### 商科代写|计量经济学代写Econometrics代考|A New Approach to Investigate Cyclical Dependence in Economic Time Series

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## 商科代写|计量经济学代写Econometrics代考|Harmonic Regression Models and Laplace Periodograms

For a given time series $\left{Y_{t}\right}$ of length $n$ and its frequency $\omega \in(0, \pi)$, the ordinary periodogram is defined as
$$G_{n}(\omega):=\frac{1}{n}\left|\sum_{t=1}^{n} Y_{t} \exp (-i t \omega)\right|^{2}$$

In the above equation, if $\omega=2 \pi k / n$, where $k$ is a certain integer, it can also be expressed as
$$G_{n}(\omega)=\frac{1}{4} n\left|\tilde{\boldsymbol{\beta}}{n}(\omega)\right|^{2}=\frac{1}{4} n \tilde{\boldsymbol{\beta}}{n}^{\prime}(\omega) \tilde{\boldsymbol{\beta}}{n}(\omega),$$ where $|\cdot|$ denotes the Euclidian norm, and $\tilde{\boldsymbol{\beta}}{n}(\omega)$ denotes the least squares estimator in the linear model with regressors $\boldsymbol{x}{t}(\omega)=[\cos (\omega t), \sin (\omega t)]^{\prime}$, corresponding to an $L{2}$-projection of the observed series onto the harmonic basis, which are obtained as the solution of the following equation.
$$\left{\tilde{\lambda}{n}(\omega), \tilde{\boldsymbol{\beta}}{n}(\omega)\right}:=\operatorname{argmin}{\lambda \in \mathbb{R}, \beta \in \mathbb{R}^{2}} \sum{t=1}^{n}\left(Y_{t}-\lambda-\boldsymbol{x}{t}^{\prime}(\omega) \boldsymbol{\beta}\right)^{2}$$ When the OLS criterion is replaced by the least absolute deviation (LAD) criterion in the harmonic regression, the LAD coefficient $\ddot{\boldsymbol{\beta}}{n}(\omega)$ is obtained as follows:
$$\left{\ddot{\lambda}{n}(\omega), \ddot{\boldsymbol{\beta}}{n}(\omega)\right}:=\operatorname{argmin}{\lambda \in \mathbb{R}, \beta \in \mathbb{R}^{2}} \sum{t=1}^{n}\left|Y_{t}-\lambda-\boldsymbol{x}{t}^{\prime}(\omega) \boldsymbol{\beta}\right|$$ By using $\ddot{\boldsymbol{\beta}}{n}(\omega), \mathrm{Li}(2008)$ has defined the Laplace periodogram as
$$L_{n}(\omega):=\frac{1}{4} n\left|\ddot{\boldsymbol{\beta}}{n}(\omega)\right|^{2}$$ Therefore, both $G{n}(\omega)$ and $L_{n}(\omega)$ are obtained by the squared norm (or sum of squares) of harmonic regression coefficients multiplied by some constant terms. In particular, the Laplace periodogram inherits the robustness properties of linear LAD regression. Just as the OLS estimator is used to characterize the sample mean, the LAD estimator applied captures the behavior of the observation around the median $(0.5$ quantile)
$\mathrm{Li}(2008)$ has derived the asymptotic normality and useful related theorems of the Laplace periodogram, which are very useful to consider asymptotic behaviors of several periodograms. His results are based on the concept of zero-crossings.

Definition (Stationarity in zero-crossings) The lagged zero-crossing rate of a random process $\left{\varepsilon_{t}\right}$ between $t$ and $s$ is defined as $\gamma_{t s}:=P\left(\varepsilon_{t} \varepsilon_{s}<0\right)$, and $\left{\varepsilon_{t}\right}$ is called to be stationary in zero-crossings if and only if $\gamma_{t s}$ depends only on $t-s$, that is, $\gamma_{t s}=\gamma_{t-s}$ for all $t$ and $s . \gamma_{\tau}$ is called as the lag-zero-crossing rate of $\left{\varepsilon_{t}\right}$ and $S(\omega):=\sum_{\tau=-\infty}^{\infty}\left(1-2 \gamma_{\tau}\right) \cos (\omega \tau)$ is called as the zero-crossing spectrum of $\left{\varepsilon_{t}\right}$.

