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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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 Asymptotic Distribution of Test Statistics

We are now ready to find the asymptotic distribution of the test statistic (12.04) under the family of drifting DGPs (12.06). In order for our asymptotic analysis to be valid, we must assume that various regularity conditions hold. Thus we will assume that $n^{-1} \boldsymbol{X}_0^{\top} \boldsymbol{X}_0, n^{-1} \boldsymbol{Z}_0^{\top} \boldsymbol{Z}_0$, and $n^{-1} \boldsymbol{Z}_0^{\top} \boldsymbol{X}_0$ all tend to finite limiting matrices with ranks $k, r$, and $\min (k, r)$, respectively, as $n \rightarrow \infty$. We will further assume that there exists an $N$ such that, for all $n>N$, the rank of the matrix $\left[\begin{array}{ll}\boldsymbol{X}_0 & \boldsymbol{Z}_0\end{array}\right]$ is $k+r$, that $n^{-1} \boldsymbol{a}^{\top} \boldsymbol{a}$ tends to a finite limiting scalar, and that $n^{-1} \boldsymbol{a}^{\top} \boldsymbol{X}_0$ and $n^{-1} \boldsymbol{a}^{\top} \boldsymbol{Z}_0$ both tend to finite limiting vectors of dimensions $1 \times k$ and $1 \times r$, respectively. Here $\boldsymbol{X}_0$ denotes $\boldsymbol{X}\left(\boldsymbol{\beta}_0\right)$ and $\boldsymbol{Z}_0$ denotes $\boldsymbol{Z}\left(\boldsymbol{\beta}_0\right)$. Whether these regularity conditions will in fact hold depends on the vector $\boldsymbol{a}$, the null hypothesis (12.01), the alternative hypothesis (whether or not it is explicit), and the simple null (12.07).

We begin by rewriting the test statistic (12.04) so that it is a product of four factors, each of which is $O(1)$ :
$$\frac{1}{\tilde{s}^2}\left(n^{-1 / 2}(\boldsymbol{y}-\overline{\boldsymbol{x}})^{\top} \tilde{\boldsymbol{Z}}\right)\left(n^{-1} \tilde{\boldsymbol{Z}}^{\top} \tilde{\boldsymbol{M}}X \tilde{\boldsymbol{Z}}\right)^{-1}\left(n^{-1 / 2} \tilde{\boldsymbol{Z}}^{\top}(\boldsymbol{y}-\overline{\boldsymbol{x}})\right)$$ What we must do now is to replace the quantities $\tilde{s}, n^{-1 / 2}(\boldsymbol{y}-\tilde{\boldsymbol{x}})^{\top} \tilde{\boldsymbol{Z}}$, and $n^{-1} \tilde{\boldsymbol{Z}}^{\top} \tilde{\boldsymbol{M}}_X \tilde{\boldsymbol{Z}}$ by what they tend to asymptotically under (12.06). We state the following results without proof. They can all be derived by suitable modification of arguments used in Chapter 5 : $$\begin{gathered} \bar{s}^2 \stackrel{p}{\longrightarrow} \sigma_0^2, \ n^{-1} \tilde{\boldsymbol{Z}}^{\top} \tilde{\boldsymbol{M}}_X \tilde{\boldsymbol{Z}} \stackrel{p}{\longrightarrow} \operatorname{plim}{n \rightarrow \infty}\left(n^{-1} \boldsymbol{Z}_0^{\top} \boldsymbol{M}_X \boldsymbol{Z}_0\right), \ n^{-1 / 2}(\boldsymbol{y}-\overline{\boldsymbol{x}})^{\top} \tilde{\boldsymbol{Z}} \stackrel{a}{=} n^{-1 / 2}\left(\boldsymbol{u}+\alpha n^{-1 / 2} \boldsymbol{a}\right)^{\top} \boldsymbol{M}_X \boldsymbol{Z}_0, \end{gathered}$$
where $\boldsymbol{M}_X \equiv \mathbf{I}-\boldsymbol{X}_0\left(\boldsymbol{X}_0^{\top} \boldsymbol{X}_0\right)^{-1} \boldsymbol{X}_0^{\top}$.
The intuition behind the results (12.11) and (12.12) is straightforward. The drifting DGP (12.06) approaches the simple null (12.07) fast enough that the limits of $\tilde{s}^2$ and $n^{-1} \tilde{\boldsymbol{Z}}^{\top} \tilde{\boldsymbol{M}}_X \tilde{\boldsymbol{Z}}$ are exactly the same as they would be under (12.07). These limits, $\sigma_0^2$ and plim $\left(n^{-1} \boldsymbol{Z}_0^{\top} \boldsymbol{M}_X \boldsymbol{Z}_0\right)$, are nonstochastic because the difference between $\overline{\boldsymbol{\beta}}$ and $\boldsymbol{\beta}_0$, which is $O\left(n^{-1 / 2}\right)$, has no effect on either $\bar{s}^2$ or $n^{-1} \tilde{\boldsymbol{Z}}^{\top} \overline{\boldsymbol{M}}_X \tilde{\boldsymbol{Z}}$ asymptotically. It is thus not surprising that the difference between the drifting DGP (12.06) and the simple null (12.07), which is likewise $O\left(n^{-1 / 2}\right)$, also has no effect on $\bar{s}^2$ and $n^{-1} \tilde{\boldsymbol{Z}}^{\top} \tilde{\boldsymbol{M}}_X \tilde{\boldsymbol{Z}}$ asymptotically.

