Consider the following linear regression model:
yi=α+βxi+εi,
Assume i.i.d. data and 𝔼[εi|xi]=0. To estimate α and β by GMM, we use the three theoretical moment conditions
𝔼[yi−α−βxi]=0
𝔼[(yi−α−βxi)xi]=0
𝔼[(yi−α−βxi)x
2
i
]=0
To compute the variance of the GMM estimator we need the matrices Γ0 and Φ0.

Question

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We are given a linear regression model yi = α + β xi + εi with i.i.d. data and E[εi|xi] = 0, and three theoretical moment conditions used in GMM: E[yi − α − β xi] = 0, E[(yi − α − β xi) xi] = 0, and E[(yi − α − β xi) x_i^2] = 0. To compute the variance of the GMM estimator, we need the matrix Γ0, which is the Jacobian (the derivative) of the moment conditions with respect to the parameter vector θ = (α, β), evaluated at the true parameter values. Here, the moment vector can be written as g_i(θ) = [1, xi]·(yi − α − β xi) for the first two moments and g_3(θ) = xi^2·(yi − α − β xi) for the third moment, so we differentiate each moment with respect to α and β:

- For the first moment m1 = yi − α − β xi: ∂m1/∂α = −1, ∂m1/∂β = −xi. Thus the first row of the Jacobian is [−1, −xi].
- For the second moment m2 = xi·(yi − α − β xi): ∂m2/∂α = −xi, ∂m2/∂β = −xi^2. Thus the second row is [−xi, −xi^2].
- For the third moment m3 = xi^2·(yi − α − β xi): ∂m3/∂α = −xi^2, ∂m3/∂β = −xi^3. Thus the third row is [−xi^2, −xi^3].

Therefore, Γ0 = E[ [−1, −xi]; [−xi, −xi^2]; [−xi^2, −xi^3] ]. This is a 3×2 matrix whose entries are the negative derivatives of each moment with respect to α and β, averaged over the distribution of xi.

Now evaluate each answer option:

- Option 1: “There is not enough information to compute the matrix Γ0.” This is inaccurate because, given the moment conditions and the model specification, the derivatives with respect to α and β can be computed symbolically as shown, and Γ0 is precisely the expectation of those derivatives. The availability of xi from the data suffices to form each row of the Jacobian.

- Option 2: “The matrix Γ0 is: Γ0 = E[1, xi; xi, x^2_i].” Here the signs are incorrect and the rows do not match the actual derivatives with respect to α and β. The first row should be [−1, −xi], not [1, xi], and the second row should involve [−xi, −xi^2], not [xi, x^2_i], so this option misstates both the signs and the functional form.

- Option 3: “The matrix Γ0 is: Γ0 = E[−1, −xi; −xi, −x^2_i; −x^2_i, −x^3_i].” This matches the computed derivatives exactly: the three rows correspond to the derivatives of m1, m2, and m3 with respect to (α, β). The signs are all negative as required by the derivative of (yi − α − β xi) with respect to α and β, and the lower rows align with derivatives for the xi and xi^2 weighting in the second and third moments. This is the correct structure for Γ0.

- Option 4: “The matrix Γ0 is: Γ0 = E[−1, −xi; −xi, −x^2_i].” This only includes two rows, effectively omitting the third moment's derivative. Since the model uses three moment conditions, Γ0 must be a 3×2 matrix incorporating the third row [−xi^2, −xi^3]. Therefore, this option is incomplete and incorrect.

- Option 5: “The matrix Γ0 is: Γ0 = E[−1, −xi; −x^2_i, −xi^3_i; −xi, −x^2_i].” This presents a misordered set of derivatives and does not align with the correct row structure derived from differentiating each moment with respect to (α, β). The second row should be [−xi, −xi^2], not [−x^2_i, −xi^3_i], and the first row is correctly [−1, −xi], but the third row should be [−xi^2, −xi^3]. Overall, this option misplaces or mislabels entries and is not the correct Jacobian.

In summary, the third option provides the Jacobian with the correct three rows corresponding to the three moment conditions, using the appropriate negative derivatives with respect to α and β, and thus matches the theoretical Γ0 required for the GMM variance calculation.

BU.232.630.W1.SP25 Quiz 2 solutions

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