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 use the following moment conditions:
𝔼[yt−αx
β
t
]=0
𝔼[(yt−αx
β
t
)xt]=0
To compute the variance of the estimates, we need to estimate the matrices Γ0 and Φ0.

Question

BlackTom AI · Accepted Answer

We start by restating what is given and what needs to be computed.

Question context: A nonlinear regression model is yt = α x_t^β + ε_t with E[ε_t | x_t] = 0. The moments used in GMM are:
1) E[y_t − α x_t^β] = 0
2) E[(y_t − α x_t^β) x_t] = 0
We want to estimate α and β and, in order to compute the asymptotic variance of the GMM estimates, we need to estimate the matrices Γ0 and Φ0. The provided option proposes an estimate for Γ0:
Γ0_hat = [ − ∑ x_t^β / T  − α ∑ x_t^β log(x_t) / T  − ∑ x_t^{β+1} / T  − α ∑ x_t^{β+1} log(x_t) / T ]

Now, let’s analyze how Γ0 should be formed in GMM.

- Step 1: Define the moment vector g_t(θ) with θ = (α, β). Given the two moments, a natural compact form is:
  g_t(θ) = [ g1_t(θ); g2_t(θ) ] where
  g1_t(θ) = y_t − α x_t^β,
  g2_t(θ) = (y_t − α x_t^β) x_t.

- Step 2: Γ0 is the expectation of the Jacobian of the moment conditions with respect to θ, evaluated at the true θ (or its consistently estimated version). Specifically, with θ = (α, β) and g_t(θ) as above, the partial derivatives are:
  ∂g1_t/∂α = − x_t^β,
  ∂g1_t/∂β = − α x_t^β log(x_t),
  ∂g2_t/∂α = − x_t^{β+1},
  ∂g2_t/∂β = − α x_t^{β+1} log(x_t).

- Step 3: Γ0 is E[ ∂g_t(θ)/∂θ' ]. If we write ∂g_t/∂θ as a 2×2 matrix, Γ0 is the expected value of that matrix:
  Γ0 = E[ [ − x_t^β,  − α x_t^β log(x_t) ]; [ − x_t^{β+1},  − α x_t^{β+1} log(x_t) ] ].
  In components, the four entries are:
  Γ0[1,1] = − E[x_t^β],
  Γ0[1,2] = − E[α x_t^β log(x_t)],
  Γ0[2,1] = − E[x_t^{β+1}],
  Γ0[2,2] = − E[α x_t^{β+1} log(x_t)].

- Step 4: If you replace expectations with sample analogs, the consistent estimator is:
  Γ0_hat = [ − (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) ],
  where α̂ and β̂ are the estimated values used to form the residuals and plug-ins. Note that the second column entries involve log(x_t) and the corresponding powers, weighted by α (or α̂) and by the appropriate x_t power.

- Step 5: About the proposed Γ0_hat in the option: the expression shown appears to mix terms into a single row vector and includes log(xt) factors in a way that does not align with the standard Jacobian structure above. Specifically, the term
  − (1/T) ∑ x_t^β log(x_t)
  multiplied by α and the presence of a separate log(xt) term in a way that would place them into a simple 1×2 or 2×2 block without clear alignment to the Jacobian is not how Γ0 is conventionally formed. Moreover, Γ0 is a 2×2 matrix (since there are two θ parameters), not a single row, so the provided boxed expression seems misformatted or incomplete for a correct Γ0 estimate.

- Step 6: In practice, to compute Γ0_hat correctly, you should:
  1) use the residuals u_t(θ̂) = y_t − α̂ x_t^{β̂},
  2) compute the Jacobian matrix J_t(θ̂) = ∂g_t(θ)/∂θ' evaluated at θ̂,
  3) average over t: Γ0_hat = (1/T) ∑ J_t(θ̂).
  Concretely, J_t(θ̂) = [ − x_t^{β̂}, − α̂ x_t^{β̂} log x_t; − x_t^{β̂+1}, − α̂ x_t^{β̂+1} log x_t ]. Therefore, Γ0_hat = [ − (1/T) ∑ x_t^{β̂},  − (1/T) ∑ α̂ x_t^{β̂} log x_t;  − (1/T) ∑ x_t^{β̂+1},  − (1/T) ∑ α̂ x_t^{β̂+1} log x_t ].

- Step 7: Common pitfalls and misconceptions:
  • Forgetting to use the residuals in the Jacobian calculation, or using y_t − α x_t^β without the dependence on θ can lead to incorrect Γ0.
  • Treating Γ0 as a diagonal or scalar quantity when it is a 2×2 matrix in this two-parameter setting.
  • Mixing up the order or placement of the partial derivatives; the correct arrangement puts ∂g1/∂α and ∂g1/∂β in the first row, and ∂g2/∂α and ∂g2/∂β in the second row.

- Summary: The correct expression for Γ0_hat is a 2×2 matrix formed from the average of the Jacobian of the moment functions with respect to (α, β), evaluated at the estimated parameters, namely:
Γ0_hat = (1/T) ∑ [ [ − x_t^{β̂}, − α̂ x_t^{β̂} log x_t ]; [ − x_t^{β̂+1}, − α̂ x_t^{β̂+1} log x_t ] ].

Thus, the proposed single-row, log(x_t)-weighted formula in the option is not the standard or correct way to construct Γ0_hat for this GMM setup; the proper construction yields a 2×2 matrix as shown above, using the residuals and the Jacobian evaluated at the estimated parameters. If any of α̂ or β̂ are replaced by true θ in a theoretical derivation, the same functional form applies with θ substituted accordingly.

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

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