An Empirical Likelihood-based Serial Correlation Test In ARX Models

The ARX model is widely adopted in time series analysis,its structure of combining autoregressive items and exogenous variables brings it much more application scenarios than autoregressive models or simple linear regressions,relevant disciplines include but not limited to financial statistical modeling,environmental monitoring,mechanical engineering,etc.In regression analysis,specially in model fitting with respect to time series,testing for serial correlation is an important procedure.The error terms which exhibit serial correlations often imply the existence of model inadequacy or model misspecification.Although there have been much outstanding work on this issue,but in recent years,empirical economists have found that in real financial data,there are widespread problems such as volatility clustering,heteroskedasticity,heavy tails,and strong correlations between different error sequences.These phenomena are extremely destructive for the robustness of existing serial correlation test methods.Facing these new challenges,both the academia and the industry urgently need a test method that is applicable under all the above special conditions.In view of this,this paper proposes a new serial correlation test for ARX models based on Profile Empirical Likelihood(PEL),and compares its numerical simulation results with existing Plug-In(PI)test and Breusch-Godfrey(BG)test.The results show that,in comparison with the serious undersized performance of the PI method in the case of strong correlation between the error sequences,and the severe oversized performance of the BG method in the case of conditional heteroscedasticity,the PEL method proposed in this paper has the best robustness due to its excellent performance of maintaining exact size under all the above considerations.The introduction of the PEL serial correlation test method expands the limitation of the existing test methods,which brings it great practical significance.In addition,we also applied the PEL serial correlation test to the empirical research.We respectively conducted ARX model fitting and serial correlation tests on a set of financial data including stock returns,price-earning ratios,dividend-yield ratios and a set of environmental data including ozone concentrations,wind speeds,humidities.The first example shows that,compared with the traditional predictive regression model,the ARX model with autoregressive items is more applicable and appropriate in fitting financial variables such as stock returns.The second example shows that the four sets of exogenous meteorological variable data used in this paper are insufficient to fit the indicator of ozone concentration,and there may be more variables involved in pollution assessment,or there exist problems in model selection,empirical researchers need to consider more complex models.