# Prediction of High-Frequency Economic Data Based on Stochastic Fluctuation Model

Authors: Xiaoyang Zhang; Tianxiang Qi; Dong-Joo Kim
Journal: Security and Communication Networks
 (2022)

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## Abstract

In order to improve the effect of economic high-frequency data analysis, this paper combines the stochastic fluctuation model to carry out the forecast analysis of economic high-frequency data. Moreover, this paper uses the spider web model for data processing and makes a preliminary judgment on the extent to which futures/stock prices lead the spot through error correction model, impulse response, and variance decomposition. Furthermore, this paper studies the contribution of futures/stock price changes to the effective price through the common factor model so as to obtain its price discovery efficiency. In addition, this paper judges the efficiency of price discovery by studying the contribution of futures/stock price changes to the effective price. Finally, based on the research idea of futures/stock price discovery efficiency, an intelligent data analysis model is constructed. The experimental study shows that the economic high-frequency data prediction model based on the stochastic fluctuation model proposed in this paper has a good effect on the analysis and prediction of economic high-frequency data.


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## Body

## 1. Introduction

The focus of the research on the market microstructure is the influence of the market trading system on the price formation process, so it is necessary to carefully examine the trading data and order book data generated during the trading day. This type of data is called high-frequency data or intraday data. Generally speaking, the information in the financial market is a continuous process of affecting price changes, and the discrete model will inevitably lead to the loss of information. The lower the data frequency is, the more information is lost.Risk and reward are what people care about most when investing in assets. After a comprehensive analysis of risk and return, investors choose suitable financial assets for rational investment according to their affordability. Therefore, accurate and reasonable risk aversion has become the focus of research [1]. Theorists have always used the volatility of assets as a measure of risk. Under the assumption of satisfying the efficient market environment, when investors choose financial products, they have certain expectations about their price trends and future income. However, some uncertain market factors often cause the real price trend to deviate from this expectation, which leads to market risks and makes the price deviate from investors’ expectations [2]. Volatility is a measure of risk, and the magnitude of volatility can express the uncertainty of asset prices. The greater the volatility, the greater the deviation between the expected return and the actual return. Therefore, there is a greater investment risk. The smaller the volatility, the smaller the deviation between the expected return of the asset and the actual return, and the smaller the corresponding risk [3].There are two very classical methods for measuring fluctuation in time series analysis: one is Engle’s ARCH model and its extended form, and the other is Taylor’s SV model and its extended form. These two kinds of methods have remarkable effect on low-frequency time series data, and their application is very successful. Due to the successful performance of ARCH model and SV model in the field of low-frequency financial time series modeling, people will naturally think: can ARCH model and SV model be directly applied to high-frequency time series? If not, we can modify and improve these two models. Can we apply them to high-frequency field? Andersen and Bollerslev (1997), the research pioneers in the field of high-frequency data volatility measurement in the stock market, have conducted a large number of empirical studies. Afterwards, it was found that the ARCH model and the SV model were unable to explain the driving factors of fluctuations and could not explain the reasons for continuous fluctuations. They could describe some features of high-frequency time series to a certain extent, but their accuracy was not high and the effect was unsatisfactory. Therefore, it is necessary to consider the statistical characteristics of high-frequency time series that are different from low-frequency time series and build a reasonable model for them according to the statistical characteristics.In order to improve the effect of high-frequency economic data analysis, this paper combines the stochastic fluctuation model to carry out the prediction and analysis of high-frequency economic data and constructs an intelligent economic data analysis model to improve the effect of subsequent economic data analysis.


