### 统计代写|金融统计代写Financial Statistics代考|GRA6518

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|金融统计代写Financial Statistics代考|High Frequency Data

In this section we further elaborate on high frequency data and introduce the series that will be analyzed later. High frequency data are very important in the financial environment, mainly because there exist large movements in short intervals of time. This aspect represents an interesting opportunity for trading. Furthermore, it is well known that volatilities in different frequencies have significant cross-correlation. We can even say that coarse volatility predicts fine volatility better than the inverse, as shown in Dacorogna et al. (2001).

As an example, take the tick by tick foreign exchange (FX) time series Euro-Dollar, from January First 1999 to December 31, 2002. Returns are calculated using bid and ask prices, as
$$r_{t}=\ln \left(\left(p_{t}^{b i d}+p_{t}^{a s k}\right) / 2\right)-\ln \left(\left(p_{t-1}^{b i d}+p_{t-1}^{a s k}\right) / 2\right)$$
We discard Saturdays and Sundays, and we replace holidays with the means of the last ten observations of the returns for each respective hour and day. After cleaning the data (see Dacorogna et al. (2001), for details) we will consider equally spaced returns, with sampling interval $\Delta t=15 \mathrm{~min}$. This seems to be adequate, as many studies indicate.

Figure 2 shows Euro-Dollar returns calculated as above. The length of this time series is 95,317 . The figure shows that the absolute returns present a seasonal pattern. This is due to the fact that physical time does not follow, necessarily, the same pattern as the business time. This is a typical behavior of a financial time series and we will use a seasonal adjustment procedure similar to that of Martens et al. (2002). However, we will use absolute returns instead of squared returns; that is, we will compute the seasunal patturn as
$$S_{d_{,}, h}=\frac{1}{s} \sum_{j=1}^{s} \mid\left(r_{d_{t}, j, t} \mid,\right.$$
where $r_{d, s, t}$ is the return in the weekday $d$, week $s$ and hour $h$, and $s$ is the number of weeks from the beginning of the series. Therefore, $S_{d, N}, N_{t}$ is the rolling window mean of the absolute returns with the beginning fixed.

In Figure 3 we have the autocorrelation function of these returns and of squared returns. The seasonality pattern is no longer present.

FX data has some distinct characteristics, mainly because they are produced twenty four hours a day, seven days a week. In particular, Euro-Dollar is the most liquid FX in the world. However, there are periods where the activity is greater or smaller, causing seasonal patterns to occur, as seen above.
Let us analyze some facts about these returns that we will denote simply by rt. We can see in Figure 4 the histogram fitted with a non-parametric density kernel estimate, using unbiased cross-validation method to estimate the bandwidth. It shows fat tails and high kurtosis, namely, 121 , while its skewness coefficient is $-0.079$, showing almost symmetry. A normality test (Jarque-Bera) rejects the hypothesis that these returns are normal.

## 统计代写|金融统计代写Financial Statistics代考|Introduction and Motivation

A financial asset is referred to as a “safe haven” asset if it provides hedging benefits during periods of market turbulence. In other words, during periods of market stress, “safe haven” assets are supposed to be uncorrelated, or negatively correlated, with large markets slumps experienced by more traditional financial assets (typically stock or bond prices).

The financial literature identifies various asset classes exhibiting “safe haven” features: gold and other precious metals, the exchange rates of some key international currencies against the US dollar, oil and other important agricultural commodities, and US long-term government bonds.

This paper contributes to the existing literature focusing on some of the most representative “safe haven” assets, namely gold, the Swiss Franc/US dollar exchange rate, and oil. The main motivation behind this choice is twofold.

First, empirical research on these assets have attracted major attention in recent years, both from academia and from institutional investors. Second, there are some weaknesses in the applied literature that need to be addressed.

The hedging properties of gold and its monetary role as a store of value are widely documented. Jaffe $(1989)$ and Chua et al. (1990) find that gold yields significant portfolio diversification benefits. Moreover, the “safe haven” properties of gold in volatile market conditions are widely documented: See, among others, Baur and McDermott (2010), Hood and Malik (2013), Reboredo (2013), and Ciner et al. (2013).
The popular views of gold as a store of value and a “safe haven” asset are well described in Baur and McDermott (2010). As reported by these authors, the 17 th Century British Mercantilist Sir William Petty described gold as “wealth at all times and all places” (Petty 1690). This popular perception of gold spreads over centuries, reinforced by its historic links to money, and even today gold is described as “ant attractive each way bet” against risks of financial losses or inflation (Economist 2005, 2009).

Turning to the role of the Swiss Franc as a “safe haven” asset, Ranaldo and Söderlind (2010) documented that the Swiss currency yields substantial hedging benefits against a decrease in US stock prices and an increase in forex volatility. These findings corroborate earlier results (Kugler and Weder 2004; Campbell et al. 2010). More recent research documented that increased risk aversion after the 2008 global financial turmoil strengthened the “safe haven” role of the Swiss currency (Tamakoshi and Hamori 2014).

