = −α x_t^β log(x_t)
- For g2,t = (y_t − α x_t^β) x_t:
  ∂g2,t/∂α = −x_t^(β+1)
  ∂g2,t/∂β = −α x_t^(β+1) log(x_t)
Thus, the population Γ0 is E[∂g_t/∂θ'] =
[ [−E[x_t^β], −E[α x_t^β log x_t]]
  [−E[x_t^(β+1)], −E[α x_t^(β+1) log x_t]] ]
If we replace expectations by sample averages and plug in α with its estimator α̂ (or use the current parameter estimates when evaluating Γ0 in a one-step or two-step GMM), the estimator is
Γ̂0 = [ [−(1/T) ∑ x_t^β, −(1/T) ∑ α̂ x_t^β log x_t]
          [−(1/T) ∑ x_t^(β+1), −(1/T) ∑ α̂ x_t^(β+1) log x_t] ]
Now, evaluating why each presented option is correct or incorrect:
- Option that shows Γ̂0 as a 2×2 matrix with top-left entry −(1/T) ∑ x_t^β, top-right entry −(1/T) ∑ α̂ x_t^β log x_t (and similarly the second row with −(1/T) ∑ x_t^(β+1) and −(1/T) ∑ α̂ x_t^(β+1) log x_t) correctly matches the derived Jacobian structure and uses the estimated parameters in the nonlinear terms. This is the correct form in a practical GMM implementation.
- Any option that misses the x_t^(β+1) term in the (2,1) position is incorrect because ∂g2,t/∂α = −x_t^(β+1) is required by the chain rule applied to g2,t = (y_t − α x_t^β) x_t.
- Any option that omits the log(x_t) factors in the derivatives with respect to β (i.e., missing log x_t where it should appear) is incorrect, since ∂(α x_t^β)/∂β = α x_t^β log x_t and similarly appears in ∂g2,t/∂β via the −α x_t^(β+1) log x_t term.
- Any option that places plus signs instead of minus signs in the Jacobian entries would be mathematically inconsistent with the derivatives of g1,t and g2,t with respect to α and β, and hence incorrect.
- Some answers might present a simplified or incorrectly scaled version (for example dropping the 1/T averaging, or mixing terms between g1 and g2 inappropriately). These would not reflect the true population Jacobian or its sample analogue and are therefore inaccurate.
- There could be an option claiming there is not enough information to compute Γ0. That would be incorrect because, given the model and moment conditions, Γ0 is identified as the expected Jacobian, which is well-defined and estimable (up to the use of α̂ and β in the nonlinear terms).
- If an option presents a matrix with entries that do not align with the derivatives of g1,t and g2,t with respect to α and β, or uses the wrong powers of x_t in the derivatives, it is incorrect for the same mathematical reasons described above.

In summary, the correct expression for the estimated Γ0, incorporating the standard GMM Jacobian with the two moment conditions and using estimated parameters where needed, is:
Γ̂0 = [ [−(1/T) ∑ x_t^β, −(1/T) ∑ α̂ x_t^β log x_t], [−(1/T) ∑ x_t^(β+1), −(1/T) ∑ α̂ x_t^(β+1) log x_t] ]
This form faithfully represents the derivatives of the moment conditions with respect to (α, β) and their sample analogs, and it aligns with the standard GMM theory for a two-moment, two-parameter setting.

BU.232.630.F3.SP25 Quiz 2 2025 - all questions

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