## 2. Related Work

With the continuous improvement of data collection and processing capabilities and the increasing availability of high-frequency data, the research on the realized volatility measurement based on high-frequency financial data has become a hot research topic in academia. Reference [4] proposes the concept of realized volatility (RV) and believes that the real volatility can be estimated by the sum of squares of intraday high-frequency returns. The realized variance is a consistent asymptotic estimator of the integral volatility as the sampling tends to infinity. This method makes the volatility change from an unobservable hidden variable to an explicit variable that can be directly measured by nonparametric methods, without the time lag brought by the classical algorithm, and can better measure the change of volatility in the time series. It has set off a wave of research on high-frequency data in the field of volatility. Due to the characteristics of the realized volatility variable, an important use is to use it as a reference for evaluating various previous conditional variance models, such as the ARCH model and its various extended forms. Reference [5] evaluated the accuracy of intra day performance data prediction of the arch model. Reference [6] compared the prediction effect of 330 ARCH extension models on IBM stocks. Another important use of realized volatility is that it can be used as time series observations of volatility to test various characteristics of volatility and predict the future. The traditional ARCH model needs to fit the model in order to obtain the potential volatility variables, which has great limitations in analysis. Reference [7] summarizes various characteristics of realized volatility based on the high-frequency data of the Dow Jones Industrial Index and proposes an ARFIMAX model for this sequence. Reference [8] proves ARFIMAX by comparing the forecasting effect of this model with FIGARCH and FIEGARCH. The model is significantly better than the latter two models. Reference [9] uses the Vector Error Correction Model (VECM) to forecast the volatility of the exchange rate market, which proves that the prediction effect of the model algorithm is significantly better than that of ARCH(1), GARCH(1,1), and the exponential smoothing method in RiskMetrics. However, the realized measure has to face the problems caused by the noise of market microstructure, the instantaneous jumping behavior of stock price, and the nontrading time period in practical application. One angle to address these issues is to investigate methods of various realized measures to reduce biased estimates of volatility. Reference [10] proposed Bipower varimion (BPV) as an estimate of the continuous component of volatility to overcome the impact of instantaneous and high-intensity jumping behavior of prices in high-frequency data on returns. Reference [11] conducted a Monte Carlo simulation test with BPV and found that the volatility jump component accounted for about 7% of the overall volatility. Literature [12] found that there are obvious price jumps in foreign exchange, stock indexes, and bonds, and combined RV and BPV to extract the volatility price jump component, and established HAR. The RV-CJ model proves that the model can improve the accuracy of out-of-sample prediction. Reference [13] proposes a realized kernel variance (realized kernel), which considers the correlation between intraday returns, which can effectively reduce the impact of microscopic noise on traditional RV. Reference [14] proposed the realized range volatility (Realized Range, RR) based on the range theory, which improved the utilization of high-frequency information and effectively controlled the influence of microscopic noise. Reference [15] proposed that Median Realized Volatility (MedRV) can effectively control the impact of stock price jumping behavior on volatility estimation and proved that this estimator has better small sample adaptability and convergence than BPV. Reference [16] proposed an optimal combination of realized volatility and overnight squared return for the non-24-hour stock market, forming an optimal linear combination realized volatility (LCRV). Reference [17] proposed a two-scale RV (two-scale Realized Volatility) based on different time scales and used subsampling for bias correction to weaken the negative impact of market microstructure noise. Reference [18] proposed a weighting method of adding the sample variance of the overnight rate of return and the sample variance of the rate of return calculated by the “opening price and closing price” of the day into the RV to eliminate the influence of the overnight rate of return. Another way to solve the noise is to study the optimal sampling frequency of high-frequency data and reduce the problem of microstructure noise by reducing the frequency. Reference [19] proposed the volatility characteristic map method; the principle is that the minimum sampling frequency that makes the characteristic curve stable is the optimal frequency. Reference [20] proposed a method for calculating the optimal sampling frequency of realized volatility based on minimizing MSE. Reference [21] studied the optimal frequency selection problem based on RV linear prediction in time-varying microstructure noise processes.


