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

The question asks to identify the matrix Γ0 used in the variance computation for the GMM estimator under the two moment conditions. To approach this, recall that for GMM with moments g_i(θ) where θ = (α, β)', Γ0 is the expected Jacobian of the moment conditions with respect to θ, i.e., Γ0 = E[∂g_i(θ)/∂θ']. Given the two moments
g1,i(α,β) = y_i − α − e^{β x_i}
 g2,i(α,β) = (y_i − α − e^{β x_i}) x_i,
we compute the partial derivatives with respect to α and β:
- For g1,i: ∂g1,i/∂α = −1 and ∂g1,i/∂β = − x_i e^{β x_i}
- For g2,i: ∂g2,i/∂α = − x_i and ∂g2,i/∂β = − x_i^2 e^{β x_i}
Thus the Jacobian matrix for a given i is
[ [ −1, − x_i e^{β x_i} ], [ − x_i, − x_i^2 e^{β x_i} ] ].
Taking expectations over the data-generating process gives Γ0 = E[ [ −1, − x_i e^{β x_i} ]; [ − x_i, − x_i^2 e^{β x_i} ] ].

Option analysis:
- The provided expression in the option matches exactly this computed Jacobian structure, with the same entries in corresponding positions and the same exponential term e^{β x_i} multiplying the x_i terms. This alignment confirms that the option correctly represents Γ0 under the stated model and moment conditions.

Why this is correct in context: the two moments yield a 2×2 Γ0 since θ has two components, and each entry is the expectation of the corresponding derivative. The (1,1) entry is ∂g1/∂α = −1, the (1,2) entry is ∂g1/∂β = − x_i e^{β x_i}, the (2,1) entry is ∂g2/∂α = − x_i, and the (2,2) entry is ∂g2/∂β = − x_i^2 e^{β x_i}, all averaged over i.

Potential misunderstandings to avoid:
- One might mistakenly omit the cross-derivative terms or misplace the signs; keeping track that each derivative comes directly from the corresponding moment function is essential.
- It’s important to note that Γ0 is the expectation of the Jacobian of g_i with respect to θ, not the variance of g_i; confusing these leads to incorrect matrices.
- If someone uses an alternative parameterization or defines g_i differently, the exact entries could change; here, with g1 and g2 as specified, the entries above are correct.

Similar Questions

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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