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

We begin by identifying the three moment conditions used in the GMM estimation:
- g1(θ) = y_i − α − β x_i − γ x_i^2
- g2(θ) = (y_i − α − β x_i − γ x_i^2) x_i
- g3(θ) = (y_i − α − β x_i − γ x_i^2) x_i^2
Here θ = (α, β, γ).
To compute the variance of the GMM estimator, we need Γ0, which is the expected Jacobian matrix of the moment functions with respect to θ, i.e., Γ0 = E[ ∂g(θ)/∂θ' ], where g(θ) stacks g1, g2, g3.
Now we differentiate each moment with respect to α, β, γ:
- For g1: ∂g1/∂α = −1, ∂g1/∂β = −x_i, ∂g1/∂γ = −x_i^2.
- For g2: ∂g2/∂α = −x_i, ∂g2/∂β = −x_i^2, ∂g2/∂γ = −x_i^3.
- For g3: ∂g3/∂α = −x_i^2, ∂g3/∂β = −x_i^3, ∂g3/∂γ = −x_i^4.
Thus the analytical form of Γ0 is the expectation of the 3×3 matrix whose (row, column) entry is the corresponding derivative, i.e.,
Γ0 = E[ [ −1, −x_i, −x_i^2 ]; [ −x_i, −x_i^2, −x_i^3 ]; [ −x_i^2, −x_i^3, −x_i^4 ] ].

Option by option:
Option 1: The proposed matrix has a first row [1, x_i, x_i^2] and appears to mix signs and include positive entries for derivatives that should be negative. Since each derivative computed above has a negative sign, this option is inconsistent with the correct gradient signs, so this is incorrect.

Option 2: This candidate shows a matrix with all entries negative, but the specific pattern of entries does not align with the derivative structure above. In particular, the (1,1) entry should be −1, but the rest of the arrangement and higher-moment terms (x_i^2, x_i^3, x_i^4) do not match the full Jacobian as derived. This mismatch makes it incorrect.

Option 3: Here the entries again display an inconsistent arrangement of negatives and powers of x that does not line up with the full set of derivatives. The (2,2) and (3,3) entries would need to be −x_i^2 and −x_i^4 respectively, but the presented pattern does not consistently reflect that across all rows, so this is not correct.

Option 4: This option presents
Γ0 = E[ [ −1, −x_i, −x_i^2 ]; [ −x_i, −x_i^2, −x_i^3 ]; [ −x_i^2, −x_i^3, −x_i^4 ] ].
This matches exactly the derived derivatives for ∂g1/∂θ, ∂g2/∂θ, and ∂g3/∂θ, arranged in the 3×3 Jacobian form and then averaged via the expectation. The signs are all negative as expected, and the pattern of x_i, x_i^2, x_i^3, and x_i^4 aligns with the cross-terms in the derivatives.

Option 5 states there is not enough information to compute the matrix. In this context, the functional form of the moment conditions provides the exact derivatives needed to assemble Γ0, so this statement is incorrect.

In summary, the only option whose structure and sign pattern exactly reflect the computed derivatives of all three moment conditions with respect to (α, β, γ) is Option 4.

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Similar Questions

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