## 商科代写|计量经济学代写Econometrics代考|Sample and Smoothed Laplace Periodogram

Define the following new variable of interest called a quantile crossing indicator:
$$V_{t}(\tau, q(\tau))=\tau-I\left{Y_{t}<q(\tau)\right}$$
If the distribution function of $Y_{t}$ is continuous and increasing at
$$q(\tau):=\inf {y: P(Y \leq y)}$$
$V_{t}(\tau)$ is bounded, stationary and mean zero random variable. Using Koenker and Basset’s approach, we define an estimate of $V_{t}(\tau)$ as follows:
$$\widehat{V}{t}(\tau)=V{t}\left(\tau, \hat{q}_{n}(\tau)\right)$$

where $\hat{q}{n}(\tau)=\operatorname{argmin}{q \in \mathbb{R}} \sum_{t=1}^{n} \rho_{\tau}\left(Y_{t}-y\right), \rho_{\tau}(x)=x{\tau-I(x<0)} . \hat{q}{n}(\tau)$ is the estimate of the $\tau$ th quantile. The $\tau$ th quantile periodogram is given by $$Q{n, \tau}(\omega):=\frac{1}{2 \pi}\left|\frac{1}{\sqrt{n}} \sum_{t=1}^{n} \widehat{V}{t}(\tau) \mathrm{e}^{-i t \omega}\right|=\frac{1}{2 \pi} \sum{|j|<n} \hat{r}{n, \tau}(j) \cos (\omega j)$$ where $i^{2}=1$ and $\hat{r}{u, r}(j)=\frac{1}{n} \sum_{t}^{n}|j|+1 \widehat{V}{l}(\tau) \widehat{V}{t-|/|}(\tau), \quad|j| \propto n . Q_{n, r}(\omega)$ is an unbiased estimate of the $\tau$ th spectral density, but is not consistent. A consistent estimator is obtained by smoothing the periodogram using kernel functions (all the results below are taken from Hagemann 2013).
We obtained a smoothed $\tau$ th quantile periodogram as
$$\widehat{Q}{n, \tau}(\omega)=\frac{1}{2 \pi} \sum{|j|<n} \lambda\left(j / b_{n}\right) \hat{r}{n, \tau}(j) \cos (\omega j)$$ $\lambda\left(j / b{n}\right)$ is a lag window and $b_{n}$ is a bandwidth parameter. It is known from the literature on spectral analysis that an optimal lag window leading a non-negative periodogram is the so-called quadratic spectral window defined as
$$\lambda_{Q S}(x)=\frac{25}{12 \pi^{2} x^{2}}\left{\frac{\sin \left(\frac{6 \pi x}{5}\right)}{\frac{6 \pi x}{5}}-\cos \left(\frac{6 \pi x}{5}\right)\right}$$

## 商科代写|计量经济学代写Econometrics代考|Copula-Based Periodogram and Rank-Based Laplace Periodogram

Laplace periodograms can be used to estimate copula spectra density kernels. We briefly present the methodology here since copula models have become widely used in economics and finance (see Patton 2012 for a review of theory and empirical estimation). One important advantage of copulas is that they do not require any distributional assumption, such as for instance the existence of finite moments.
Let us consider again a strictly stationary time series $\left{Y_{t}\right}_{t \in \mathbb{Z}}$ and its marginal distribution function $F$. In the traditional approach, the spectral density kernels are defined associated with autocovariance kernels of the series. To capture more general features of pairs of $Y_{t}$ and $Y_{t-k}$, the clipped processes $\left(I\left{Y_{t} \leq q\right}\right){t \in \mathbb{Z}}$ and $\left(I\left{U{t} \leq \tau\right}\right){t \in \mathbb{Z}}$, where $U{t}:=F\left(Y_{t}\right)$ are introduced; then, the spectral density kernels are defined associated with covariance kernels of these clipped processes, which are shown below.
$$\gamma_{k}\left(q_{1}, q_{2}\right):=\operatorname{Cov}\left(I\left{Y_{t} \leq q_{1}\right}, I\left{Y_{t-k} \leq q_{2}\right}\right), \quad q_{1}, q_{2} \in \overline{\mathbb{R}}, k \in \mathbb{Z}$$