## 3. Stochastic Volatility Data Analysis

Any probability distribution on the parameter spaceΘ is called a prior distribution. In this paper, πθ is used to represent the prior distribution of θ. Among them, πθ is the probability function of the random variable θ.After obtaining the samplex, the posterior distribution of θ is the conditional distribution of θ under the given condition of X=x, which is denoted as πθ|x. For the case of density, its density function is(1)πθ|x=hx,θmx=fx|θπθ∫Θfx|θπθdθ.Among them,hx,θ=fx|θπθ, which is the joint distribution of X and θ. However, mx=∫Θhx,θdθ=∫Θfx|θπθdθ is the marginal distribution of X.Equation (1) is the density function form of the Bayesian formula, which concentrates all the information about θ in the three kinds of information: population, sample, and prior. However, it is the result obtained after excluding all information unrelated to θ. If θ is a discrete random variable, the prior distribution is represented by the prior distribution sequence πθi,i=1,2,…. At this point, the posterior distribution has the following form:(2)πθi|x=fx|θiπθi∑ifx|θiπθii=1,2,….If the populationX from which the sample comes is also discrete, we only need to regard the density function fx|θi as the probability PX=x|θi of the event X=x|θi. At this time, the sample X is in the form of a joint density function, and formula (2) is the Bayesian formula.When the sample distribution is known, for theoretical needs, the prior distribution of the parameters is usually selected as the conjugate prior distribution, which is defined as follows.We assume thatℱ represents a family of distributions consisting of the prior distribution π0 of θ. If for any π∈ℱ and sample value x, the posterior distribution πθi|x still belongs to ℱ, then ℱ is said to be a conjugate prior distribution family. In this paper, the calculation of the posterior distribution can be simplified by using the conjugate prior distribution, and the calculation formula of the posterior density is given by(3)πθ|x=fx|θπθfmx=fx|θπθ∫Θfx|θπθdθ.Among them,πθ is the prior density of θ, fmx is the marginal density of X, and fx|θ is the density function of the sample. In some cases, fx|θ can be replaced by the likelihood function Lθ|x. If it is regarded as the probability density of random variable X, it is replaced by fx|θ. If it is regarded as the likelihood function of θ, it is replaced by Lθ|X. Since fmx has nothing to do with θ, this paper treats fmx as a constant independent of θ; thus, we get(4)πθ|x=Lθ|xπθfmx∝Lθ|xπθ.Among them, “∝” means “proportional to”; that is, the left- and right-hand sides of (4) differ only by a constant factor, and this constant has nothing to do with θ.Therefore, for the case of the conjugate prior distribution, the solution of the posterior density is carried out according to the following steps:(1)
In this paper, the kernel of the likelihood functionLθ|x of θ is written, that is, the factors in Lθ|x that are only related to the parameter θ, and then the kernel of the prior density πθ is written, that is, the factors in πθ that are only related to the parameter θ.(2)
Similar to equation (3), the kernel of the posterior density is written; that is,(5)πθ|x∝Lθ|xπθ∝Lθ|xNucleus of⋅πθNucleus of.That is to say, the “kernel of the posterior density” is the product of the “kernel of the likelihood function” and the “kernel of the prior density.”(3)
A regularization constant factor (which can be related tox) is added to the right-hand side of equation (4), and the posterior density can be obtained.It is worth noting that the above simplified method is only valid for the case where the prior distribution is a conjugate distribution, and it does not consider the case where the prior part is a nonconjugate prior. After obtaining the kernel of the posterior distribution, if you cannot judge the type of the posterior distribution, you will not know how to add a regularization constant factor, and you will not be able to calculate the posterior density. At this time, the posterior density can only be calculated by formula (1).After obtaining the posterior distribution of the parameterθ, the posterior mean of θ can be used as an estimator for the parameter θ, as follows:(6)θ^=Eθ|x=∫Θθπθ|xdθ=∫Θθfx|θπθdθ∫Θfx|θπθdθ.This process involves evaluating integrals in expressions. However, in some practical problems, the abovementioned integrals are high-dimensional integrals, and the posterior distribution of the parameterθ does not show an expression, which makes the numerical integration algorithm difficult to work. In recent years, a series of advanced computing methods have been developed to solve the difficult problem of Bayesian computing to a large extent. Among them, the most widely used is the Markov Chain Monte Carlo (MCMC) sampling method, such as Gibbs sampling method. Before introducing the MCMC method, we first give the posterior density functions of the parameters of the standard SV model and SV-T model.When performing Bayesian inference on the model, we need to set the prior distribution of the parameters first. When choosing a prior distribution, two aspects should be considered: rationality and ease of computation. In Bayesian inference, conjugate prior distributions are often used. In the standard SV model, considering the value range of the parameterϕ, we set up ϕ1=ϕ+1/2. We refer to the priors on the parameters of the standard SV model given by Kim et al. (1998) as follows:(7)μ∼N0,100,ϕ1∼Be20,1.5,τ∼Ga2.5,0.025,h0∼Nμ,τ−1.Among them,Be represents the beta distribution and Ga represents the gamma distribution. It is not difficult to obtain that the prior mean of the persistence parameter ϕ is equal to 0.86, and the prior mean of the precision parameter τ is equal to 100. Compared with the parameters of the SV model in practice, the set prior information is closer to the actual situation, which has a certain rationality. We deduce the posterior density of each parameter in the model according to the method in formula (5), as follows:(1)
The posterior density ofμ is as follows:(8)πμ|ϕ,τ,h,y=πμ,ϕ,τ,h|yπϕ,τ,h|y∝exp−μ22exp−τ2∑t=1nht−μ−ϕht−1−μ2∝exp−121+n1−ϕ2τμ−1−ϕτ2+2n1−ρ2τ∑t=1nht−ϕht−12∝N1−ϕτ2+2n1−ϕ2τ∑t=1nht−ϕht−12,1+n1−ϕ2τ.(2)
The posterior density ofϕ is as follows:(9)πϕ|μ,τ,h,y=πμ,ϕ,τ,h|yπμ,τ,h|y∝1+ϕ2191−ϕ2exp−τ2∑t=1nht−μ−ϕht−1−μ2∝1+ϕ2191−ϕ20.5exp−τ2∑t=1nht−1−μ2ϕ−∑t=1nht−μht−1−μ∑t=1nht−1−μ22.(3)
The posterior density ofτ is as follows:(10)πτ|μ,ϕ,h,y=πμ,ϕ,τ,h|yπμ,ϕ,h|y∝τ1.5cxp−0.025ττn+1/2exp−τ2∑t=1nht−μ−ϕht−1−μ2+h02∝τn+4/2exp−τ2∑t=1nht−μ−ϕht−1−μ2+h02+120.(4)
The posterior density ofht is as follows:(11)πht|μ,ϕ,h−t,y∝exp−yt22expht−121+ϕ2τht2−2τμ+ϕht−1−μ+ϕht−1−ϕμ1−ϕht+ht.Compared with the unknown parametersμ, φ, τ, and h of the SV model, the unknown parameters of the SV-T model have more degrees of freedom parameters ω. The prior of the degree of freedom ω of the SV-T model is ω∼χ28, and the priors of other parameters μ, φ, τ, and h are consistent with those of the standard SV model. The posterior distribution density of each parameter of the SV-T model is as follows.(1)
The posterior density ofμ is as follows:(12)πμ|ϕ,τ,ω,h,y=πμ,ϕ,τ,ω,h|yπϕ,τ,ω,h|y∝exp−121+n1−ϕ2τμ−1−ϕτ2+2n1−ρ2τ∑t=1nht−ϕht−12∝N1−ϕτ2+2n1−ϕ2τ∑t=1nht−ϕht−12,1+n1−ϕ2τ.(2)
The posterior density ofϕ is as follows:(13)πϕ|μ,τ,ω,h,y=πμ,ϕ,τ,ω,h|yπμ,τ,ω,h|y∝1+ϕ2191−ϕ20.5exp−τ2∑t=1nht−1−μ2ϕ−∑t=1nht−μht−1−μ∑t=1nht−1−μ22.(3)
The posterior density ofτ is as follows:(14)πτ|μ,ϕ,ω,h,y=πμ,ϕ,τ,ω,h|yπμ,ϕ,ω,h|y∝τn+4/2exp−τ2∑t=1nht−μ−ϕht−1−μ2+h02+120.(4)
The posterior density ofω is as follows:(15)πω|μ,ϕ,τ,h,y=πμ,ϕ,τ,ω,h|yπμ,ϕ,τ,h|y∝ω6−n/2exp−ω2Γω+1/2Γω/2n∏t=1n1+yt2exp−htω−ω+1/2.(5)
The posterior density ofht is as follows:(16)πht|μ,ϕ,τ,ω,h−t,y∝1+yt2exp−htω−ω+1/2exp−12τ1+ϕ2ht2−2τμ+ϕht−1−μ+ht+1−μϕht+ht.It can be seen that the posterior distribution density of each parameter of the standard SV model and the SV-T model is a high-dimensional multivariate distribution, and the Bayesian calculation can be realized by the Markov Chain Monte Carlo (MCMC) sampling method. In the following, we briefly introduce the Markov chain Monte Carlo method and the Gibbs sampling method.We assume thatXn,n≥0 is a random process that takes only countable values. If Xn=i, it means that the state of the process at time n is i, and S=0,1,2,… means the state set. If the following conditions are satisfied for any n, then Xn,n≥0 is called a discrete-time Markov chain, which is abbreviated as Markov chain.(17)PXn+1=j|X0=i0,…,Xn−1=in−1,Xn=i=PXn+1=j|Xn=i.It can be seen from the above formula that for the random processXn,n≥0, the future state Xn+1=j is only related to the current state Xn=i, and it has nothing to do with the past state Xk=ik,k≤n−1.The conditional probabilityPXn+1=j|Xn=i is called the single-step transition probability of the Markov chain. If the transition probability has nothing to do with n and is a fixed value, then the Markov chain is said to have a stable transition probability, which is denoted as pij. A Markov chain with a smooth transition probability is also called a time-homogeneous Markov chain, and P=pij,∀i,j∈S is called the transition probability matrix of the Markov chain.The conditional probabilityPXn+1=j|Xn=i is called the single-step transition probability of the Markov chain. If the transition probability has nothing to do with n and is a fixed value, then the Markov chain is said to have a stable transition probability, which is denoted as pij. A Markov chain with a smooth transition probability is also called a time-homogeneous Markov chain, and P=pij,∀i,j∈S is called the transition probability matrix of the Markov chain.


