### 统计代写|金融统计代写financial statistics代考|Financial econometrics

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• Statistical Inference 统计推断
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• (Generalized) Linear Models 广义线性模型
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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|A survey of volatility

In recent years, the field of financial econometrics has seen tremendous gains in the amount of data available for use in modeling and prediction. Much of this data is very high frequency, and even “tick-based,” and hence falls into the category of what might be termed “big data.” The availability of such data, particularly that available at high frequency on an intra-day basis, has spurred numerous theoretical advances in the areas of volatility/risk estimation and modeling. In this chapter, we discuss key such advances, beginning with a survey of numerous nonparametric estimators of integrated volatility. Thereafter, we discuss testing for jumps using said estimators. Finally, we discuss recent advances in testing for co-jumps. Such co-jumps are important for a number of reasons. For example, the presence of co-jumps, in contexts where data has been partitioned into continuous and discontinuous (jump) components, is indicative of (near) instantaneous transmission of financial shocks across different sectors and companies in the markets; and hence represents a type of systemic risk. Additionally, the presence of co-jumps across sectors, say, suggests that if jumps can be predicted in one sector, then such predictions may have useful information for modeling variables such as returns and volatility in another sector. As an illustration of the methods discussed in this chapter, we carry out an empirical analysis of DOW and NASDAQ stock price returns.

## 统计代写|金融统计代写financial statistics代考|Introduction

The importance of integrated volatility, jumps, and co-jumps in the financial econometrics literature and in terms of successful risk management by investors is quite obvious now, given the amount of research that has gone into this field. Measures of integrated volatility are crucial given the advent of numerous volatility-based derivative products traded in financial markets while tests for jumps are essential in modeling and predicting volatility and returns. Tests of co-jumps on the other hand are meaningful indicators of transmission of financial shocks across different sectors, companies, and markets. The rationale behind this chapter is to discuss some of recent advances in jump and co-jump testing methodology and measurement of integrated volatility, and the properties thereof, in a way which would help both researchers and practitioners in application of such econometric methods in finance. We begin by surveying the most widely used integrated volatility measures, jump and co-jump tests, followed by an empirical analysis using high-frequency intraday stock prices of DOW 30 companies and ETFs.

Daily integrated volatility is unobservable. Econometricians have developed numerous measures which estimate price fluctuations in a variety of ways. One of the earliest measures is the realized volatility $(R V)$ in Andersen et al. (2001). However, this measure does not separate jump variation from variation due to continuous components. Barndorff-Nielsen and Shephard (2004) use the product of adjacent intra-day returns to develop jump robust measures bipower variation $(B P V)$ and tripower variation $(T P V)$. One of the more recent techniques of separating out the jump component is the truncation methodology which essentially eliminates returns which are above a given threshold as in Corsi et al. (2010) and Aït-Sahalia et al. (2009). One important caveat of high-frequency data is the existence of market microstructure noise which creates a bias in the estimation procedure. Zhang et al. (2005, 2006) and Kalnina and Linton (2008) solved this problem with noise robust volatility estimators.
In Duong and Swanson (2011), the authors find that $22.8 \%$ of the days during the 1993-2000 period had jumps while $9.4 \%$ of the days during the 2001-2008 period had jumps. The existence of jumps in financial markets is obvious, which has led many researches to develop techniques which can test for jumps. Jump diffusion is pivotal in analyzing asset movement in financial econometrics and developing jump tests to identify jumps has been

the focus for many theoretical econometricians in past few years. Using the ratio of $B P V$ and estimated quadratic variation, Barndorff-Nielsen and Shephard (2006) construct a nonparametric test for the existence of jumps. Lee and Mykland (2007) on the other hand propose tests to detect the exact timing of jumps at the intra-day level while Jiang and Oomen (2008) provide a “swap variance” approach to detect the presence of jumps. Instead of the more widely used “fixed time span” tests, Corradi et al. (2014, 2018) develop “long time span” jump test, building on earlier work by Aït-Sahalia (2002).
Co-jump tests which are instrumental in identifying systemic risk across multiple sectors and markets are relatively new in the literature. Co-jumps reflect market correlation and have important implication for portfolio management and risk hedging. There are tests which utilize univariate jump tests to identify co-jumps among multivariate processes (Gilder et al., 2014), while co-jump tests can also be directly applied to multiple price processes (see, e.g., Jacod and Todorov, 2009, Bandi and Reno, 2016, Bibinger and Winkelmann, 2015, Caporin et al., 2017). Gnabo et al. (2014) propose a co-jump test based on bootstrapping methods, Bandi and Reno (2016) develop a nonparametric infinitesimal moments method to detect co-jumps between asset returns and volatilities and Caporin et al. (2017) build a co-jump test based on the comparison between smoothed realized variance and smoothed random realized variation.

