....¹

An interesting side point can be studied by extending the Taylor series approximation to a third term, $\frac{1}{2}(b-\beta)'g''(\beta)(b-\beta)$, and calculating $E(\hat Y^p)$. The expected value of the second term is zero, and the entire approximation now becomes $E(\hat Y^p)\approx g(\beta) + \frac{1}{2}A'V(b)A$, where $A'A=g''(\beta)$, the matrix of second derivatives. $A$ can be calculated by taking the Cholesky decomposition of the second derivative matrix (in Gauss, for example: A=CHOL(D), where D is the second derivative.) This calculation shows the details of the familiar result that $g(E(T))\not=E(g(T))$, for nonlinear functional forms $g(\cdot)$. However, since the term $\frac{1}{2}A'V(b)A$ goes to zero as the sample size increases, $g(\beta)$ is a reasonable approximation to $E(g(b))\equiv E(\hat Y^p)$ in large samples.

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... element.²

In Gauss, one would use X.*exp(X*b).

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