Serial Correlation Tests In ARMA Models And The Related Applications In Empirical Studies

A large number of different types of time series models have been proposed in the past few decades due to their wide application in depicting the relevant characteristics of interdependent data in the fields of finance and economics.Among them,the ARMA model(Autoregressive moving average model)has attracted widespread attention because of its more accurate spectral estimation and better spectral resolution performance than the AR model and the MA model,and has become the most commonly used stationary sequence fitting model.It is widely used in the analysis and prediction of sales volume and market size in economic life,stock price in financial market,and environmental atmospheric laws.In making relevant statistical inferences and applications to ARMA models,standard assumptions about model errors include that the sequence of model errors must be sequence-independent.The violation of the correlation here may cause the correlation inference of the ARMA model to be biased or inefficient.Therefore,before actually using the ARMA model,it is generally necessary to check whether there is a serial correlation in the model error sequence.Note that there have been many discussions on the serial correlation test of ARMA models in the literature,such as Mc Leod?Li(1983)^[46],Pe(?)a?Rodr(?)guez(2002,2006)^[53][54],Li(2003)^[41]et al.Among them,Zhu(2016)^[72]deduced the critical values of?(?)_M and?(?)_M in the ARMA model under weak assumptions by the random weighted method,and relaxed the test conditions to only irrelevance,which attracted widespread attention.However,the test method proposed by Zhu(2016)^[72]may lead to misjudgment due to randomness in the case of small sample size.In addition,the numerical simulation results in this thesis show that the?(?)_M statistic in Zhu(2016)^[72]is not stable for different model coefficients.Therefore,it is worth thinking deeply about how to test the serial correlation of the ARMA model under weak assumptions.In view of this,this thesis considers two test statistics based on the empirical likelihood method:(i)The test statistic EL1 can be used to test the serial correlation of the ARMA model under weak assumptions,and the test results are more stable for different coefficients in the model.The method in(i)does not consider the issue of whether the variance of the model error is infinite.It is noted that in the case where the variance may be infinite,the method in(i)has a certain defect,that is,it has a tendency to over-reject,and as the sample size increases increases this problem will be more serious.Therefore,on the basis of(i),this thesis further proposes(ii)the test statistic EL2,which is suitable for testing the serial correlation of ARMA models under the possibility of infinite variance.The biggest advantage of the two empirical likelihood methods is that their asymptotic distributions are standard chi-square distributions under relatively loose conditions,which may bring greater convenience to their practical applications.And it is worth mentioning that the two test statistics based on the empirical likelihood method in(i)and(ii)overcome the main problems of the method in Zhu(2016)^[72]to a certain extent.Therefore,the new method has limited limitations.The sample properties are better.After deriving the large sample properties of the EL1 and EL2 statistics proposed in this thesis and conducting a large number of numerical simulations,the thesis finally applies the relevant methods to the PM2.5 index data of 20 domestic cities and the exchange rate data of8 foreign stock markets.In the research and analysis of,it is determined whether there is serial correlation in the fitted model,which provides a basis for further research and analysis.The specific chapters of this thesis are arranged as follows.Chapter 1 is the introduction,which are the introduces of the thesis,and explains the origin and development of serial correlation test through literature review and commentary at home and abroad,as well as the current research status of this test at home and abroad.In addition,the research ideas and framework are given,and the innovation of the thesis and some difficulties encountered in the research process are explained.Chapter 2 is the prior knowledge.It summarizes the modeling research process and model diagnosis of general time series models,and compares and introduces the commonly used serial correlation test methods.The empirical likelihood function and random weighted method used in the proposed test method are compared and introduced.Chapter 3 is the serial correlation test of autoregressive model,which is also the core chapter of this thesis.Firstly,the ARMA model and the hypothesis test of serial correlation are introduced.Secondly,the empirical likelihood method is used to test the serial correlation of residual terms under the condition of limited variance,and then it is extended to the case where residual terms have infinite variance.and demonstrate the asymptotic distribution of the proposed test statistic.Finally,studies simulations were performed to verify the finite-sample nature of the proposed test statistic and compared with the test statistic in Zhu(2016)^[72].Chapter 4 is the empirical analysis.Based on the ARMA model discussed above,two empirical analyses are carried out.Among them,the first demonstration is the main empirical analysis of this thesis,which is to carry out model fitting and serial correlation test on PM2.5data of different cities.The second demonstration is the extension of the test method proposed in this thesis,and the model fitting and serial correlation test are carried out on the exchange rate data of stock market in different countries.Chapter 5 is the conclusions and limitations,summarizes this thesis,clarifies the purpose of the research,proposes solutions and the process of verifying its feasibility,and points out the limitations in the thesis,hoping to improve and perfect these areas in the future.