## 统计代写|金融统计代写financial statistics代考|Setup

We represent the log-price of a financial asset at continuous time $t$, as $Y_{r}$. It is assumed that the log-price is a Brownian semimartingale process with jumps and it can be denoted as ${ }^{a}$ :
$$Y_{t}=Y_{0}+\int_{0}^{t} \mu_{s} d s+\int_{0}^{t} \sigma_{s} d W_{s}+J_{t}$$
In (1) $\mu_{s}$ the drift term is a predictable process, $\sigma_{s}$ the diffusion term is a cádlág process, $W_{s}$ is a standard Brownian motion, and $J_{t}$ is a pure jump process. $J_{t}$ can be defined as the sum of all discontinuous log-price movements up to time $t$,
$$J_{t}=\sum_{s \leq t} \Delta Y_{s}$$

When this jump component is a finite activity jump process, i.e., a compound Poisson process (CPP), then
$$J_{t}=\sum_{j=1}^{N_{t}} \xi_{j}$$
where $N_{t}$ is a Poisson process with intensity $\lambda$, the jumps occur at the corresponding times given as $\left(\tau_{j}\right){j=1, \ldots, N{t}}$, and $\xi_{j}$ refers to i.i. $d$ random variables measuring the size of jumps at time $\tau_{j}$. The finite activity jump assumption has been widely used in financial econometrics literature. Log-price $Y_{t}$ can be decomposed into a continuous Brownian component $Y_{t}^{c}$ and a discontinuous component $Y_{t}^{d}$ (due to jumps). The “true variance” of process $Y_{t}$ can be given as:
$$Q V_{t}=[Y, Y]{t}=[Y, Y]{t}^{c}+[Y, Y]{t}^{d}$$ where $Q V$ stands for quadratic variation. The variation due to the continuous component is $$[Y, Y]{t}^{c}=\int_{0}^{t} \sigma_{s}^{2} d s,$$
and the variation due to the discontinuous jump component is
$$[Y, Y]{t}^{d}=\sum{j=1}^{N_{t}} \xi_{j}^{2}$$
Integrated volatility which is the continuous part of $Q V$ is denoted as:
$$I V_{t}=\int_{t-1}^{t} \sigma_{s}^{2} d s, \quad t=1, \ldots, T$$
where $I V$ is the (daily) integrated volatility at day $t$. Since $I V$ is unobservable, different realized measures of integrated volatility are used as its substitute. The presence of market frictions in high-frequency financial data has been documented in recent literature. To take care of this, the observed log-price process $X$ can then be given as
$$X=Y+\epsilon$$
where $Y$ is the latent log price and $\epsilon$ captures market microstructure noise. We consider $M$ equi-spaced intra-daily observations for each of $T$ days for process $\mathrm{X}$ which leads to a total of $M T$ observations, i.e.,
$$X_{t+j / M}=Y_{t+j / M}+\epsilon_{t+j / M}, t=0, \ldots, T \text { and } j=1, \ldots, M$$
where $\epsilon$ follows a zero mean independent process. The intra-daily return or increment of process $X$ follows:
$$\Delta_{j} X=X_{t+(j+1) / M}-X_{t+j / M}$$

## 统计代写|金融统计代写financial statistics代考|Setup

Ĵ吨=∑s≤吨Δ是s

Ĵ吨=∑j=1ñ吨Xj

[是,是]吨d=∑j=1ñ吨Xj2

X=是+ε

X吨+j/米=是吨+j/米+ε吨+j/米,吨=0,…,吨 和 j=1,…,米

ΔjX=X吨+(j+1)/米−X吨+j/米

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