# 经济代写|计量经济学代写Econometrics代考|Best27

#### Doug I. Jones

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|计量经济学代写Econometrics代考|The Gauss-Newton Regression

Associated with the method of GNLS is a version of the Gauss-Newton regression which may be used in all the ways that the original Gauss-Newton regression can be used (see Chapter 6). This GNR is
$$\boldsymbol{\eta}(\boldsymbol{y}-\boldsymbol{x}(\boldsymbol{\beta}))=\boldsymbol{\eta} \boldsymbol{X}(\boldsymbol{\beta}) \boldsymbol{b}+\text { residuals, }$$
where $\boldsymbol{b}$ is a $k$-vector of coefficients to be estimated and $\boldsymbol{\eta}$ is any $n \times n$ matrix that satisfies equation (9.08). It is not coincidental that regression (9.14) resembles regression (9.09), which was used to compute GLS estimates in the linear case. The GNR is in fact a linearization of the original nonlinear model,

with both regressand and regressors transformed so as to make the covariance matrix of the error terms proportional to an identity matrix.

If we evaluate both $\boldsymbol{x}(\boldsymbol{\beta})$ and $\boldsymbol{X}(\boldsymbol{\beta})$ at $\tilde{\boldsymbol{\beta}}$, running regression (9.14) yields $\tilde{b}=\mathbf{0}$ and the estimated covariance matrix
$$\frac{(\boldsymbol{y}-\tilde{\boldsymbol{x}})^{\top} \boldsymbol{\eta}^{\top} \boldsymbol{\eta}(\boldsymbol{y}-\tilde{\boldsymbol{x}})}{n-k}\left(\tilde{\boldsymbol{X}}^{\top} \boldsymbol{\eta}^{\top} \boldsymbol{\eta} \tilde{\boldsymbol{X}}\right)^{-1}=\frac{\operatorname{SSR}(\tilde{\boldsymbol{\beta}} \mid \boldsymbol{\Omega})}{n-k}\left(\tilde{\boldsymbol{X}}^{\top} \boldsymbol{\Omega}^{-1} \tilde{\boldsymbol{X}}\right)^{-1} .$$
The first factor on the right-hand side of (9.15) is just the OLS estimate of the variance of the GNR; as we explain in a moment, it should tend to 1 as $n \rightarrow \infty$ if the covariance matrix of $\boldsymbol{u}$ is actually $\Omega$. This first factor would normally be omitted in practice. ${ }^2$ Comparing the second factor on the right-hand side of (9.15) with the covariance matrix that appears in (9.06), it is evident that the former provides a sensible estimate of the covariance matrix of $\tilde{\boldsymbol{\beta}}$.

In the preceding discussion, we asserted that $(n-k)^{-1} \operatorname{SSR}(\tilde{\boldsymbol{\beta}} \mid \Omega)$ should tend to 1 as $n \rightarrow \infty$. In doing so we implicitly made use of the result that
$$\operatorname{plim}_{n \rightarrow \infty}\left(\frac{1}{n} \tilde{\boldsymbol{u}}^{\top} \Omega^{-1} \tilde{\boldsymbol{u}}\right)=1 .$$
This result requires justification. First of all, we must assume that the eigenvalues of $\Omega$, which are all strictly positive since $\Omega$ is assumed to be positive definite, are bounded from above and below as $n \rightarrow \infty$. These assumptions imply that the eigenvalues of $\boldsymbol{\eta}$ have the same properties. Next, we use the result that
$$\tilde{\boldsymbol{u}}=\boldsymbol{M}_0^{\Omega} \boldsymbol{u}+o\left(n^{-1 / 2}\right) .$$
Here $\boldsymbol{M}_0^{\Omega}$ is an oblique projection matrix essentially the same as (9.13) but depending on the matrix of derivatives $\boldsymbol{X}_0 \equiv \boldsymbol{X}\left(\boldsymbol{\beta}_0\right)$ rather than on a regressor matrix $\boldsymbol{X}$. The result (9.17) is clearly the GNLS analog of the result (5.57) for ordinary NLS, and we will therefore not bother to derive it.

## 经济代写|计量经济学代写Econometrics代考|Feasible Generalized Least Squares

In practice, the covariance matrix $\Omega$ is rarely known, but it is often assumed to depend in a particular way on a vector of unknown parameters $\boldsymbol{\alpha}$. In such a case, there are two ways to proceed. One is to obtain a consistent estimate of $\boldsymbol{\alpha}$, say $\check{\boldsymbol{\alpha}}$, by some auxiliary procedure. This then yields an estimate of $\boldsymbol{\Omega}$, $\Omega(\check{\alpha})$, that is used in place of the true covariance matrix $\Omega_0 \equiv \Omega\left(\alpha_0\right)$ in what is otherwise a standard GLS procedure. This approach, which will be the topic of this section, is called feasible GLS because it is feasible in many cases when ordinary GLS is not. The other approach is to use maximum likelihood to estimate $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ jointly, generally under the assumption of normality; it will be discussed in Section 9.6. ${ }^3$

Under reasonable conditions, feasible GLS yields estimates that are not only consistent but also asymptotically equivalent to genuine GLS estimates, and therefore share their efficiency properties. However, even when this is the case, the performance in finite samples of feasible GLS may be much inferior to that of genuine GLS if $\check{\alpha}$ is a poor estimator of $\boldsymbol{\alpha}$.

