Consider the following nonlinear regression model:
𝑦
𝑡
=
𝛼
𝑥
𝑡
𝛽
+
𝜀
𝑡
Assume i.i.d. data and
𝔼
[
𝜀
𝑡
|
𝑥
𝑡
]
=
0
. To estimate
𝛼
and
𝛽
by GMM, we chose among the following moment conditions:
𝔼
[
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
]
=
0

𝔼
[
(
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
)
𝑥
𝑡
]
=
0

𝔼
[
(
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
)
1
𝑥
𝑡
]
=
0
Choose the most appropriate answer below:

Question

Consider the following nonlinear regression model:
𝑦
𝑡
=
𝛼
𝑥
𝑡
𝛽
+
𝜀
𝑡
Assume i.i.d. data and 
𝔼
[
𝜀
𝑡
|
𝑥
𝑡
]
=
0
. To estimate 
𝛼
 and 
𝛽
 by GMM, we chose among the following moment conditions:
𝔼
[
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
]
=
0

𝔼
[
(
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
)
𝑥
𝑡
]
=
0

𝔼
[
(
𝑦
𝑡
−
𝛼
𝑥
𝑡
𝛽
)
1
𝑥
𝑡
]
=
0
Choose the most appropriate answer below:

BlackTom AI · Accepted Answer

To tackle the question, I’ll lay out the context and then inspect each proposed moment condition in turn.

Option 1: "Only the first and third equations are valid moment conditions to estimate α and β by GMM."
- The first moment condition is E[y_t − α x_t β] = 0. This corresponds to setting the mean of the error term e_t = y_t − α x_t β to zero, which is a standard starting point for GMM if e_t is exogenous with respect to the instruments used.
- The third moment condition is E[(y_t − α x_t β) · (1/x_t)] = 0. Since 1/x_t is a function of x_t and if E[e_t | x_t] = 0, then E[e_t · (1/x_t)] = 0 as long as x_t ≠ 0 with non-negligible probability. This is a valid moment condition as it uses an instrument (1/x_t) that is uncorrelated with the error term under the exogeneity assumption.
- However, the claim that the second condition is not valid contradicts the logic that any function of the exogenous regressor used in the model can serve as a valid instrument, provided the instrument is uncorrelated with the error and has finite variance. Consequently, ruling out the second moment condition without a specific counterexample or violation of exogeneity weakens this option.

Option 2: "Only the second and third equations are valid moment conditions to estimate α and β by GMM."
- The second moment condition is E[(y_t − α x_t β) x_t] = 0. This uses x_t as an instrument. If E[e_t | x_t] = 0, then E[e_t x_t] = 0, so this is a valid instrumented moment. However, this claim excludes the first and would imply they are invalid, which clashes with the standard exogeneity of the error term.
- The third condition, E[(y_t − α x_t β) (1/x_t)] = 0, is similarly valid under E[e_t | x_t] = 0 and x_t ≠ 0, as discussed above. Nonetheless, excluding the first condition is not warranted unless there is a specific issue with E[e_t] ≠ 0 or with the intercept-like term. Thus, this option is overly restrictive.

Option 3: "All equations are valid moment conditions to estimate α and β by GMM."
- If the true data-generating process satisfies E[e_t | x_t] = 0, then any instrument that is a measurable function of x_t (and not perfectly correlated with e_t) can be used to form valid moment conditions. This includes 1, x_t, and 1/x_t as long as x_t ≠ 0 with positive probability and the moments have finite variance. Under standard exogeneity assumptions for e_t, all three proposed moments are legitimately valid.
- Practically, this means we can use the stack of moments E[e_t], E[e_t x_t], and E[e_t (1/x_t)] simultaneously to identify α and β, provided the instruments satisfy the usual relevance and exogeneity requirements and no division by zero occurs.

Option 4: "Only the first and second equations are valid moment conditions to estimate α and β by GMM."
- This scenario would exclude the third condition that uses 1/x_t as an instrument. While E[e_t] = 0 and E[e_t x_t] = 0 are standard, dropping E[e_t (1/x_t)] = 0 would unnecessarily discard a valid instrument under E[e_t | x_t] = 0 and x_t ≠ 0. The remaining two do not inherently threaten identifiability, but eliminating a valid instrument reduces efficiency and is not warranted purely from exogeneity premises.

Option 5: "None of the equations above are valid moment conditions to estimate α and β by GMM."
- Given the usual exogeneity condition E[ε_t | x_t] = 0 and the structure of e_t = y_t − α x_t β, taking moments with respect to functions of x_t such as 1, x_t, and 1/x_t tends to be valid as long as x_t is not degenerately distributed (e.g., x_t = 0 almost surely would break the 1/x_t moment). Therefore, there is a reasonable case for at least some of these moment conditions to be valid under standard assumptions. Declaring none of them valid would require a violation of the core exogeneity assumption or a problematic distribution of x_t.

Overall, each option hinges on whether the stated set of moments aligns with the exogeneity assumption E[e_t | x_t] = 0 and with the mathematical caveats (like x_t not being zero with probability one). The common thread is that, under standard exogeneity and regularity conditions, multiple moment conditions based on functions of x_t are legitimate, and excluding valid ones would be unnecessary. The collection of moments that uses 1, x_t, and (1/x_t) collectively leverages the information in x_t in different scales to identify α and β more efficiently, assuming the data meet the regularity conditions.}

BU.232.630.W1.SP25 Quiz 2 solutions

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Similar Questions

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