Variance Reduction And Bias Correction For Non-Stationary High-Frequency Data

Statistical inference of the mathematical models regarding financial problems has been a hot issue in statistical study.In the literature,the randomness of the finan-cial markets and economic phenomena are often described by diffusion processes.The classical diffusion process,as the solution of stochastic differential equation dXt = ?(Xt)dt + ?(Xt)dWt,has already been applied in derivative pricing and used to characterize the asset price changing in the markets.For instance,jump diffusion models can be used to explain the non-smoothing movements of the stock price and measure the influence of major risk events on the stock price.In general,concerning the pricing problems,Most dif-fusion models have a requirement of stationarity.However,in the real markets,data is often non-stationary.Like the relationship between the implied volatility and option price.The latter is non-stationary while The former is more desired but unobservable.So,it's unrealistic to consider these problems under the stationary setting.Recently,scholars' attentions has been focused on the non-stationary diffusion process.In the study of non-stationary financial data,there is a situation that is worth noting:that is,the data will exhibit stability(for example,GDP)after differencing.In discrete time,some scholars study non-stationary financial data through the unit root process,such as(random walk with the drift of the stock price).Nicolau(2007 Econometric theory)proposes a second-order diffusion model dYt = Xtdt dXt = ?(Xt)dt + ?(Xt)dWt to study non-stationary financial data.Under the assumption of high-frequency data,he studied the nonparametric estimation of two unknown coefficients.His research be-longs to the category of nonparametric statistical inference.The basic task of statistical inference is to infer the population from the characteristics of sample data.In fact,some of the existing statistical inference theories used directly to deal with actual finan-cial model problems often do not work well.For example,classical statistical inference theory often requires the assumptions of a given model,such as the assumption of the distribution of the sample's population,whether the samples are independent of each other,the reproducibility of the data,and so on.These conditions are not yet complete for the financial market to achieve.Even though the non-parametric approach has the advantage of not having to assume the density function in advance,the model setting is still very high.Nicolau(2007)used the most common N-W estimation method to estimate ?(x),?2(x),Wang and Lin(2011)used local linear estimation to improve the results of Nicolau(2007).However,the classical kernel smoothing methods tend to have poor symmetry distribution.Considering the statistical inference of the diffusion process,since the model itself is not a regression model but an approximate regression model,there are endogenous variables.The second-order diffusion process cannot di-rectly obtain the sampling value of the process.Therefore,there is almost no symmetry in the process marginal distribution.When using symmetrized kernel method estimates,marginal results tend to be poor.At the same time,when considering the large-sample property of local linear estimation,it is often difficult to establish the asymptotic distri-bution property when considering the dependency sample because of the destruction of the predictability structure.Wang and Lin(2011)only gave consistency,so an approxi-mate confidence interval cannot be established.The main task of this paper is to combine asymmetric kernels with local linear estimation methods to study nonparametric estima-tion methods for drift terms and diffusion terms in second-order diffusion models.The main innovations of this paper are:First,the non-stationary financial high-frequency data is processed for the first time by combining asymmetric kernels with local linear methods;therefore,the method of this paper guarantees a small overall estimation bias and guarantees the marginal results.Second,asymptotic distribution and its proof are given for local linear estimators of asymmetric kernels.The structure of this paper is as follows:In the first chapter,we will briefly intro-duce the required stochastic analysis basis and some classical non-parametric estimation methods.The second chapter introduces the second-order diffusion model.The reason why this model is concerned is mainly the consideration of finance.Some of the key pro-cess models in the market and economy can be influenced by the cumulative disturbances of all the past processes,such as stock prices and exchange rates,so their time series do not satisfy the stationarity,but the difference is stable,Since the diffusion process is almost inevitably Undifferentiable,so consider introducing a second-order diffusion model.In Chapter three,we will construct a nonparametric estimation method for drift terms and diffusion terms for second-order diffusion models and give proofs of their large sample properties.In Chapter four,we will conduct simulation and empirical analysis to test the validity and robustness of the non-parametric estimators we construct.