### 商科代写|计量经济学代写Econometrics代考|ECOM20001

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• Longitudinal Data Analysis 纵向数据分析
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## 商科代写|计量经济学代写Econometrics代考|General Conditions and the Functions ψˆ n,αˆ and ψα

We give general conditions that we assume to hold in the remainder of the paper here and give graphical comparisons of the functions $\hat{\psi}{n, \alpha}$ and $\psi{\alpha}$, where $\psi_{\alpha}$ is defined in Definition $1 .$

Example 1 As an illustrative example, we take $d=2, \psi_{0}(x)=x^{3}, \alpha_{0}=(1 / \sqrt{2}$, $1 / \sqrt{2})^{T}, Y_{i}=\psi_{0}\left(\alpha_{0}^{T} X_{i}\right)+\varepsilon_{i}$, where the $\varepsilon_{i}$ are i.i.d. standard normal random variables, independent of the $\boldsymbol{X}{i}$, which are i.i.d. random vectors, consisting of two independent Uniform $(0,1)$ random variables. In this case, the conditional expectation function (5) is a rather complicated function of $\alpha$ which we shall not give here but can be computed by a computer package such as Mathematica or Maple. The loss functions: $L^{\mathrm{LSE}}: \alpha{1} \mapsto \mathbb{E}\left{Y-\psi_{\alpha}\left(\alpha^{T} \boldsymbol{X}\right)\right}^{2} \quad$ and $\quad \widehat{L}{n}^{\mathrm{LSE}}: \alpha{1} \mapsto n^{-1} \sum_{i=1}^{n}\left{Y_{i}-\hat{\psi}{n, \alpha}\left(\alpha^{T} \boldsymbol{X}{i}\right)\right}^{2}$
where the loss function $\widehat{L}{n}^{L S E}$ is for sample sizes $n=10,000$ and $n=100,000$, and $\alpha=\left(\alpha{1}, \alpha_{2}\right)^{T}$. For $\alpha_{1} \in[0,1]$ and $\alpha_{2}$ equal to the positive root $\left{1-\alpha_{1}^{2}\right}^{1 / 2}$, we get Fig. 1. The function $L^{\mathrm{LSE}}$ has a minimum equal to 1 at $\alpha_{1}=1 / \sqrt{2}$, and $\widehat{L}{n}^{\mathrm{LSE}}$ has a minimum at a value very close to $1 / \sqrt{2}$ (furnishing the profile LSE $\hat{\alpha}{n}$ ), which gives a visual evidence for consistency of the profile LSE.

In order to show the $\sqrt{n}$-consistency and asymptotic normality of the estimators in the next sections, we now introduce some conditions, which correspond to those in Balabdaoui et al. (2019b). We note that we do not need conditions on reparameterization.
(A1) $\boldsymbol{X}$ has a density w.r.t. Lebesgue measure on its support $\mathcal{X}$, which is a convex set $\mathcal{X}$ with a nonempty interior, and satisfies $\mathcal{X} \subset\left{x \in \mathbb{R}^{d}:|x| \leq R\right}$ for some $R>0$.
(A2) The function $\psi_{0}$ is bounded on the set $\left{u \in \mathbb{R}: u=\alpha_{0}^{T} \boldsymbol{x}, \boldsymbol{x} \in \mathcal{X}\right}$.
(A3) There exists $\delta>0$ such that the conditional expectation $\tilde{\psi}{\alpha}$, defined by (5), is nondecreasing on $I{\alpha}=\left{u \in \mathbb{R}: u=\alpha^{T} \boldsymbol{x}, x \in \mathcal{X}\right}$ and satisfies $\bar{\psi}{\alpha}=\psi{\alpha}$, so minimizes $$\left|\mathbb{E}\left{Y-\psi\left(\boldsymbol{\alpha}^{T} \boldsymbol{X}\right)\right} \boldsymbol{X}\right|^{2}$$
over nondecreasing functions $\psi$, if $\left|\boldsymbol{\alpha}-\boldsymbol{\alpha}_{0}\right| \leq \delta$.