### 3.1. Stationarity

We assume that the Markov chain has a transition probability matrixP=pij. If a probability distribution π=πi,i≥0 satisfies π=∑iπipij, then it is called the stationary distribution of the Markov chain.It is not difficult to see that if the initial stateX0 of the process has a stationary distribution π=πi,i≥0, that is, PX0=j=πj, then there is(18)PX1=j=∑iPX1=j|X0=iPX0=i=∑iπipij=πj.By induction, we get(19)PXn=j=∑iPXn=j|Xn−1=iPXn−1=i=∑iπipij=πj.Thus, for alln,Xn has the same distribution π; that is, Xn,n≥0 is stationary as a random process.


### 3.2. Irreducibility

We assume that a Markov chainXn with countable state space S and transition probability matrix P=pij is called irreducible. If for any two states i,j∈S, the probability of this chain transitioning from state i to state j is positive. That is, for a certain n≥1, there is(20)Pijn=PXn=j|X0=i>0.By definition, a Markov chain with “irreducibility” means that any other state can always be reached from any state.


### 3.3. Aperiodic

We assume that a statei of a Markov chain has period k. If it is possible to return to state i after a multiple of k steps, that is(21)ki=gcdn:PXn=j|X0=i>0.Among them,gcd denotes the “maximum number of conventions.” If the maximum number of times it returns to any state is 1, a Marshall chain is acyclic. A nonperiodic Marschall chain is guaranteed not to get trapped in a loop.


### 3.4. Normal Relapse

For always-returning statei, we assume that Ti=infn≥1:Xn=i|X0=i is the moment of returning to state i for the first time. If it satisfies the following formula, state i is said to be normal. When μi=∞, the state i is said to be zero-returning.(22)μi=ETi<∞.


