Modeling Realized Covariance Using Smooth Transition Multivariate Heterogeneous Autoregressive Models

Investment portfolio and risk management demand refinement of the portfolio selection theory,while investors' desire to avoid risk demands models that have better predicting ability of the risk in financial markets.Realized covariance(RCOV)is constructed using high-frequency data and serves as a proxy for portfolio risk,thus to accurately predict realized covariance has become a noticeable question for study.So far,there has been a great amount of literature on realized volatility models,but realized covariance models,such as the multivariate HAR model and WAR model,are relatively few in comparison,due to restrictions imposed on the modeling of realized covariance models.It's long been known that asymmetric volatility,usually in the form of the so-called"leverage effect" where negative shocks are succeeded by higher volatility,is present in financial markets.Besides the leverage effect or sign asymmetry,other forms like size asymmetry have also been discovered.To allow for asymmetric volatility,there has been a great amount of work on models that use smooth transition functions,and these studies show that allowing for smooth transition functions in univariate models can boost the fitting and forecasting abilities of the models.However,there is a serious lack of literature on introducing smooth transition functions in multivariate models,and no empirical study on smooth transition models modeling realized covariance.This paper introduces a smooth transition function into the Multivariate Heterogeneous Autoregressive model(MHAR)model,and conducts an empirical study by using 7 individual stocks of the SSE 50 ETF and the SSE 50 index as data and modeling the realized covariance of the stocks.Since positivity is required for the realized covariance matrix,the common practice is to transform the matrix using positive definite transformation methods,vectorizes the upper triangular transformed matrix,and models on the resulting vector instead.However since the autocorrelations and asymmetries of the elements of the transformed matrix can be highly affected and are not similar,this paper proposes 2 approaches.The first approach is to model the diagonal elements and non-diagnonal elements separately with the MHAR-diag model,and also model the differences between asymmetries of diagonal elements and non-diagonal elements with different smooth transition models based on MHAR-diag.The second approach is to use a transformation called lcor transformation,which doesn't guarantee positivity with inverse transformation,and thus restrictions should be applied on the models to guarantee positivity.This paper conducts the DM test and the MCS test to test the predicting ability of the models under different loss functions.The empirical study shows that,compared to the MHAR model,the MHAR-diag model can improve the forecasting ability under all loss functions used.Moreover,under the same model form,modeling an lcor-transformed RCOV matrix can get better forecasting results than modeling a matrix-logarithm-transformed RCOV matrix.Finally,by performing an MCS test upon all models,we find that the MHAR-splitST model,modeling the lcor-transformed RCOV matrix,has the best overall predicting ability among all models in normal stock price volatility time periods.