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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).

经济代写|计量经济学代写Econometrics代考|Further Reading

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|Best22

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

在本节中,我们将讨论如何将高斯-牛顿回归用作最小化平方和函数的有效算法的一部分。这实际上是 GNR 的最初动机。“Gauss-Newton”这个术语实际上取自关于应用于非线性最小二乘问题的数值优化的文献,并且这种人工回归在计量经济学中的大多数其他用途都是相对较新的,正如我们在第 6.9 节中讨论的那样。
试图最大化或最小化两个或多个变量的平滑函数的最有效算法,例如问(一世), 操作方式基本相同。这样的算法经历了一系列主要的迭代,在每一次迭代中,它都以一个特定的值开始一世, 说一世(j),并试图找到一个更好的。该算法首先选择一个方向来搜索更好的值一世然后决定朝那个方向移动多远。无约束优化算法之间的主要区别在于选择搜索方向的方式以及确定该方向上最终步长大小的方式。有多种选择。

请注意,任何最小化算法都可以很容易地用于最大化,因为最小化问(一世)相当于最大化−问(一世). 按照大多数文献中使用的约定,我们将处理最小化的情况,这就是我们希望对平方和函数所做的事情。2在本节中,我们将尝试概述数值最小化算法的工作原理以及高斯-牛顿回归如何用作其中的一部分,但我们不会讨论许多与计算机相关的重要问题,这些问题会显着影响性能的计算机算法。Gill、Murray 和 Wright (1981) 是关于数值优化艺术和科学的优秀参考资料;另见 Bard (1974)、Quandt (1983)、Press、Flannery、Teukolsky 和 ​​Vetterling (1986,第 10 章),以及 Seber 和 Wild (1989,第 14 章)。

经济代写|计量经济学代写Econometrics代考|Further Reading

如上所述,高斯-牛顿回归多年来一直作为高斯-牛顿法的关键部分,实际上是非线性最小二乘估计的几种相关算法。Bard (1974) 讨论了其中的许多问题。顾名思义,牛顿的方法非常古老,通过仅依赖一阶导数的矩阵来逼近 Hessian 矩阵的想法可以追溯到 Gauss (1809)。然而,由于非线性估计在数字计算机广泛使用之前通常不实用,因此该领域的大多数工作都是相对较新的。Gauss-Newton 方法的后计算机发展的重要论文包括 Hartley (1961) 和 Marquardt (1963)。Quandt(1983)的调查文章提供了许多其他参考资料,Seber 和 Wild(1989,第 14 章)也是如此。

与其在估计方面的悠久历史相比,将 GNR 用于规范测试是最近才出现的。计量经济学文献中的第一篇论文似乎是 Durbin (1970),它提出了相当于 GNR 的一个特例,作为在存在滞后因变量时测试线性回归模型的序列相关性的一种方法。然而,这个过程以相当粗略的方式处理,因为德宾在同一篇论文中提出了他的著名H测试。后来被称为德宾的“替代程序”,它实际上是 GNR 的一个特例,几年来在很大程度上被理论计量经济学家忽略了,也被实践者完全忽略了。所有这些将在本章中进一步讨论10.

高斯-牛顿回归作为一种生成检验统计量的方式的兴趣主要是从晚期1970 s. 戈弗雷(1978一个,1978 b)和 Breusch (1978) 极大地推广了 Durbin 的替代程序,并展示了如何使用 GNR 计算序列相关的 LM 检验。许多其他作者处理了其他特殊情况,有助于加深对我们在本章中讨论的一般情况的理解,并开发了相关测试。著名的文章包括 Breusch 和 Pagan (1980) 和 Engle (1982a)。这些文献中的大部分都明确地假设了正常误差,并在最大似然估计的框架内将测试开发为 LM 测试。这可能会有点误导,因为正如我们所看到的,对于非线性最小二乘估计或基于 GNR 的测试渐近有效,实际上不需要正态性假设。最近的论文,例如 Pagan (1984a)、Davidson and MacKinnon (1985a) 和 MacKinnon (1992),专注于回归模型的案例,并试图统一和澄清以前的文献。在本书的其余部分,我们将看到大量的高斯牛顿回归,以及具有相似性质的相关人工回归。

经济代写|计量经济学代写Econometrics代考

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