### 3.5. Ergodicity

We assume that the state of a Markov chain is called ergodic. If it is aperiodic and normal, and all states of the Markov chain are traversal, then the Markov chain is said to be ergodic.To sum up, from the basic theory of Markov chain, we can know that the Markov chain we need to construct must be irreducible, normal recursive, and aperiodic. However, a Markov chain that satisfies these regular conditions has a unique stationary distribution.The theoretical basis for Bayesian analysis using the Markov Chain Monte Carlo MCMC method is based on some of the following limit theorems.Theorem 1 is as follows. We assume thatXn,n≥0 is a Markov chain with a countable state space S whose transition probability matrix is P. Further, we assume that it is irreducible, aperiodic, and has a stationary distribution π=πi:i∈S. Then, for any initial distribution π of X0, we have(23)∑j∈SPXn=j−πj⟶0,n⟶∞.In other words, for largern, the distribution of Xn will be close to π. For general state spaces, similar results hold. Under suitable conditions, when n⟶∞, the distribution of Xn will converge to π.Theorem 2 (the theorem of large numbers for the Markov chain) is as follows: We assume thatXn,n≥0 is a Markov chain with countable state space S, and its transition probability matrix is P. Further, we assume that it is irreducible and has a flat distribution with a stationary distribution π=πi:i∈S. Then, for any initial distribution of any bounded function h:S⟶R and initial value X0, we have(24)1n∑i=0n−1hXi⟶∑jhjπj,n⟶∞.When the state space is uncountable, the Markov chainXn,n≥0 is irreducible and there is a stationary distribution π; there is also(25)1n∑i=0n−1hXi⟶∫Shxdπx,n⟶∞.The conclusion of this theorem is very useful. For example, we assume a probability distributionπ on a set S and a real function hθ on S. We assume that we want to calculate the integral μ=∫Shθdπθ|x ; then, we can construct a Markov chain whose state space is S. At the same time, its stationary distribution π is the target posterior distribution π⋅|x starting from an initial value θ0, running this chain for a period of time, such as 0,1,2,…,n−1, and then generating random samples θ0,θ1,θ2,…,θn−1. From the Markov chain theorem of large numbers, it can be known that(26)μ¯n=1n∑i=0n−1hθi.The above formula is a consistent estimate of the required integralμ. When it is difficult to sample directly from the posterior distribution πθ|x, a suitable Markov chain is constructed, and its stationary distribution is the target sampling distribution. Furthermore, the sample paths of the Markov chain are used to calculate the distribution features of interest. This method is called the Markov Chain Monte Carlo (MCMC) method.In practical computing, most Bayesian computing problems need to deal with a high-dimensional multivariate posterior distribution. Randomly generating samples from such high-dimensional distributions is often difficult. However, Gibbs sampling is particularly suitable for this situation. The most attractive part of this method is that it can generate an irreducible and aperiodic Markov chain with the target high-dimensional posterior distribution as a stationary distribution only by sampling from a series of univariate distributions.We assume thatX=X1,…,Xd is a random variable in ℝd, and its joint distribution fx is the target sampling distribution. It is a random variable of dimension d-1, as follows:(27)X−j=X1,…,Xj−1,Xj+1,…,Xd.Subsequently, the conditional density ofXj|X−j is denoted as fxj|x−j; then, the Gibbs sampling method is to extract candidate points from these d distributions, thus solving the difficulty of direct sampling from fx. The specific algorithm is as follows:(1)
The algorithm gives the initial value ofX0.(2)
Fort=1,2,…,T:(i)
The algorithm sets upt=1,2,…,T.(ii)
For each componentx1=X1t−1,(a)
The algorithm extracts the candidate pointXj∗t from fXj|x−j;(b)
xj=Xj∗t is updated.(iii)
The algorithm establishesXt=X1∗t,…,Xd∗t.(iv)
Ast increases, the algorithm repeats step (i).In algorithm step (ii), each component is updated sequentially:(28)x1t∼fx1|x2t−1,…,xdt−1,x2t∼fx2|x1t,x3t−1,…,xdt−1,…xdt∼fxd|x1t,…,xd−1t.It is relatively straightforward to draw samples from the univariate distributionfxj|x1t,x2t,…,xj−1t,xj+1t−1,…,xdt−1. Because fxj|x−j∝fx, except xj which is a variable, all other variables are constants.What is special about Gibbs sampling is that a fully conditional distribution can determine a unique joint distribution. The well-known Hammersley–Clifford theorem reveals this law.We assume that the joint density distribution of the random variableX=X1,…,Xd is fx1,…,xd, and the edge density function of Xi is fixii=1,…,d. If fixi>0i=1,…,d, which means fx1,…,xd>0, then the joint density distribution f is said to satisfy the positive condition.Theorem 3 (Hammersley–Clifford theorem) is as follows: Under positive conditions, the joint density distributionf satisfies the following:(29)fx1,…,xd∝∏j=1dfxj|x1,…,xj−1,xj+1′,…,xd′fxj′x1,…,xj−1,xj+1′,…,xd′.For anyx and x′ in the support set S of f, we have(30)fx1,…,xd=fxd|x1,…,xd−1fx1,…,xd−1=fxd|x1,…,xd−1fxd′|x1,…,xd−1fx1,…,xd−1,xd′=fxd|x1,…,xd−1fxd′|x1,…,xd−1fxd−1|x1,…,xd−2,xd′fxd−1′|x1,…,xd−2,xd′×fx1,…,xd−1′,xd′=…=∏j=1dfxj|x1,…,xj−1,xj+1′,…,xd′fxj′|x1,…,xj−1,xj+1′,…,xd′fx1′,…,xd′.Taking the Gibbs sampling of the SV model as an example, we assume that the parameter setθ=μ,ϕ,τ and the initial value of the logarithmic volatility h, which is denoted as Ω0=μ0,ϕ0,τ0,h0. Then, a single Gibbs sampling is as follows:(1)
The algorithm extractsh1 from πht|μ0,ϕ0,τ0,h−t0,y.(2)
The algorithm extractsτ1 from πτ|μ0,ϕ0,h1,y.(3)
The algorithm extractsφ1 from πϕ|μ0,τ1,h1,y.(4)
The algorithm extractsμ1 from πμ|ϕ1,τ1,h1,y.After the above iterations, the initial value is updated toΩ1=μ1,ϕ1,τ1,h1, and by repeating the Gibbs sampling process k times, we can get Ω1,Ω2,…,Ωk. Under the condition of regularity, when the number of iterations k is large enough, Ωk is equivalent to a random draw in the posterior joint distribution. In practice, Gibbs sampling requires enough iterations to reach a stationary state. However, the sample composed of m data before reaching the stationary state after discarding Gibbs sampling is Ωm,Ωm+1,…,Ωk. In the Monte Carlo estimation of parameters, the posterior mean of the parameter is estimated by the mean of the following k-m samples.