where $I{\cdot}$ denotes the indicator function and $\overline{\mathbb{R}}:=\mathbb{R} \cup{-\infty, \infty}$ the extended real line. The definition descriked above is the I añlace cross-covariance. The copula cross-covariance is
$$\gamma_{k}^{U}\left(\tau_{1}, \tau_{2}\right):=\operatorname{Cov}\left(I\left{U_{t} \leq \tau_{1}\right}, I\left{U_{t-k} \leq \tau_{2}\right}\right), \quad \tau_{1}, \tau_{2} \in[0,1], k \in \mathbb{Z}$$
By using the Laplace cross-covariance and the copula cross-covariance, researchers can consider more general dependence structures of $Y_{t}$ and $Y_{t-k}$ that traditional covariance-based methods unable to deal with, such as time-irreversibility, tail dependence, varying conditional skewness or kurtosis, and so on, though various extensions and revisions have been proposed in the $L_{2}$-periodograms (Kleiner et al. 1979; Klüppelberg and Mikosch 1994; Mikosch 1998; Katkovnik 1998; Hong 1999; Hill and McCloskey 2014).

Under summability conditions on $\gamma_{k}$ and $\gamma_{k}^{U}$, the population spectral densities are defined as follows.
$$\begin{gathered} f_{q_{1}, q_{2}}(\omega):=\frac{1}{2 \pi} \sum_{k=-\infty}^{\infty} \gamma_{k}\left(q_{1}, q_{2}\right) \mathrm{e}^{-i k \omega}, \quad q_{1}, q_{2} \in \overline{\mathbb{R}}, \omega \in \mathbb{R}, \ f_{q_{t_{1}}, q_{\tau_{2}}}(\omega):=\frac{1}{2 \pi} \sum_{k=-\infty}^{\infty} \gamma_{k}^{U}\left(\tau_{1}, \tau_{2}\right) \mathrm{e}^{-i k \omega}, \quad \tau_{1}, \tau_{2} \in[0,1], \omega \in \mathbb{R}, \end{gathered}$$

## 商科代写|计量经济学代写Econometrics代考|Harmonic Regression Models and Laplace Periodograms

Gn(ω):=1n|∑吨=1n是吨经验⁡(−一世吨ω)|2

Gn(ω)=14n|b~n(ω)|2=14nb~n′(ω)b~n(ω),在哪里|⋅|表示欧几里得范数，并且b~n(ω)表示带有回归量的线性模型中的最小二乘估计量X吨(ω)=[因⁡(ω吨),罪⁡(ω吨)]′，对应于大号2-将观察到的序列投影到谐波基础上，作为以下方程的解获得。

\left{\tilde{\lambda}{n}(\omega), \tilde{\boldsymbol{\beta}}{n}(\omega)\right}:=\operatorname{argmin}{\lambda \in \ mathbb{R}, \beta \in \mathbb{R}^{2}} \sum{t=1}^{n}\left(Y_{t}-\lambda-\boldsymbol{x}{t}^ {\prime}(\omega) \boldsymbol{\beta}\right)^{2}\left{\tilde{\lambda}{n}(\omega), \tilde{\boldsymbol{\beta}}{n}(\omega)\right}:=\operatorname{argmin}{\lambda \in \ mathbb{R}, \beta \in \mathbb{R}^{2}} \sum{t=1}^{n}\left(Y_{t}-\lambda-\boldsymbol{x}{t}^ {\prime}(\omega) \boldsymbol{\beta}\right)^{2}当OLS准则被调和回归中的最小绝对偏差（LAD）准则代替时，LAD系数b¨n(ω)得到如下：

