Predictability Test For ARMAX Models And Its Applications

The ARMAX model(Autoregressive Moving Average with Exogenous Inputs)is a commonly used model in time series analysis.It extends the ARMA model by incorporating exogenous variables to consider the influence of external factors on the time series.This enables a better description of the dynamic changes in time series data and offers strong flexibility and adaptability,making it highly valuable in practical applications.However,using covariates as predictors when their impact on the dependent variable is not significant can lead to overfitting,increased model complexity,and biased predictions.Additionally,endogeneity is a prevalent challenge in real-world data.For example,when using the ARMAX model to predict future housing prices,the population size of a city may affect housing prices,but at the same time,the population size may be influenced by housing prices,resulting in endogeneity.In the ARMAX model,endogeneity is manifested by the nonzero correlation between the covariates and the model errors.Therefore,it is worth discussing and researching how to test the significance and effectiveness of covariates in the ARMAX model under the presence of endogeneity.However,much of the existing literature on the ARMAX model is primarily focused on its applications.Studies on the relationship between exogenous variables and the dependent variable typically analyze whether the introduction of exogenous variables improves the predictive accuracy or reduces prediction errors in the ARMAX model.There is a lack of theoretical tests for the predictability of the ARMAX model.Traditional model testing methods,such as the Wald test,examine whether parameter estimates are significantly different from specified parameter values,while the likelihood ratio test compares the goodness of fit between two nested models.In theory,these tests can be applied to test the predictability of the ARMAX model.However,this thesis specifically discusses the predictability test of the ARMAX model under endogeneity,and further exploration is needed to determine if traditional model testing methods are still applicable.In light of this,this thesis proposes an empirical likelihood method to address the predictability test of the ARMAX model under endogeneity and demonstrates that its test statistic asymptotically follows a chi-square distribution.Additionally,numerical simulations are conducted to compare the size and power of the proposed empirical likelihood method with traditional Wald tests and likelihood ratio tests.The results show that the proposed empirical likelihood method outperforms traditional methods in the predictability test of the ARMAX model under endogeneity.Moreover,both the Wald test and likelihood ratio test tend to be significantly biased in size,indicating that traditional methods may not have an advantage in testing the predictability of the ARMAX model with endogeneity.Furthermore,Google Trends is a free tool provided by Google for analyzing the search trends of specific keywords or topics on a global scale.It is widely used to assist research in various fields due to its convenient data accessibility and low cost.Many current studies have discussed whether Google Trends improves the prediction of tourist demand by establishing ARMAX models,and they have obtained consistent conclusions that Google Trends enhances the accuracy of tourist demand prediction.However,most of these studies are based on data prior to the pandemic.Therefore,this thesis is concerned with whether Google Trends remains significantly effective in the ARMAX models established with tourist demand data during the pandemic.Accordingly,this thesis empirically establishes ARMAX models using Google Trends and tourist demand indicators for 73 countries and conducts predictability tests.However,the conclusions differ from those before the pandemic,as there is insufficient evidence to suggest that Google Trends,as a covariate,is significantly effective in the ARMAX models established with tourist demand data for most countries.Therefore,during the pandemic,the ability of Google Trends to assist in predicting and analyzing tourist demand may be affected,and caution should be exercised when using Google Trends data to supplement tourist demand data for prediction and analysis.In conclusion,this study proposes an empirical likelihood method to address the predictability test of the ARMAX model under endogeneity,providing a new complement and improvement to the theoretical research of the ARMAX model.It also examines the significance and effectiveness of Google Trends as a covariate in ARMAX models constructed with tourism demand data during the pandemic.Thus,this method offers important reference value for practical applications in various fields.