In most cases, the estimates of $\boldsymbol{\alpha}$ that are used by feasible GLS are based on OLS or NLS residuals, of which a typical one is $\hat{u}_t \equiv y_t-x_t(\hat{\boldsymbol{\beta}})$. It is possible to use these residuals for the purposes of estimating $\alpha$ because, in many circumstances, they consistently estimate the error terms $u_t$, despite being based on an estimation procedure that uses the wrong covariance matrix. It is obvious that if the OLS or NLS estimates $\hat{\boldsymbol{\beta}}$ consistently estimate $\boldsymbol{\beta}$, the residuals will consistently estimate the error terms. What is not so obvious (and is not always true) is that $\hat{\boldsymbol{\beta}}$ will consistently estimate $\boldsymbol{\beta}$.

A rigorous treatment of the conditions under which NLS estimates are consistent when the error terms $u_t$ do not satisfy the i.i.d. assumption is beyond the scope of this book. See Gallant (1987) for such a treatment. However, it is worth seeing how the consistency proof of Section $5.3$ would be affected if we relaxed that assumption. Recall that the consistency of $\hat{\boldsymbol{\beta}}$ depends entirely on the properties of $n^{-1}$ times the sum-of-squares function:
$$s s r(\boldsymbol{y}, \boldsymbol{\beta}) \equiv \frac{1}{n} \sum_{t=1}^n\left(y_t-x_t(\boldsymbol{\beta})\right)^2=\frac{1}{n} \sum_{t=1}^n\left(x_t\left(\boldsymbol{\beta}_0\right)-x_t(\boldsymbol{\beta})+u_t\right)^2$$

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|高斯-牛顿回归

$$\boldsymbol{\eta}(\boldsymbol{y}-\boldsymbol{x}(\boldsymbol{\beta}))=\boldsymbol{\eta} \boldsymbol{X}(\boldsymbol{\beta}) \boldsymbol{b}+\text { residuals, }$$
，其中$\boldsymbol{b}$是待估计系数的$k$ -向量，$\boldsymbol{\eta}$是满足方程(9.08)的任意$n \times n$矩阵。回归(9.14)与回归(9.09)相似，这并非巧合，后者被用于计算线性情况下的GLS估计值。GNR实际上是对原始非线性模型的线性化，

，同时对回归量和回归量进行变换，使误差项的协方差矩阵与单位矩阵成正比 如果我们在$\tilde{\boldsymbol{\beta}}$对$\boldsymbol{x}(\boldsymbol{\beta})$和$\boldsymbol{X}(\boldsymbol{\beta})$进行评估，运行回归(9.14)得到$\tilde{b}=\mathbf{0}$和估计的协方差矩阵
$$\frac{(\boldsymbol{y}-\tilde{\boldsymbol{x}})^{\top} \boldsymbol{\eta}^{\top} \boldsymbol{\eta}(\boldsymbol{y}-\tilde{\boldsymbol{x}})}{n-k}\left(\tilde{\boldsymbol{X}}^{\top} \boldsymbol{\eta}^{\top} \boldsymbol{\eta} \tilde{\boldsymbol{X}}\right)^{-1}=\frac{\operatorname{SSR}(\tilde{\boldsymbol{\beta}} \mid \boldsymbol{\Omega})}{n-k}\left(\tilde{\boldsymbol{X}}^{\top} \boldsymbol{\Omega}^{-1} \tilde{\boldsymbol{X}}\right)^{-1} .$$
(9.15)右边的第一个因子只是GNR方差的OLS估计;我们稍后会解释，如果$\boldsymbol{u}$的协方差矩阵实际上是$\Omega$，那么它应该趋向于1作为$n \rightarrow \infty$。这第一个因素在实践中通常会被忽略。${ }^2$将(9.15)右边的第二个因子与(9.06)中出现的协方差矩阵进行比较，很明显前者提供了$\tilde{\boldsymbol{\beta}}$的协方差矩阵的合理估计。

$$\operatorname{plim}_{n \rightarrow \infty}\left(\frac{1}{n} \tilde{\boldsymbol{u}}^{\top} \Omega^{-1} \tilde{\boldsymbol{u}}\right)=1 .$$

$$\tilde{\boldsymbol{u}}=\boldsymbol{M}_0^{\Omega} \boldsymbol{u}+o\left(n^{-1 / 2}\right) .$$

## 经济代写|计量经济学代写Econometrics代考|可行广义最小二乘

$$s s r(\boldsymbol{y}, \boldsymbol{\beta}) \equiv \frac{1}{n} \sum_{t=1}^n\left(y_t-x_t(\boldsymbol{\beta})\right)^2=\frac{1}{n} \sum_{t=1}^n\left(x_t\left(\boldsymbol{\beta}_0\right)-x_t(\boldsymbol{\beta})+u_t\right)^2$$

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