## 经济代写|计量经济学代写Econometrics代考|Nonlinear Estimation Using the GNR

In this section, we discuss how the Gauss-Newton regression can be used as part of an effective algorithm for minimizing sum-of-squares functions. This was actually the original motivation for the GNR. The term “Gauss-Newton” is in fact taken from the literature on numerical optimization as applied to nonlinear least squares problems, and most of the other uses of this artificial regression in econometrics are relatively recent, as we discuss in Section 6.9.
Most effective algorithms that attempt to maximize or minimize a smooth function of two or more variables, say $Q(\boldsymbol{\theta})$, operate in basically the same way. Such an algorithm goes through a series of major iterations, at each of which it starts with a particular value of $\boldsymbol{\theta}$, say $\boldsymbol{\theta}^{(j)}$, and tries to find a better one. The algorithm first chooses a direction in which to search for a better value of $\boldsymbol{\theta}$ and then decides how far to move in that direction. The main differences among unconstrained optimization algorithms are in the way in which the direction to search is chosen and in the way in which the size of the ultimate step in that direction is determined. Numerous choices are available.

Note that any algorithm for minimization can just as easily be used for maximization, since minimizing $Q(\boldsymbol{\theta})$ is equivalent to maximizing $-Q(\boldsymbol{\theta})$. Following the convention used in most of the literature, we will deal with the case of minimization, which is what we wish to do with sum-of-squares functions anyway. ${ }^2$ In this section, we will attempt to give an overview of how numerical minimization algorithms work and how the Gauss-Newton regression may be used as part of them, but we will not discuss many of the important computer-related issues that substantially affect the performance of computer algorithms. An excellent reference on the art and science of numerical optimization is Gill, Murray, and Wright (1981); see also Bard (1974), Quandt (1983), Press, Flannery, Teukolsky, and Vetterling (1986, Chapter 10), and Seber and Wild (1989, Chapter 14).

As noted above, the Gauss-Newton regression has been used for many years as the key part of the Gauss-Newton method, which is actually several related algorithms for nonlinear least squares estimation. Bard (1974) discusses many of these. Newton’s method, as its name implies, is very old, and the idea of approximating the Hessian by a matrix that depends only on first derivatives dates back to Gauss (1809). However, because nonlinear estimation was generally not practical until digital computers became widely available, most work in this area has been relatively recent. Important papers in the post-computer development of the Gauss-Newton method include Hartley (1961) and Marquardt (1963). The survey article by Quandt (1983) provides numerous other references, as does Seber and Wild (1989, Chapter 14).

In contrast to its long history in estimation, the use of the GNR for specification testing is quite recent. The first paper in the econometric literature appears to be Durbin (1970), which proposed what amounts to a special case of the GNR as a way of testing linear regression models for serial correlation when there are lagged dependent variables. This procedure was treated in a rather cursory fashion, however, since it was in the same paper that Durbin proposed his well-known $h$ test. What came to be known as Durbin’s “alternative procedure,” which is really a special case of the GNR, was for some years largely ignored by theoretical econometricians and entirely ignored by practitioners. All this will be discussed further in Chapter $10 .$

Interest in the Gauss-Newton regression as a way of generating test statistics dates principally from the late $1970 \mathrm{~s}$. Godfrey $(1978 \mathrm{a}, 1978 \mathrm{~b})$ and Breusch (1978) greatly generalized Durbin’s alternative procedure and showed how to calculate LM tests for serial correlation using the GNR. Numerous other authors dealt with other special cases, contributed to the increased understanding of the general case we have discussed in this chapter, and developed related tests. Notable articles include Breusch and Pagan (1980) and Engle (1982a). Much of this literature explicitly assumes normal errors and develops the tests as LM tests within the framework of maximum likelihood estimation. This may be slightly misleading because, as we have seen, no assumption of normality is in fact needed for either nonlinear least squares estimation or tests based on the GNR to be asymptotically valid. More recent papers, such as Pagan (1984a), Davidson and MacKinnon (1985a), and MacKinnon (1992), have focused on the case of regression models and have tried to unify and clarify the previous literature. We will be seeing a great deal of the GaussNewton regression, and also of related artificial regressions that have similar properties, throughout the remainder of the book.

## 经济代写|计量经济学代写Econometrics代考|Nonlinear Estimation Using the GNR

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