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

Question

BlackTom AI · Accepted Answer

Let's unpack what Φ0 represents in the GMM setup for the given nonlinear regression model. The moment conditions are g_i(θ) = [ y_i − α − x_i^β ; (y_i − α − x_i^β) x_i ], evaluated at the true parameter vector θ = (α, β). The matrix Φ0 is the population covariance matrix of these moment conditions: Φ0 = E[g_i g_i^T]. This means each element of Φ0 comes from taking expectations of products of the components of g_i.

- The (1,1) entry is E[(y_i − α − x_i^β)^2], which is the variance of the first moment when the true parameters are used. This corresponds to how dispersed the residual is around zero in the first equation.
- The (1,2) and (2,1) entries are E[(y_i − α − x_i^β) · (y_i − α − x_i^β) x_i] = E[(y_i − α − x_i^β)^2 x_i], which captures how the first residual interacts with x_i in the second moment.
- The (2,2) entry is E[( (y_i − α − x_i^β) x_i )^2] = E[(y_i − α − x_i^β)^2 x_i^2], which reflects the variance of the second moment in the presence of x_i.

Therefore, Φ0 should be the 2×2 matrix with entries [ E[(ei)^2], E[(ei)^2 xi]; E[(ei)^2 xi], E[(ei)^2 xi^2] ], where ei = yi − α − x_i^β. The proposed expression matches this structure exactly, listing the corresponding expectations in each position of the matrix.

Why this form is appropriate: since the moment conditions are g_i = [ei; ei xi], the outer product g_i g_i^T yields a matrix with elements ei^2, ei^2 xi, and ei^2 xi^2, and taking expectations at the true θ gives Φ0 as shown. This aligns with standard GMM theory for a two-moment, two-parameter setup under i.i.d. data and E[ε_i | x_i] = 0, which supports a well-defined Φ0 used in the asymptotic covariance of the GMM estimator.

Potential pitfalls to consider (to avoid misinterpretation): if one mistakenly uses E[e_i^2 x_i^2] in a different position or replaces e_i with the residuals evaluated at incorrect parameter guesses, the matrix would be mis-specified. Also, note that the expression assumes evaluation at the true α and β; in practice, one uses the estimated parameters to form the sample analog of Φ0. Finally, while E[ε_i | x_i] = 0 implies exogeneity and allows E[e_i] = 0 at the true parameters, it does not by itself impose homoskedasticity; the (1,1) entry E[(ei)^2] can depend on x_i if the error variance is heteroskedastic, but Φ0 as defined accommodates that through the expectation of e_i^2 without assuming constant variance across observations.

BU.232.630.W4.SP25 sample_quiz_2

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