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# 经济代写|计量经济学代写Introduction to Econometrics代考|Sample and Smoothed Laplace Periodogram

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## 经济代写|计量经济学代写Introduction to 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 $\hat{r}{n, \tau}(j)=\frac{1}{n} \sum_{t=|j|+1}^n \widehat{V}t(\tau) \widehat{V}{t-|j|}(\tau), \quad|j|<n . Q_{n, \tau}(\omega)$ 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).

## 经济代写|计量经济学代写Introduction to 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 $\overline{\mathbb{R}}:=\mathbb{R} \bigcup{-\infty, \infty}$ the extended real line. The definition described above is the Laplace 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).

## 经济代写|计量经济学代写Introduction to Econometrics代考|Sample and Smoothed Laplace Periodogram

$$V_t(\tau, q(\tau))=\tau-I\left{Y_t<q(\tau)\right}$$

$$q(\tau):=\inf {y: P(Y \leq y)}$$
$V_t(\tau)$是有界的、平稳的、平均为零的随机变量。使用Koenker和Basset的方法，我们定义$V_t(\tau)$的估计值如下:
$$\widehat{V}_t(\tau)=V_t\left(\tau, \hat{q}_n(\tau)\right)$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Copula-Based Periodogram and Rank-Based Laplace Periodogram

$$\gamma_k\left(q_1, q_2\right):=\operatorname{Cov}\left(I\left{Y_t \leq q_1\right}, I\left{Y_{t-k} \leq q_2\right}\right), \quad q_1, q_2 \in \overline{\mathbb{R}}, k \in \mathbb{Z}$$

$$\gamma_k^U\left(\tau_1, \tau_2\right):=\operatorname{Cov}\left(I\left{U_t \leq \tau_1\right}, I\left{U_{t-k} \leq \tau_2\right}\right), \quad \tau_1, \tau_2 \in[0,1], k \in \mathbb{Z}$$

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