\left{\ddot{\lambda}{n}(\omega), \ddot{\boldsymbol{\beta}}{n}(\omega)\right}:=\operatorname{argmin}{\lambda \in \ mathbb{R}, \beta \in \mathbb{R}^{2}} \sum{t=1}^{n}\left|Y_{t}-\lambda-\boldsymbol{x}{t}^ {\prime}(\omega) \boldsymbol{\beta}\right|\left{\ddot{\lambda}{n}(\omega), \ddot{\boldsymbol{\beta}}{n}(\omega)\right}:=\operatorname{argmin}{\lambda \in \ mathbb{R}, \beta \in \mathbb{R}^{2}} \sum{t=1}^{n}\left|Y_{t}-\lambda-\boldsymbol{x}{t}^ {\prime}(\omega) \boldsymbol{\beta}\right|通过使用b¨n(ω),大号一世(2008)已将拉普拉斯周期图定义为

## 商科代写|计量经济学代写Econometrics代考|Sample and Smoothed Laplace Periodogram

V_{t}(\tau,q(\tau))=\tau-I\left{Y_{t}<q(\tau)\right}V_{t}(\tau,q(\tau))=\tau-I\left{Y_{t}<q(\tau)\right}

q(τ):=信息是:磷(是≤是)

\lambda_{Q S}(x)=\frac{25}{12 \pi^{2} x^{2}}\left{\frac{\sin \left(\frac{6 \pi x}{5} \right)}{\frac{6 \pi x}{5}}-\cos \left(\frac{6 \pi x}{5}\right)\right}\lambda_{Q S}(x)=\frac{25}{12 \pi^{2} x^{2}}\left{\frac{\sin \left(\frac{6 \pi x}{5} \right)}{\frac{6 \pi x}{5}}-\cos \left(\frac{6 \pi x}{5}\right)\right}

## 商科代写|计量经济学代写Econometrics代考|Copula-Based Periodogram and Rank-Based Laplace Periodogram

\gamma_{k}\left(q_{1}, q_{2}\right):=\operatorname{Cov}\left(I\left{Y_{t} \leq q_{1}\right}, I\左{Y_{tk} \leq q_{2}\right}\right), \quad q_{1}, q_{2} \in \overline{\mathbb{R}}, k \in \mathbb{Z}\gamma_{k}\left(q_{1}, q_{2}\right):=\operatorname{Cov}\left(I\left{Y_{t} \leq q_{1}\right}, I\左{Y_{tk} \leq q_{2}\right}\right), \quad q_{1}, q_{2} \in \overline{\mathbb{R}}, k \in \mathbb{Z}

\gamma_{k}^{U}\left(\tau_{1}, \tau_{2}\right):=\operatorname{Cov}\left(I\left{U_{t} \leq \tau_{1 }\right}, I\left{U_{tk} \leq \tau_{2}\right}\right), \quad \tau_{1}, \tau_{2} \in[0,1], k \在 \mathbb{Z}\gamma_{k}^{U}\left(\tau_{1}, \tau_{2}\right):=\operatorname{Cov}\left(I\left{U_{t} \leq \tau_{1 }\right}, I\left{U_{tk} \leq \tau_{2}\right}\right), \quad \tau_{1}, \tau_{2} \in[0,1], k \在 \mathbb{Z}

Fq1,q2(ω):=12圆周率∑ķ=−∞∞Cķ(q1,q2)和−一世ķω,q1,q2∈R¯,ω∈R, Fq吨1,qτ2(ω):=12圆周率∑ķ=−∞∞Cķ在(τ1,τ2)和−一世ķω,τ1,τ2∈[0,1],ω∈R,

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