Consider the following nonlinear 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 need at least two moment conditions, and we use
𝔼
[
𝑦
𝑖
−
𝛼
−
𝛽
𝑥
𝑖
]
=
0
𝔼
[
(
𝑦
𝑖
−
𝛼
−
𝛽
𝑥
𝑖
)
𝑥
𝑖
𝛽
𝑥
𝑖
−
1
]
=
0
Chose the correct answer below.

Question

Consider the following nonlinear 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 need at least two moment conditions, and we use
𝔼
[
𝑦
𝑖
−
𝛼
−
𝛽
𝑥
𝑖
]
=
0
𝔼
[
(
𝑦
𝑖
−
𝛼
−
𝛽
𝑥
𝑖
)
𝑥
𝑖
𝛽
𝑥
𝑖
−
1
]
=
0
Chose the correct answer below.

BlackTom AI · Accepted Answer

To approach this problem, I’ll lay out the question and the answer choices, then examine each option in turn so you can see why it makes sense or not.
Option 1: 'The following equation is also a valid moment condition for estimation of this model and can be added as a third equation, E[(y_i − α − β x_i) x_i] = 0'
This is indeed a valid moment condition. Under the assumption E[ε_i | x_i] = 0, we have E[ε_i g(x_i)] = 0 for any function g for which the expectation exists. Since ε_i = y_i − α − β x_i, choosing g(x_i) = x_i yields E[(y_i − α − β x_i) x_i] = 0 as another moment condition. So this option correctly states a legitimate additional moment.
Option 2: 'The following equation is also a valid moment condition for estimation of this model and can be added as a third equation, E[(y_i − α − β x_i) log(x_i)] = 0'
This is also valid provided the log transform is defined and x_i is positive (so log(x_i) is finite). Given E[ε_i | x_i] = 0, the orthogonality E[ε_i h(x_i)] = 0 holds for any measurable h with finite expectation, so E[(y_i − α − β x_i) log(x_i)] = 0 is a legitimate moment condition when log x_i is well-defined. The essential caveat is the domain of x_i; if x_i can be nonpositive, log x_i is not defined and this moment would be invalid in that sample. Within the standard positive x_i setting, this is a valid moment.
Option 3: 'All of the answers are correct.'
This option asserts that both of the previous moment conditions are valid (and any other implied ones) and thus that all statements presented are correct. Since option 1 and option 2 can both be valid moment conditions under appropriate assumptions (positivity of x for the log transform, and existence of moments for the x-weighted term), this meta-statement could be considered correct if you accept those conditions as satisfied.
Option 4: 'We can substitute the second equation above with the following moment condition E[(y_i − α − β x_i) x_i] = 0'
This option essentially repeats the first moment condition and suggests a substitution of the second one with the same form. It is not introducing a fundamentally different moment; it is just restating the same orthogonality condition. If interpreted as removing the log-based moment (option 2) and keeping the x-based moment, it remains a valid moment, but it does not conflict with the idea that you can have multiple valid moments. However, as a standalone statement, it’s true only insofar as you accept that you can replace the second moment with an equivalent one that uses x_i; it does not add a new, independent moment beyond what option 1 already provides.

Putting the reasoning together, we see:
- Option 1 is a valid moment condition for this model under E[ε|x]=0.
- Option 2 is valid if log(x_i) is well-defined (i.e., x_i > 0) and E[ε_i log(x_i)] is finite.
- Option 3 claims all answers are correct, which is consistent if you accept the conditions in options 1 and 2 (including the domain assumption for x_i in option 2).
- Option 4 is a restatement of option 1, not a new moment, so it remains valid as a substitution but does not introduce a distinct additional moment beyond what option 1 provides.

In summary, each option hinges on the same core principle: with E[ε_i | x_i] = 0, orthogonality to functions of x_i holds. The first and fourth options reflect the same moment condition using x_i; the second option reflects a moment with log(x_i) which is valid only if x_i is strictly positive. The third option asserts that all statements are correct, which aligns with the validity of the first two under the stated domain assumptions. The exact interpretation depends on the domain of x_i and the finiteness of the expectations involved, but conceptually all the presented moment conditions are compatible with the standard GMM identification when E[ε_i|x_i]=0 is assumed.

BU.232.630.W1.SP25 Quiz 2 solutions

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