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

Question

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

BlackTom AI · Accepted Answer

To tackle this, I’ll first restate the setup and what Γ0 represents in GMM. We have a linear regression model y_i = α + β x_i + ε_i with E[ε_i | x_i] = 0, and three moment conditions used for GMM:
- E[y_i − α − β x_i] = 0
- E[(y_i − α − β x_i) x_i] = 0
- E[(y_i − α − β x_i) x_i^2] = 0
Γ0 is the matrix of expected derivatives (the Jacobian) of the moment functions g_i(θ) with respect to the parameter vector θ = (α, β), evaluated at the true θ. In other words, Γ0 = E[ ∂g_i(θ)/∂θ ] where g_i(θ) contains the three moments above.

Step-by-step derivation of ∂g_i/∂θ for each moment:
- For g1(θ) = y_i − α − β x_i:
  ∂g1/∂α = −1,  ∂g1/∂β = −x_i.
- For g2(θ) = (y_i − α − β x_i) x_i:
  ∂g2/∂α = −x_i,  ∂g2/∂β = −x_i^2.
- For g3(θ) = (y_i − α − β x_i) x_i^2:
  ∂g3/∂α = −x_i^2,  ∂g3/∂β = −x_i^3.
Stacking these derivatives gives a 3×2 matrix of partials for each i:
[ [−1, −x_i],
  [−x_i, −x_i^2],
  [−x_i^2, −x_i^3] ].
Taking expectations across i yields Γ0 = E[ this matrix ] =
[ [−1, −E[x_i]],
  [−E[x_i], −E[x_i^2]],
  [−E[x_i^2], −E[x_i^3]] ].

Now evaluate each answer option:
- Option 1: There is not enough information to compute the matrix Γ0. This is inaccurate because Γ0 is determined by the model structure and the moments; while we may not know the numerical moments E[x_i], E[x_i^2], E[x_i^3], the form of Γ0 in terms of expectations of x_i is well-defined. The statement ignores the explicit derivative structure and the moments we can write down from the assumptions.
- Option 2: The matrix Γ0 is: Γ0 = E[ [1, x_i], [x_i, x_i^2] ]. This is incorrect in several ways: (a) signs are wrong relative to the derivatives (the first row should have −1 and −x_i, not 1 and x_i); (b) the arrangement of entries does not correspond to the Jacobian blocks derived above; (c) the expected cross-terms are not matching the third-order moment structure (x_i^2, x_i^3) that appear in the third row.
- Option 3: The matrix Γ0 is: Γ0 = E[ [−1, −x_i], [−x_i, −x_i^2], [−x_i^2, −x_i^3] ]. This mirrors the calculated Jacobian blocks for each moment and takes expectations correctly: the first row has (−1, −x_i), the second row (−x_i, −x_i^2), and the third row (−x_i^2, −x_i^3). This aligns with the derivatives we listed and their expectations, so as an expression it matches the proper Γ0 structure.
- Option 4: The matrix Γ0 is: Γ0 = E[ [−1, −x_i], [−x_i, −x_i^2] ]. This option only includes two rows and omits the third moment condition entirely; it does not represent the full 3×2 Jacobian matrix needed for the three moment conditions, making it incomplete and incorrect.
- Option 5: The matrix Γ0 is: Γ0 = E[ [−1, −x_i], [−x_i, −x_i^2], [−x_i, −x_i^2], [−x_i^2, −x_i^3] ]. This option appears to mix blocks in a way that does not correspond to the proper 3×2 Jacobian structure (it seems to duplicate entries and arrange them improperly), so it does not match the correct Γ0 form.

In summary, Option 3 is the correct representation of Γ0 for the three moment conditions, because it faithfully captures the Jacobian structure of all three moments with the appropriate signs and the correct cross-moments, and then takes expectations of those derivatives. The other options either misstate signs, misorder entries, omit a moment condition, or propose an incomplete/inaccurate matrix form.

BU.232.630.W1.SP25 Quiz 2 solutions

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类似问题

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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