## 3.1. Stationarity

We assume that the Markov chain has a transition probability matrixP=pij. If a probability distribution π=πi,i≥0 satisfies π=∑iπipij, then it is called the stationary distribution of the Markov chain.It is not difficult to see that if the initial stateX0 of the process has a stationary distribution π=πi,i≥0, that is, PX0=j=πj, then there is(18)PX1=j=∑iPX1=j|X0=iPX0=i=∑iπipij=πj.By induction, we get(19)PXn=j=∑iPXn=j|Xn−1=iPXn−1=i=∑iπipij=πj.Thus, for alln,Xn has the same distribution π; that is, Xn,n≥0 is stationary as a random process.


## 3.2. Irreducibility

We assume that a Markov chainXn with countable state space S and transition probability matrix P=pij is called irreducible. If for any two states i,j∈S, the probability of this chain transitioning from state i to state j is positive. That is, for a certain n≥1, there is(20)Pijn=PXn=j|X0=i>0.By definition, a Markov chain with “irreducibility” means that any other state can always be reached from any state.


## 3.3. Aperiodic

We assume that a statei of a Markov chain has period k. If it is possible to return to state i after a multiple of k steps, that is(21)ki=gcdn:PXn=j|X0=i>0.Among them,gcd denotes the “maximum number of conventions.” If the maximum number of times it returns to any state is 1, a Marshall chain is acyclic. A nonperiodic Marschall chain is guaranteed not to get trapped in a loop.


## 3.4. Normal Relapse

For always-returning statei, we assume that Ti=infn≥1:Xn=i|X0=i is the moment of returning to state i for the first time. If it satisfies the following formula, state i is said to be normal. When μi=∞, the state i is said to be zero-returning.(22)μi=ETi<∞.