## 统计代写|金融统计代写Financial Statistics代考|A Multivariate Garch Model of Asset Returns

This section employs a well-known approach belonging to the class of Multivariate Garch estimators, namely Engle (2002) Dynamic Conditional Correlation model, in order to compute time-varying conditional correlations between asset returns. The first sub-section provides a short outline of this econometric framework. The latter sub-section presents parameters estimates and analyzes pair-wise correlation patterns between asset returns.
3.1. Engle (2002) Dynamic Conditional Correlation Model
Let $r_{t}=\left(r_{1 t}, \ldots, r_{n t}\right)$ represent a $(n \times 1)$ vector of financial assets returns at time (t). Moreover, let $\varepsilon_{t}$ $=\left(\varepsilon_{1 t}, \ldots, \varepsilon_{n t}\right)$ be a $(n \times 1)$ vector of error terms obtained from an estimated system of mean equations for these return series.

Engle (2002) proposes the following decomposition for the conditional variance-covariance matrix of asset returns:
$$H_{t}=D_{t} R_{t} D_{t}$$
where $D_{t}$ is a $(n \times n)$ diagonal matrix of time-varying standard deviations from univariate Garch models, and $R_{t}$ is a $(n \times n)$ time-varying correlation matrix of asset returns $\left(\rho_{i j}, t\right)$.

The conditional variance-covariance matrix $\left(\mathrm{H}{t}\right)$ displayed in equation [1] is estimated in two steps. In the first step, univariate Garch $(1,1)$ models are applied to mean returns equations, thus obtaining conditional variance estimates for each financial asset ( $\sigma{i t}^{2} ;$ for $\left.i=1,2, \ldots ., n\right)$, namely:
$$\sigma_{i t}^{2}=\sigma_{U i t}^{2}\left(1-\lambda_{1 i}-\lambda_{2 i}\right)+\lambda_{1 i} \sigma_{i, t-1}^{2}+\lambda_{2 i} \varepsilon_{i, t-1}^{2}$$

where $\sigma^{2}$ uit is the unconditional variance of the $i$ th asset return, $\lambda_{1 i}$ is the volatility persistence parameter, and $\lambda_{2 i}$ is the parameter capturing the influence of past errors on the conditional variance.
In the second step, the residuals vector obtained from the mean equations system $\left(\varepsilon_{t}\right)$ is divided by the corresponding estimated standard deviations, thus obtaining standardized residuals (i.e., $u_{i t}=$ $\varepsilon_{i t} / \sqrt{\sigma_{i, t}^{2}}$ for $\left.\mathrm{i}=1,2, \ldots ., n\right)$, which are subsequently used to estimate the parameters governing the time-varying correlation matrix.

More specifically, the dynamic conditional correlation matrix of asset returns may be expressed as:
$$Q_{t}=\left(1-\delta_{1}-\delta_{2}\right) \overline{\mathrm{Q}}+\delta_{1} Q_{t-1}+\delta_{2}\left(u_{t-1} u_{t-1}^{\prime}\right)$$
where $\overline{\mathrm{Q}}=\mathrm{E}\left[u_{t} u_{t}^{\prime}\right]$ is the $(n \times n)$ unconditional covariance matrix of standardized residuals, and $\delta_{1}$ and $\delta_{2}$ are parameters (capturing, respectively, the persistence in correlation dynamics and the impact of past shocks on current conditional correlations).

## 统计代写|金融统计代写Financial Statistics代考|High Frequency Data

$$r_{t}=\ln \left(\left(p_{t}^{b i d}+p_{t}^{a s k}\right) / 2\right)-\ln \left(\left(p_{t-1}^{\text {bid }}+p_{t-1}^{a s k}\right) / 2\right)$$

$$S_{d, h}=\frac{1}{s} \sum_{j=1}^{s} \mid\left(r_{d_{t}, j, t} \mid\right.$$

## 统计代写|金融统计代写Financial Statistics代考|Introduction and Motivation

Baur 和 McDermott (2010) 很好地描述了黄金作为价值储存和”避风港”盗产的流行观点。正如这些作者所报道的那样， 17 世纪的英国重商主义者威廉·㑉蒂爵士将黄金描述为“随时随地的财富”(佩蒂 1690 年)。这种对黄金的普遍看法传播 了几个世纪，并因其与货币的历史联系而得到加强，即使在今天，黄金也被描述为“对金融损失或通货膨胀风险进行”单向 押注” (Economist 2005, 2009)。

## 统计代写|金融统计代写Financial Statistics代考|A Multivariate Garch Model of Asset Returns

3.1。Engle (2002) 动态条件相关模型

Engle (2002) 对资产收益的条件方差-协方差矩阵提出了以下分解:
$$H_{t}=D_{t} R_{t} D_{t}$$

$$\sigma_{i t}^{2}=\sigma_{U i t}^{2}\left(1-\lambda_{1 i}-\lambda_{2 i}\right)+\lambda_{1 i} \sigma_{i, t-1}^{2}+\lambda_{2 i} \varepsilon_{i, t-1}^{2}$$

$$Q_{t}=\left(1-\delta_{1}-\delta_{2}\right) \overline{\mathrm{Q}}+\delta_{1} Q_{t-1}+\delta_{2}\left(u_{t-1} u_{t-1}^{\prime}\right)$$

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