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 use the two theoretical moment conditions
𝔼[yi−α−eβxi]=0
𝔼[(yi−α−eβxi)xi]=0
To compute the variance of the GMM estimator we need the matrices Γ0 and Φ0.

Question

BlackTom AI · Accepted Answer

We begin by restating the setup: a nonlinear regression yi = α + e^{β xi} + εi with moment conditions E[yi − α − e^{β xi}] = 0 and E[(yi − α − e^{β xi}) xi] = 0. For GMM, the matrix Γ0 is the expectation of the Jacobian of the moment conditions with respect to the parameters θ = (α, β) evaluated at the true parameter values. Specifically, if g_i(θ) = [ yi − α − e^{β xi}; (yi − α − e^{β xi}) xi ], then the partial derivatives are:
- ∂/∂α of the first moment is −1, and ∂/∂β of the first moment is − xi e^{β xi}.
- For the second moment, ∂/∂α is − xi, and ∂/∂β is − xi · (∂/∂β e^{β xi}) = − xi · xi e^{β xi} = − xi^2 e^{β xi}.
Collecting these into the 2×2 Jacobian matrix gives:
Γ0 = E[ [ −1, − xi e^{β xi} ]; [ − xi, − xi^2 e^{β xi} ] ].
Now evaluate the provided options:
Option 1: 'There is not enough information to compute the matrix Γ0.' This is incorrect because, given the model and moment conditions, the Jacobian with respect to (α, β) is fully determined and the expression for Γ0 is identifiable in terms of E of known functions of xi. The statement ignores the standard form for Γ0 and overstates the information requirement.
Option 2: 'The matrix Γ0 is: Γ0 = E[ [ −1, − x_i e^{β x_i} ]; [ − x_i, − x_i^2 e^{β x_i} ] ].' This matches the correct Jacobian-derived form described above. It correctly places −1 in the (1,1) entry, − x_i e^{β x_i} in the (1,2) entry, − x_i in the (2,1) entry, and − x_i^2 e^{β x_i} in the (2,2) entry. This is the correct representation of Γ0.
Option 3: 'The matrix Γ0 is: Γ0 = E[ [ −1, β e^{β x_i} ]; [ − x_i, − x_i e^{β x_i} ] ].' This misstates the derivatives: the (1,2) entry should be − x_i e^{β x_i}, not β e^{β x_i}, and the (2,2) entry should be − x_i^2 e^{β x_i}, not − x_i e^{β x_i}.
Option 4: 'The matrix Γ0 is: Γ0 = E[ [ −1, − x_i e^{β x_i} ]; [ − x_i, − x^2_i e^{β x_i} ; −1] ].' This has several formatting and placement issues: the (2,2) term is written with an extraneous '; −1' and misordered elements; it does not present a valid 2×2 matrix, and the −1 in the bottom-right is incorrect. Overall, this option misrepresents the matrix.
Option 5: 'The matrix Γ0 is: Γ0 = E[ [ − x_i, − x_i e^{β x_i} ]; [ − x_i, − x_i^2 e^{β x_i} ] ].' This swaps the (1,1) entry from −1 to − x_i, which is incorrect because the derivative of the first moment with respect to α is −1, not − x_i. The resulting matrix is inconsistent with the proper Jacobian.
In summary, the only option that faithfully encodes the correct Jacobian of the moment conditions with respect to (α, β) and matches the standard GMM Γ0 form is Option 2. The other choices either misstate derivatives, misplace entries, or introduce inconsistent terms, leading to incorrect Γ0 expressions.

BU.232.630.F2.SP25 Sample Quiz 2 2025

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类似问题

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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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