## 3.5. Ergodicity

We assume that the state of a Markov chain is called ergodic. If it is aperiodic and normal, and all states of the Markov chain are traversal, then the Markov chain is said to be ergodic.To sum up, from the basic theory of Markov chain, we can know that the Markov chain we need to construct must be irreducible, normal recursive, and aperiodic. However, a Markov chain that satisfies these regular conditions has a unique stationary distribution.The theoretical basis for Bayesian analysis using the Markov Chain Monte Carlo MCMC method is based on some of the following limit theorems.Theorem 1 is as follows. We assume thatXn,n≥0 is a Markov chain with a countable state space S whose transition probability matrix is P. Further, we assume that it is irreducible, aperiodic, and has a stationary distribution π=πi:i∈S. Then, for any initial distribution π of X0, we have(23)∑j∈SPXn=j−πj⟶0,n⟶∞.In other words, for largern, the distribution of Xn will be close to π. For general state spaces, similar results hold. Under suitable conditions, when n⟶∞, the distribution of Xn will converge to π.Theorem 2 (the theorem of large numbers for the Markov chain) is as follows: We assume thatXn,n≥0 is a Markov chain with countable state space S, and its transition probability matrix is P. Further, we assume that it is irreducible and has a flat distribution with a stationary distribution π=πi:i∈S. Then, for any initial distribution of any bounded function h:S⟶R and initial value X0, we have(24)1n∑i=0n−1hXi⟶∑jhjπj,n⟶∞.When the state space is uncountable, the Markov chainXn,n≥0 is irreducible and there is a stationary distribution π; there is also(25)1n∑i=0n−1hXi⟶∫Shxdπx,n⟶∞.The conclusion of this theorem is very useful. For example, we assume a probability distributionπ on a set S and a real function hθ on S. We assume that we want to calculate the integral μ=∫Shθdπθ|x ; then, we can construct a Markov chain whose state space is S. At the same time, its stationary distribution π is the target posterior distribution π⋅|x starting from an initial value θ0, running this chain for a period of time, such as 0,1,2,…,n−1, and then generating random samples θ0,θ1,θ2,…,θn−1. From the Markov chain theorem of large numbers, it can be known that(26)μ¯n=1n∑i=0n−1hθi.The above formula is a consistent estimate of the required integralμ. When it is difficult to sample directly from the posterior distribution πθ|x, a suitable Markov chain is constructed, and its stationary distribution is the target sampling distribution. Furthermore, the sample paths of the Markov chain are used to calculate the distribution features of interest. This method is called the Markov Chain Monte Carlo (MCMC) method.In practical computing, most Bayesian computing problems need to deal with a high-dimensional multivariate posterior distribution. Randomly generating samples from such high-dimensional distributions is often difficult. However, Gibbs sampling is particularly suitable for this situation. The most attractive part of this method is that it can generate an irreducible and aperiodic Markov chain with the target high-dimensional posterior distribution as a stationary distribution only by sampling from a series of univariate distributions.We assume thatX=X1,…,Xd is a random variable in ℝd, and its joint distribution fx is the target sampling distribution. It is a random variable of dimension d-1, as follows:(27)X−j=X1,…,Xj−1,Xj+1,…,Xd.Subsequently, the conditional density ofXj|X−j is denoted as fxj|x−j; then, the Gibbs sampling method is to extract candidate points from these d distributions, thus solving the difficulty of direct sampling from fx. The specific algorithm is as follows:(1)
The algorithm gives the initial value ofX0.(2)
Fort=1,2,…,T:(i)
The algorithm sets upt=1,2,…,T.(ii)
For each componentx1=X1t−1,(a)
The algorithm extracts the candidate pointXj∗t from fXj|x−j;(b)
xj=Xj∗t is updated.(iii)
The algorithm establishesXt=X1∗t,…,Xd∗t.(iv)
Ast increases, the algorithm repeats step (i).In algorithm step (ii), each component is updated sequentially:(28)x1t∼fx1|x2t−1,…,xdt−1,x2t∼fx2|x1t,x3t−1,…,xdt−1,…xdt∼fxd|x1t,…,xd−1t.It is relatively straightforward to draw samples from the univariate distributionfxj|x1t,x2t,…,xj−1t,xj+1t−1,…,xdt−1. Because fxj|x−j∝fx, except xj which is a variable, all other variables are constants.What is special about Gibbs sampling is that a fully conditional distribution can determine a unique joint distribution. The well-known Hammersley–Clifford theorem reveals this law.We assume that the joint density distribution of the random variableX=X1,…,Xd is fx1,…,xd, and the edge density function of Xi is fixii=1,…,d. If fixi>0i=1,…,d, which means fx1,…,xd>0, then the joint density distribution f is said to satisfy the positive condition.Theorem 3 (Hammersley–Clifford theorem) is as follows: Under positive conditions, the joint density distributionf satisfies the following:(29)fx1,…,xd∝∏j=1dfxj|x1,…,xj−1,xj+1′,…,xd′fxj′x1,…,xj−1,xj+1′,…,xd′.For anyx and x′ in the support set S of f, we have(30)fx1,…,xd=fxd|x1,…,xd−1fx1,…,xd−1=fxd|x1,…,xd−1fxd′|x1,…,xd−1fx1,…,xd−1,xd′=fxd|x1,…,xd−1fxd′|x1,…,xd−1fxd−1|x1,…,xd−2,xd′fxd−1′|x1,…,xd−2,xd′×fx1,…,xd−1′,xd′=…=∏j=1dfxj|x1,…,xj−1,xj+1′,…,xd′fxj′|x1,…,xj−1,xj+1′,…,xd′fx1′,…,xd′.Taking the Gibbs sampling of the SV model as an example, we assume that the parameter setθ=μ,ϕ,τ and the initial value of the logarithmic volatility h, which is denoted as Ω0=μ0,ϕ0,τ0,h0. Then, a single Gibbs sampling is as follows:(1)
The algorithm extractsh1 from πht|μ0,ϕ0,τ0,h−t0,y.(2)
The algorithm extractsτ1 from πτ|μ0,ϕ0,h1,y.(3)
The algorithm extractsφ1 from πϕ|μ0,τ1,h1,y.(4)
The algorithm extractsμ1 from πμ|ϕ1,τ1,h1,y.After the above iterations, the initial value is updated toΩ1=μ1,ϕ1,τ1,h1, and by repeating the Gibbs sampling process k times, we can get Ω1,Ω2,…,Ωk. Under the condition of regularity, when the number of iterations k is large enough, Ωk is equivalent to a random draw in the posterior joint distribution. In practice, Gibbs sampling requires enough iterations to reach a stationary state. However, the sample composed of m data before reaching the stationary state after discarding Gibbs sampling is Ωm,Ωm+1,…,Ωk. In the Monte Carlo estimation of parameters, the posterior mean of the parameter is estimated by the mean of the following k-m samples.


