ARMA Models With Asymmetric Laplace Noise

This paper modifies the basic foundational assumption of the classic time series analysis to adapt the time series models to more fields such as finance.Classic time series analysis theory assumes that the noise follows a normal dis-tribution.This paper puts forward a new distribution of noise instead of normal distribution,asymmetric Laplace distribution,which is leptokurtosis and has fat tails.We present the definition of asymmetric Laplace distribution and de-rive the main properties further.We also derive the method to produce random number following asymmetric Laplace distribution.Afterwards,we focus on AR-MA models which are the most basic model in time series models,and research the properties of ARMA models with asymmetric Laplace noise.Xt of ARMA models can be represented by the linear combination of noise terms.So we can combine the character functions of noise terms to get the character function of Xt.Then we use some skill to turn product to sum,which makes it easier to get the explicit expression of the marginal distribution of ARMA models by Fourier inversion formula of the character function.And then,interval estimation is con-ducted to research the forecast ability of ARMA models with asymmetric Laplace noise.Both one-side confidence interval and two-sides confidence interval are de-rived.Furtherly,we fit ARMA models with conditional maximum likelihood estimation and make a numerical methods to maximize the likelihood function Some random simulations are made to confirm the correction of our methods At last,three most significant indexes in Chinese stock markets are chosen to fit the ARMA models with either Gaussian noise or asymmetric Laplace noise These models with two different noises are comparable using Akaike Information Criterion(AIC).The comparison shows that the ARMA model with asymmetric Laplace noise is consistently better than the one with Gaussian noise.This paper put forward a new model which is better than the existing models of the same kind when researching Chinese stock markets.