## 经济代写|计量经济学代写Econometrics代考|The Geometry of Test Power

The NCP (12.18) is not very illuminating as it stands. It can, however, be rewritten in a much more illuminating way. Consider, first of all, the vector $\alpha n^{-1 / 2} \boldsymbol{M}X \boldsymbol{a}$, the squared length of which, asymptotically, is $$\alpha^2 \operatorname{plim}{n \rightarrow \infty}\left(\frac{1}{n} \boldsymbol{a}^{\top} \boldsymbol{M}_X \boldsymbol{a}\right) .$$
This quantity is $\alpha^2$ times the plim of the sum of squared residuals from a regression of $n^{-1 / 2} \boldsymbol{a}$ on $\boldsymbol{X}_0$. Suppose that for fixed $n$ the DGP corresponding to that sample size is represented by the vector $\boldsymbol{x}\left(\boldsymbol{\beta}_0\right)+\alpha n^{-1 / 2} \boldsymbol{a}$ in $E^n$. If the null hypothesis is represented as in Section $2.2$ by the manifold $x$ generated by the vectors $\boldsymbol{x}(\boldsymbol{\beta})$ as $\boldsymbol{\beta}$ varies, then the above sum of squared residuals is the square of the Euclidean distance from the point representing the DGP to the linear approximation $\mathcal{S}\left(\boldsymbol{X}_0\right)$ to the manifold $X$ at the point $\boldsymbol{\beta}_0$. It thus provides a measure of the discrepancy, for given $n$, between the model being tested and the data-generating process.
Now consider the artificial regression
$$\left(\alpha / \sigma_0\right) n^{-1 / 2} \boldsymbol{M}_X \boldsymbol{a}=\boldsymbol{M}_X \boldsymbol{Z} \boldsymbol{d}+\text { residuals, }$$
where $\boldsymbol{d}$ is simply an $r$-vector of coefficients that will be chosen by least squares to make this regression fit as well as possible. The plim of the total sum of squares for this regression is expression (12.19) divided by $\sigma_0^2$. The plim of the explained sum of squares is the NCP (12.18). Thus the asymptotic uncentered $R^2$ from regression (12.20) is
$$\frac{\operatorname{plim}\left(n^{-1} \boldsymbol{a}^{\top} \boldsymbol{M}_X \boldsymbol{Z}\right) \operatorname{plim}\left(n^{-1} \boldsymbol{Z}^{\top} \boldsymbol{M}_X \boldsymbol{Z}\right)^{-1} \operatorname{plim}\left(n^{-1} \boldsymbol{Z}^{\top} \boldsymbol{M}_X \boldsymbol{a}\right)}{\operatorname{plim}\left(n^{-1} \boldsymbol{a}^{\top} \boldsymbol{M}_X \boldsymbol{a}\right)}$$

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|The Geometry of Test Power

NCP (12.18) 就目前而言并不是很有启发性。然而，它可以以一种更有启发性的方式重写。首先考虑向量 $\alpha n^{-1 / 2} \boldsymbol{M X a}$, 其平方长度, 渐近地, 是
$$\alpha^2 \operatorname{plim} n \rightarrow \infty\left(\frac{1}{n} \boldsymbol{a}^{\top} \boldsymbol{M}_X \boldsymbol{a}\right) .$$

$$\left(\alpha / \sigma_0\right) n^{-1 / 2} \boldsymbol{M}_X \boldsymbol{a}=\boldsymbol{M}_X \boldsymbol{Z} \boldsymbol{d}+\text { residuals, }$$

$$\frac{\operatorname{plim}\left(n^{-1} \boldsymbol{a}^{\top} \boldsymbol{M}_X \boldsymbol{Z}\right) \operatorname{plim}\left(n^{-1} \boldsymbol{Z}^{\top} \boldsymbol{M}_X \boldsymbol{Z}\right)^{-1} \operatorname{plim}\left(n^{-1} \boldsymbol{Z}^{\top} \boldsymbol{M}_X \boldsymbol{a}\right)}{\operatorname{plim}\left(n^{-1} \boldsymbol{a}^{\top} \boldsymbol{M}_X \boldsymbol{a}\right)}$$

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## MATLAB代写

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师