### 金融代写|金融实证代写Financial Empirical 代考|CMSE11509

statistics-lab™ 为您的留学生涯保驾护航 在代写金融实证Financial Empirical方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融实证Financial Empirical股权市场金融实证Financial Empirical相关的作业也就用不着说。

• 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

## 金融代写|金融实证代写Financial Empirical 代考|Base Model

Based on the above discussion a time series $x(t)$ with possibly continuous time index $t$ in an interval $[a, b]$ will be analysed and additively decomposed in the unobservable important and interpretable components trend (and economic cycle) $x_1(t)$ and season (and calendar) $x_2(t)$. The rest $u(t)$ contains the unimportant, irregular unobservable parts, maybe containing additive outliers.

An “ideal” trend $\tilde{x}1(t)$ is represented by a polynomial of given degree $p-1$ and an “ideal” season $\tilde{x}_2(t)$ is represented by a linear combination of trigonometric functions of chosen frequencies (found by exploration) $\omega_j=2 \pi / S_j$ with $S_j=$ $S / n_j$ and $n_j \in \mathbb{N}$ for $j=1, \ldots, q$. Here $S$ is the known base period and $S_j$ leads to selected harmonics, which can be defined by Fourier analysis. Therefore holds $$\tilde{x}_1(t)=\sum{j=0}^{p-1} a_j t^j \quad \text { and } \quad \tilde{x}2(t)=\sum{j=1}^q\left(b_{1 j} \cos \omega_j t+b_{2 j} \sin \omega_j t\right), \quad t \in[a, b] .$$
In applications the components $x_1(t)$ and $x_2(t)$ won’t exist in ideal representation. They will be additively superimposed by random disturbances $u_1(t)$ and $u_2(t)$. Only at some points in time $t_1, \ldots, t_n$ in the time interval $[a, b]$ the sum $x(t)$ of components is observable, maybe flawed by further additive errors $\varepsilon_1, \ldots, \varepsilon_n$. The respective measurements are called $y_1, \ldots, y_n$.
Now we have following base model
$x_1(t)=\ddot{x}_1(t)+u_1(t)$
$x_2(t)=\tilde{x}_2(t)+u_2(t) \quad t \in[a, b] \quad$ state equation
$y_k=x_1\left(t_k\right)+x_2\left(t_k\right)+\varepsilon_k, \quad k=1, \ldots, n \quad$ observation equation,
cf. Fig. 1 .

## 金融代写|金融实证代写Financial Empirical 代考|Construction of the Estimation Principle

For evaluation of smoothness (in contrast to flexibility) the following smoothness measures are constructed (actually these are roughness measures).

By differentiation $\mathrm{D}=\frac{\mathrm{d}}{\mathrm{d} t}$ the degree of a polynomial is reduced by 1 . Therefore for a trend $x_1(t)$ as polynomial of degree $p-1$ always holds $\mathrm{D}^p x_1(t)=0$. On the

other hand, every function $x_1(t)$ with this feature is a polynomial of degree $p-1$. Therefore
$$Q_1\left(x_1\right)=\int_a^b\left|\mathrm{D}^p x_1(t)\right|^2 \mathrm{~d} t \quad \text { measure of smoothness of trend }$$
is a measure of the smoothness of an appropriately chosen function $x_1$.
For any sufficiently often differentiable and quadratically integrable function $x_1$ in interval $[a, b] Q_1\left(x_1\right)$ is zero iff $x_1$ is there a polynomial of degree $p-1$, i.e. $x_1(t)=\sum_{j=0}^{p-1} a_j t^j$, a smoothest (ideal) trend. The larger the value of $Q_1$ for a function $x_1$ in $[a, b]$ the larger is the deviation of $x_1$ from a (ideal) trend polynomial of degree $p-1$.

Two times differentiation of the functions $\cos \omega_j t$ and $\sin \omega_j t$ gives $-\omega_j^2 \cos \omega_j t$ and $-\omega_j^2 \sin \omega_j t$ such that $\prod_{j=1}^q\left(\mathrm{D}^2+\omega_j^2 \mathrm{I}\right)$ (I: identity) nullifies any linear combination $x_2(t)$ of all functions $\cos \omega_j t$ and $\sin \omega_j t, j=1, \ldots, q$. That is because the following
\begin{aligned} &\left(\mathrm{D}^2+\omega_j^2 \mathrm{I}\right)\left(b_{1 k} \cos \omega_k t+b_{2 k} \sin \omega_k t\right)= \ &=b_{1 k}\left(\omega_j^2-\omega_k^2\right) \cos \omega_k t+b_{2 k}\left(\omega_j^2-\omega_k^2\right) \sin \omega_k t \quad \text { for } \quad j, k=1, \ldots, q, \end{aligned}
nullifies for the case $j=k$ the respective oscillation. This also proves the exchangeability of the operators $\mathrm{D}^2+\omega_j^2 \mathrm{I}, j=1, \ldots, q$.

If inversely $\prod_{j=1}^q\left(\mathrm{D}^2+\omega_j^2 \mathrm{I}\right) x_2(t)=0$ holds, the function $x_2(t)$ is a linear combination of the trigonometric functions under investigation. Consequently
$Q_2\left(x_2\right)=\int_a^b\left|\prod_{j=1}^q\left(\mathrm{D}^2+\omega_j^2 \mathrm{I}\right) x_2(t)\right|^2 \mathrm{~d} t \quad$ measure of seasonal smoothness is a measure for seasonal smoothness of the chosen function $x_2$.

## 金融代写|金融实证代写Financial Empirical 代考|Base Model

“理想”的趋势 $\tilde{x} 1(t)$ 由给定次数的多项式表示 $p-1$ 和一个“理想”的李节 $\tilde{x}2(t)$ 由选定频率的三角函数的线性组合表 示 (通过探索发现) $\omega_j=2 \pi / S_j$ 和 $S_j=S / n_j$ 和 $n_j \in \mathbb{N}$ 为了 $j=1, \ldots, q$. 这里 $S$ 是已知的基期和 $S_j$ 导致选 定的谐波，可以通过傅里叶分析来定义。因此成立 $$\tilde{x}_1(t)=\sum j=0^{p-1} a_j t^j \quad \text { and } \quad \tilde{x} 2(t)=\sum j=1^q\left(b{1 j} \cos \omega_j t+b_{2 j} \sin \omega_j t\right), \quad t \in[a, b] .$$

$x_1(t)=\ddot{x}_1(t)+u_1(t)$
$x_2(t)=\tilde{x}_2(t)+u_2(t) \quad t \in[a, b] \quad$ 状态方程
$y_k=x_1\left(t_k\right)+x_2\left(t_k\right)+\varepsilon_k, \quad k=1, \ldots, n$ 观察方程，

## 金融代写|金融实证代写Financial Empirical 代考|Construction of the Estimation Principle

$$Q_1\left(x_1\right)=\int_a^b\left|D^p x_1(t)\right|^2 \mathrm{~d} t \quad \text { measure of smoothness of trend }$$
is a measure of the smoothness of an appropriately chosen function $x_1$.

$$\left(\mathrm{D}^2+\omega_j^2 \mathrm{I}\right)\left(b_{1 k} \cos \omega_k t+b_{2 k} \sin \omega_k t\right)=\quad=b_{1 k}\left(\omega_j^2-\omega_k^2\right) \cos \omega_k t+b_{2 k}\left(\omega_j^2-\omega_k^2\right) \sin \omega_k t$$

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