### 商科代写|计量经济学代写Econometrics代考|The Multivariate Case

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## 商科代写|计量经济学代写Econometrics代考|The Multivariate Case

Now we consider the multivariate case of quantile spectral densities and periodograms based on the copula- and Laplace-related concepts, which have been already introduced in the univariate case in the previous sections. Let $\left{\boldsymbol{X}{t}\right}{t \in \mathcal{Z}}$ be a $d$-variate strictly stationary process, with components $X_{t, l}, l=1, \ldots, d$; i.e., $\boldsymbol{X}{t}=\left(X{t, 1}, \ldots, X_{t, d}\right)^{\prime} . X_{t, l}$ has its marginal distribution function $F_{l}(q)$ and inverse function $q_{l}(\tau):=F_{l}^{-1}(\tau):=\inf \left{q \in \mathbb{R}: \tau \leq F_{l}(q)\right}$, where $\tau \in[0,1]$. The matrix of quantile cross-covariance, $\Gamma_{k}\left(\tau_{1}, \tau_{2}\right):=\left(\gamma_{k}^{l_{1} l_{2}}\left(\tau_{1}, \tau_{2}\right)\right){l{1}, l_{2}=1, \ldots, d}$, where $\gamma_{k}^{l_{1} l_{2}}\left(\tau_{1}, \tau_{2}\right)$ is the cross-covariance of a pair of $\left(X_{t, l_{1}}, X_{t, l_{2}}\right)$, which is as follows.
$$\gamma_{k}^{l_{1} l_{2}}\left(\tau_{1}, \tau_{2}\right):=\operatorname{Cov}\left(I\left{X_{t, l_{1}} \leq q_{l_{1}}\left(\tau_{1}\right)\right}, I\left{X_{t-k, l_{2}} \leq q_{l_{2}}\left(\tau_{2}\right)\right}\right)$$
where $l_{1}, l_{2} \in{1, \ldots, d}, k \in \mathbb{Z}$, and $\tau_{1}, \tau_{2} \in[0,1]$. The quantile-based quantities are functions of $\tau_{1}$ and $\tau_{2}$, which are quantiles of a quantile regression. In the frequency domain, under approximate mixing conditions, the quantile cross-spectral densities are
$$f_{q_{z_{1}}, q_{t_{2}}}(\omega):=\left(f_{q_{\tau_{1}}, q_{t_{2}}}^{l_{1} l_{2}}(\omega)\right){l{1}, l_{2}=1, \ldots, d},$$
where $$f_{q_{\tau_{1}}, q_{\tau_{2}}}^{l_{l_{2}}}(\omega):=\frac{1}{2 \pi} \sum_{k=-\infty}^{\infty} \gamma_{k}^{l_{1} l_{2}}\left(\tau_{1}, \tau_{2}\right) \mathrm{e}^{-i k \omega},$$
$l_{1}, l_{2} \in{1, \ldots, d}, \omega \in \mathbb{R}$, and $\tau_{1}, \tau_{2} \in[0,1]$. Each quantile cross-spectral density, i.e., $f_{q_{\tau_{1}}, q_{t_{2}}}^{l_{2}}(\omega)$, is a complex-valued function. As considered in traditional spectral analysis, its real and imaginary parts are referred to as quantile cospectrum and quantile quadrature spectrum.

To measure dynamic dependence structure of the two processes $\left{X_{t, l_{1}}\right}_{t \in \mathbb{Z}}$ and $\left{X_{t, l_{2}}\right}_{t \in \mathbb{Z}}$, the quantile coherency is defined as follows.
$$\mathcal{R}{q{t_{1}}, q_{t_{2}}}^{l_{1} l_{2}}(\omega):=\frac{f_{q_{\tau_{1}}, q_{\tau_{2}}}^{l_{1} l_{2}}(\omega)}{\left(f_{q_{t_{1}}, q_{t_{1}}}(\omega) f_{q_{t_{2}}, q_{t_{2}}}^{l_{1} l_{2}}(\omega)\right)^{1 / 2}},$$

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

This section shows an example of the quantile-based spectral analysis for the stock returns by using the $\mathrm{R}$ package “QUANTSPEC version $1.2 .1$ “. The following returns of daily stock average indexes (Dow Jones Industrial Average, CAC 40 , and Nikkei 225 ) were taken from “Factiva.com” during the post-period of global financial crisis from July 27 th 2009 to March 27 th 2020 (2516 observations).

We first plot the following three types of data for each stock index: (1) $Y_{t}$ : returns
(2) $\operatorname{Cov}\left(Y_{t+k}, Y_{t}\right)$ : autocovariances of the returns, and (3) $\operatorname{Cov}\left(Y_{t+k}^{2}, Y_{t}^{2}\right):$ autocovariances of the squared returns. Figure 2 shows the stock prices and their returns of three stock average indexes, DJ (Dow Jones Industrial Average in the United States), CAC (CAC 40 in France), and Nikkei (Nikkei 225 in Japan). Each return seems to have zero-mean with some outliers.

Their highly volatile periods correspond to “Flash crash” in May 2010 , “Black Monday” in August 2011, “China shock” in August 2015, “Brexit” in June 2016, and “VIX shock” in February 2018, and “Coronavirus shock” in March 2020. Additionally, the highly volatile period, especially limited to Nikkei (Japanese market), corresponds to the “East Japan great earthquake” in March $2011 .$

Figure 3 shows their autocovariances with lag $k$. DJ has significantly negative serial correlations $(\mathrm{Lag}=1,3,5,8$, or 19) and positive correlations (Lag $=2,9$, or 11). CAC has a significantly negative serial correlation (Lag $=5$ ) and a positive correlation $(\mathrm{Lag}=6)$. Nikkei seems to have no serial correlation. Thus, only Japanese stock market appears to be uncorrelated. This is a typical characteristic of many financial returns, as long as we use a linear measure of dependence.