## 商科代写|计量经济学代写Econometrics代考|The Limit Theory for the SSE

In this section, we derive the limit distribution of the SSE introduced above. In our derivation, the function $\psi_{\alpha}$ of Definition 1 plays a crucial role. Below, we will use the following assumptions, additionally to $(\mathrm{A} 1)-(\mathrm{A} 6)$.
(A7) There exists a $\delta>0$ such that for all $\alpha \in\left(\mathcal{B}\left(\alpha_{0}, \delta\right) \cap \mathcal{S}{d-1}\right) \backslash\left{\alpha{0}\right}$, the random variable
$$\operatorname{cov}\left(\left(\boldsymbol{\alpha}{0}-\boldsymbol{\alpha}\right)^{T} \boldsymbol{X}, \psi{0}\left(\boldsymbol{\alpha}{0}^{T} \boldsymbol{X}\right) \mid \boldsymbol{\alpha}^{T} \boldsymbol{X}\right)$$ is not equal to 0 almost surely. (A8) The matrix $$\mathbb{E}\left[\psi{0}^{\prime}\left(\boldsymbol{\alpha}{0}^{T} \boldsymbol{X}\right) \operatorname{cov}\left(\boldsymbol{X} \mid \boldsymbol{\alpha}{0}^{T} \boldsymbol{X}\right)\right]$$
has rank $d-1$.
We start by comparing (3) with the function
$$\alpha \mapsto\left|\mathbb{E}\left{Y-\psi_{\alpha}\left(\alpha^{T} \boldsymbol{X}\right)\right} \boldsymbol{X}\right|^{2}$$ As in Sect. 1, the function $\hat{\psi}_{n, \alpha}$ is just the (isotonic) least squares estimate for fixed $\alpha$.

## 商科代写|计量经济学代写Econometrics代考|The Limit Theory for ESE and Cubic Spline Estimator

The proofs of the consistency and asymptotic normality of the ESE and spline estimator are highly similar to the proofs of these facts for the SSE in the preceding section. The only extra ingredient is the occurrence of the estimate of the derivative of the link function. We only discuss the asymptotic normality.
In addition to the assumptions (A1)-(A7), we now assume the following:
(A8′) $\psi_{\alpha}$ is twice differentiable on $\left.\inf {x \in \mathcal{X}}\left(\alpha^{T} \boldsymbol{x}\right), \sup {x \in \mathcal{X}^{\prime}}\left(\boldsymbol{\alpha}^{T} \boldsymbol{x}\right)\right)$.
(A9) The matrix
$$\mathbb{E}\left[\psi_{0}^{\prime}\left(\alpha_{0}^{T} \boldsymbol{X}\right)^{2} \operatorname{cov}\left(\boldsymbol{X} \mid \alpha_{0}^{T} \boldsymbol{X}\right)\right]$$
has rank $d-1$.
An essential step is again to show that
\begin{aligned} &\int \boldsymbol{x}\left{y-\hat{\psi}{n, \hat{\alpha}{n}}\left(\hat{\boldsymbol{\alpha}}{n}^{T} \boldsymbol{x}\right)\right} \hat{\psi}{n \hat{\boldsymbol{\alpha}}{n}}^{\prime}\left(\hat{\boldsymbol{\alpha}}{n}^{T} \boldsymbol{x}\right) d \mathbb{P}{n}(\boldsymbol{x}, y) \ &=\int\left{\boldsymbol{x}-\mathbb{E}\left(X \mid \hat{\boldsymbol{\alpha}}{n}^{T} \boldsymbol{X}\right)\right}\left{y-\hat{\psi}{n, \hat{\alpha}{n}}\left(\hat{\boldsymbol{\alpha}}{n}^{T} \boldsymbol{x}\right)\right} \hat{\psi}{n \hat{\alpha}{n}}^{\prime}\left(\hat{\alpha}{n}^{T} \boldsymbol{x}\right) d \mathbb{P}{n}(\boldsymbol{x}, y) \ &+o{p}\left(n^{-1 / 2}\right)+o_{p}\left(\hat{\alpha}{n}-\boldsymbol{\alpha}{0}\right) \end{aligned}
For the ESE, this is done by defining the piecewise constant function $\bar{\rho}{n, \alpha}$ for $u$ in the interval between successive jumps $\tau{i}$ and $\tau_{i+1}$ ) of $\hat{\psi}{n \alpha}$ by $$\bar{\rho}{n, \alpha}(u)= \begin{cases}\mathbb{E}\left[\boldsymbol{X} \mid \alpha^{T} \boldsymbol{X}=\tau_{i}\right] \psi_{\alpha}^{\prime}\left(\tau_{i}\right) & \text { if } \psi_{\alpha}(u)>\hat{\psi}{n \alpha}\left(\tau{i}\right) \text { for all } u \in\left(\tau_{i}, \tau_{i+1}\right) \ \mathbb{E}\left[\boldsymbol{X} \mid \alpha^{T} \boldsymbol{X}=s\right] \psi_{\alpha}^{\prime}(s) & \text { if } \psi_{\alpha}(s)=\hat{\psi}{n \alpha}(s) \text { for some } s \in\left(\tau{i}, \tau_{i+1}\right) \ \mathbb{E}\left[\boldsymbol{X} \mid \alpha^{T} \boldsymbol{X}=\tau_{i+1}\right] \psi_{\alpha}^{\prime}\left(\tau_{i+1}\right) & \text { if } \psi_{\alpha}(u)<\hat{\psi}{n \alpha}\left(\tau{i}\right) \text { for all } u \in\left(\tau_{i}, \tau_{i+1}\right)\end{cases}$$ see Appendix E in the supplement of Balabdaoui et al. (2019b). The remaining part of the proof runs along the same lines as the proof for the SSE. For additional details, see Appendix E in the supplement of Balabdaoui et al. (2019b).