## 4. Prediction of High-Frequency Economic Data Based on Stochastic Fluctuation Model

The cobweb model is a classic model in Western economics. The model is mainly divided into three cases, and the different elasticity of supply and demand is still discussed. When the elasticity of supply of commodities is less than the elasticity of demand, the nonequilibrium price will converge according to the spider web model and finally reach the equilibrium price. In the case where the elasticity of supply of commodities is greater than the elasticity of demand, the conclusion is just the opposite: the nonequilibrium price will continue to diverge, the fluctuation will become larger and larger, and it will deviate from the equilibrium price. When the elasticity of supply of commodities is equal to the elasticity of demand of commodities, according to the cobweb model, prices will enter a cycle and cannot be adjusted to the equilibrium price level. We take the example of commodity supply elasticity equal to commodity demand elasticity for analysis. First, we consider the case where there is no futures/stock market, as shown in Figure1.Figure 1
Cobweb model. (a) Cobweb model without futures/stock market. (b) Cobweb model with futures/stock markets.
(a)(b)This paper makes a preliminary judgment on the degree to which futures/stock price changes lead the spot by means of error correction model, impulse response, and variance decomposition. Furthermore, the contribution of futures/stock price changes to the effective price is studied through the common factor model so as to obtain its price discovery efficiency. Our empirical research design ideas are shown in Figure2.Figure 2
Structure diagram of empirical research.The empirical idea to test whether futures/stocks have the function of price discovery is shown in Figure3.Figure 3
The empirical structure idea to test whether futures/stocks have price discovery function.The price discovery efficiency is judged by studying the contribution of futures/stock price changes to the effective price. The research idea of futures/stock price discovery efficiency is shown in Figure4(a).Figure 4
Financial market information processing. (a) Structure idea diagram of the empirical test of futures/stock price discovery efficiency. (b) Information and efficient markets.
(a)(b)On the basis of information and securities price reflection, three different levels of efficient markets are proposed, namely, weak efficient market, semistrong efficient market, and strong efficient market, as shown in Figure4(b).The benchmark model used is an on-frequency forecasting model with low-frequency (weekly) stock squared returns as predictors. Specifically, our analytical framework can be shown in Figure5.Figure 5
Prediction model framework.After the above model is constructed, the effect of high-frequency data fluctuation is analyzed for the economic high-frequency data prediction model based on the stochastic fluctuation model, and the prediction effect of economic high-frequency data is evaluated. Moreover, this paper carries out research in combination with the simulation test and counts the relevant test results and obtains the test results shown in Tables1 and 2 and Figures 6 and 7.Table 1
Data fluctuation analysis of economic high-frequency data prediction model based on stochastic fluctuation model.
NumberAnalysis of economic fluctuations178.5285.4383.5488.9584.7687.8791.0882.5984.91083.41188.71284.91386.41483.41579.31686.71788.21890.41985.32089.12180.12280.72381.52486.92585.62690.72780.02878.22979.23088.63178.63280.33378.53486.23585.23681.23786.13889.93987.24082.74189.74287.74385.34488.74589.04690.54787.54886.14989.85085.55187.05286.15379.25484.9Table 2
Prediction effect of economic high-frequency data prediction model based on stochastic fluctuation model.
NumberHigh-frequency data prediction179.9283.8375.8474.7587.9686.7784.4885.6983.01086.31184.61279.21376.61481.01577.51678.81786.51876.51975.72074.42181.92281.22378.82482.62581.92675.52785.32886.82974.23079.73182.93278.43384.33486.43578.83684.23774.13881.23979.84079.84181.04284.44386.14474.54574.14684.14780.44874.84986.95080.75182.95281.55383.55475.7Figure 6
Data fluctuation analysis effect diagram of economic high-frequency data prediction model based on stochastic fluctuation model.Figure 7
Prediction effect of economic high-frequency data prediction model based on stochastic fluctuation model.It can be seen from the above experimental research that the economic high-frequency data prediction model based on the stochastic fluctuation model proposed in this paper has a good effect on the analysis and prediction of economic high-frequency data.


## 5. Conclusion

High-frequency data studied in the field of financial market microstructure refers to the type of data as opposed to data on a daily or longer time interval. It is the transaction price, transaction volume, and other data collected during the trading day, and it is mainly the data collected in hours, minutes, or seconds. The UHF data refers to the data collected in real time during the transaction process of financial products such as securities and foreign exchange. Obviously, UHF data is data at unequal intervals. High-frequency and ultra-high-frequency data contain more real-time information about the securities trading process and more accurately capture every tiny change in the market. Using high-frequency and ultra-high-frequency data to study security price behavior has many advantages over using low-frequency data. In order to improve the effect of economic high-frequency data analysis, this paper combines the stochastic fluctuation model to carry out the forecast analysis of economic high-frequency data. The experimental research results show that the economic high-frequency data prediction model based on the stochastic fluctuation model proposed in this paper has a good effect on the analysis and prediction of economic high-frequency data.



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