Figure 4 shows the autocovariances of the squared returns, i.e., their volatilities. In the series of all volatilities, we can find significant and persistent autocovariances. These squared returns are clearly correlated. However, all autocovariances persist until at least lag 14 (more than 2 weeks). The persistency of their volatilities suggests that an ARCH or GARCH model will be required if we focus on the traditional approach of financial analyses. In this section. we focus on another approach. i.e.. quantile-hased spectral analysis.

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

The existing literature usually models exchange rate pass-through by considering variations of the following equation:
$$\Delta m p_{t}=\alpha+\sum_{j=1}^{n} \gamma_{j} \Delta m p_{t-j}+\rho \Delta y_{t}+\lambda \Delta m c_{t}^{}+\theta \Delta e_{t}+\epsilon_{t}$$ where $m p$ represents import prices, $y$ is a local demand factor, $m c^{}$ stands for the exporter marginal cost (i.e., the foreign production costs), $e$ is the nominal effective exchange rate, $i$ denotes the industry and $t$ refers to the period. Our primary concern in this equation is the pass-through elasticity, which corresponds to the coefficient on the exchange rate change, namely $\theta$. The case $\theta=1$ refers to a complete ERPT, corresponding to a one-for-one pass-through changes in import prices. Incomplete ERPT occurs when $\theta<1$, i.e., when exporters adjust their markup. Equation (1) is estimated at the aggregated (i.e., country) and product levels using, in the latter case, individual fixed effects. All the variables are expressed in logarithms.

To explore the global factors’ dimension of pass-through, our empirical strategy consists in extending the benchmark ERPT equation as follows:
\begin{aligned} \Delta m p_{t}=& \alpha+\beta_{t}+\sum_{j=1}^{n} \gamma_{j} \Delta m p_{t-j}+\rho \Delta y_{t}+\lambda \Delta m c_{t}^{*} \ &+\theta \Delta e_{t}+\theta^{C}\left(\Delta e_{t} \times C_{t}\right)+C_{t}+\epsilon_{t} \end{aligned}
where $C$ is an indicator of trade integration: changes in trade openness, changes in intra-industry trade, changes in tariffs for a country’s imports, changes in the weight of China in a country $i$ ‘s exports, and changes in intra-EU imports share. In Eq. (2), we interpret a significant coefficient $\theta^{C}$ as evidence that ERPT is affected by global factors.

## 商科代写|计量经济学代写Econometrics代考|The Multivariate Case

\gamma_{k}^{l_{1} l_{2}}\left(\tau_{1}, \tau_{2}\right):=\operatorname{Cov}\left(I\left{X_{t , l_{1}} \leq q_{l_{1}}\left(\tau_{1}\right)\right}, I\left{X_{tk, l_{2}} \leq q_{l_{2 }}\left(\tau_{2}\right)\right}\right)\gamma_{k}^{l_{1} l_{2}}\left(\tau_{1}, \tau_{2}\right):=\operatorname{Cov}\left(I\left{X_{t , l_{1}} \leq q_{l_{1}}\left(\tau_{1}\right)\right}, I\left{X_{tk, l_{2}} \leq q_{l_{2 }}\left(\tau_{2}\right)\right}\right)

Fq和1,q吨2(ω):=(Fqτ1,q吨2l1l2(ω))l1,l2=1,…,d,

Fqτ1,qτ2ll2(ω):=12圆周率∑ķ=−∞∞Cķl1l2(τ1,τ2)和−一世ķω,
l1,l2∈1,…,d,ω∈R， 和τ1,τ2∈[0,1]. 每个分位数的交叉谱密度，即Fqτ1,q吨2l2(ω), 是复值函数。在传统的光谱分析中，它的实部和虚部被称为分位数余谱和分位数正交谱。

$$\mathcal{R} {q {t_{1}}, q_{t_{2}}}^{l_{1} l_{2}}(\omega):=\frac{f_{q_{\tau_ {1}}, q_{\tau_{2}}}^{l_{1} l_{2}}(\omega)}{\left(f_{q_{t_{1}}, q_{t_{1} }}(\omega) f_{q_{t_{2}}, q_{t_{2}}}^{l_{1} l_{2}}(\omega)\right)^{1 / 2}},$$

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

(2)这⁡(是吨+ķ,是吨): 收益的自协方差，和 (3)这⁡(是吨+ķ2,是吨2):平方收益的自协方差。图 2 显示了 DJ（美国道琼斯工业平均指数）、CAC（法国 CAC 40）和日经指数（日本日经 225）三个股票平均指数的股价及其回报。每个回报似乎都具有一些异常值的零均值。

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

Δ米p吨=一个+∑j=1nCjΔ米p吨−j+ρΔ是吨+λΔ米C吨+θΔ和吨+ε吨在哪里米p代表进口价格，是是本地需求因素，米C代表出口商边际成本（即国外生产成本），和是名义有效汇率，一世表示行业和吨指时期。我们在这个等式中主要关注的是传递弹性，它对应于汇率变化的系数，即θ. 案子θ=1指完整的ERPT，对应进口价格的一对一传递变化。不完整的 ERPT 发生在θ<1，即当出口商调整他们的标记时。等式（1）是在综合（即国家）和产品水平上估计的，在后一种情况下，使用个体固定效应。所有变量均以对数表示。

Δ米p吨=一个+b吨+∑j=1nCjΔ米p吨−j+ρΔ是吨+λΔ米C吨∗ +θΔ和吨+θC(Δ和吨×C吨)+C吨+ε吨

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