The corresponding step in the proof for the spline estimator is given by the following lemma.

## 商科代写|计量经济学代写Econometrics代考|General Conditions and the Functions ψˆ n,αˆ and ψα

$\mathrm{~ I q u a d ~ I w i d e h a t { { \ { n }}$ 其中损失函数 $\widehat{L} n^{L S E}$ 适用于样本量 $n=10,000$ 和 $n=100,000$ ，和 $\alpha=\left(\alpha 1, \alpha_{2}\right)^{T}$. 为了 $\alpha_{1} \in[0,1]$ 和 $\alpha_{2}$ 等于正根 $\mathrm{~ l e f t { 1 – l a l p h a _ { 1 } へ { 2 } \ r i g h t }}$ $\widehat{L} n^{\mathrm{LSE}}$ 最小值非常接近 $1 / \sqrt{2}$ (提供简介 LSE $\left.\hat{\alpha} n\right)$ ，这为配置文件 LSE 的一致性提供了视觉证据。

(A1) $\boldsymbol{X}$ 在其支持上有一个密度 wrt Lebesgue 度量 $\mathcal{X}$ ，这是一个凸集 $\mathcal{X}$ 具有非空的内部，并且满足
$\mathrm{~ I m a t h c a l { X } }}$
(A2) 功能 $\psi_{0}$ 有界在集合上
\eft ${u \backslash$ in $\backslash \mathrm{~ m a t h b b b { R } : ~ u =}$
(A3) 存在 $\delta>0$ 使得条件期望 $\tilde{\psi} \alpha$ ，由 (5) 定义，在
I{\alpha $}=\backslash 1$ eft $\left{u \backslash\right.$ in $\backslash$ mathbb ${R}: u=\backslash a \mid p h a^{\wedge}{T} \backslash$ boldsymbol ${x}, x \backslash \mathrm{~ i n ~}$
Veft $\backslash \backslash$ mathbb ${$ E $} \backslash$ eft ${Y \mathrm{~ – ~ \ p s i V l e f t (}$

## 商科代写|计量经济学代写Econometrics代考|The Limit Theory for the SSE

(A7) 存在一个 $\delta>0$ 这样对于所有人
$\mathrm{~ \ a l p h a ~ \ i n \ l e f t ( \ m a t h c a l { B }}$

$$\operatorname{cov}\left((\boldsymbol{\alpha} 0-\boldsymbol{\alpha})^{T} \boldsymbol{X}, \psi 0\left(\boldsymbol{\alpha} 0^{T} \boldsymbol{X}\right) \mid \boldsymbol{\alpha}^{T} \boldsymbol{X}\right)$$

$$\mathbb{E}\left[\psi 0^{\prime}\left(\boldsymbol{\alpha} 0^{T} \boldsymbol{X}\right) \operatorname{cov}\left(\boldsymbol{X} \mid \boldsymbol{\alpha} 0^{T} \boldsymbol{X}\right)\right]$$

\alpha \mapsto\left } \backslash \backslash \text { mathbb } { E } \backslash l e f t { Y \mathrm { ~ –

## 商科代写|计量经济学代写Econometrics代考|The Limit Theory for ESE and Cubic Spline Estimator

ESE 和样条估计量的一致性和渐近正态性的证明与上一节中 SSE 的这些事实的证明非常相似。唯一的额外因素是链 接函数导数估计的出现。我们只讨论渐近正态性。

(A8′) $\psi_{\alpha}$ 是两次可微的 $\left.\inf x \in \mathcal{X}\left(\alpha^{T} \boldsymbol{x}\right), \sup x \in \mathcal{X}^{\prime}\left(\boldsymbol{\alpha}^{T} \boldsymbol{x}\right)\right)$.
(A9) 矩阵
$$\mathbb{E}\left[\psi_{0}^{\prime}\left(\alpha_{0}^{T} \boldsymbol{X}\right)^{2} \operatorname{cov}\left(\boldsymbol{X} \mid \alpha_{0}^{T} \boldsymbol{X}\right)\right]$$

\begin } { \text { 对齐 } } \text { \& Int } \backslash \text { boldsymbol } { x } \backslash l e f t { y \mathrm { ~ – ~

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