Partially Linear Multiplicative Error Model And Its Application In Researching Stock Liquidity

We often encounter a lot of nonnegative variable sequence in real life. And a growing body of research on financial markets depends on the dynamic analysis of the nonnegative financial time series variables. Liquidity is one of the most core problems in financial markets, which is the vitality of the financial markets and the key to the healthy development of the modern financial markets. If the financial markets are lack of liquidity, the value of financial assets will be underestimated and the trade is difficult to complete. In some extreme cases the transaction may be even unable to complete and there is no point in the existence of financial markets. The stock markets, as the main form of financial markets, have been the focus of academic research in the economic field. Therefore, the researches to the liquidity of the stock markets have important theoretical and application values,In the study of nonnegative time series variables, Engle (2002) proposed a new model which is called multiplicative error model. The setting of the model makes the nonnegative automatically guaranteed. Then, a lot of scholars researched the theory and application of the MEM. But these researches are more belong to the foreign, and the domestic study of the MEM is relatively lacking. And these researches are more focused on the study of the parametric MEM, which are rarely involved in the field of the nonparametric and semiparametric field.This paper makes a literature review of research on stock market liquidity, and finds that the domestic and foreign scholars were more focused on the external influence to the liquidity and ignored the inner influence of liquidity. Bid-ask spread and market depth is one of the most common measure indexes of liquidity. Because China stock market is the order-driven market, we can’t get the real bid-ask spread data. Thus this paper uses the market depth to study liquidity.Market depth belongs to the nonnegative time series variables, so this paper will use market depth as the measure index and the MEM to dynamically analyze the liquidity of China Unicom. In order to avoid the problem of "dimension disaster" of the nonparametric model, this paper will put forward the partially linear MEM on the basis of the parametric MEM and the partly linear model and give the corresponding estimation algorithm and the proof of the consistency of the estimation algorithm. Then in the parametric MEM(1,1) and the partially linear MEM(1,1), this paper will research which model can better depict the liquidity of financial markets respectively from the perspective of simulation and practice. On the one hand, this paper makes the MEM expand from the parametric field to the semiparametric field, provides a new research tool to deal with the nonnegative financial time series variables and enriches the research methods and content of the financial markets econometrics, time series analysis and nonparametric econometrics. On the other hand, the partially linear MEM is applied to the specific empirical research of stock liquidity in the first time which provides a new model and method for later scholars. Considering the current situation of China’s stock market, the empirical results have important practical application value, which can help investors to understand the status of the stock liquidity, improve liquidity regulation of China stock market and provide scientific decision basis for improving the liquidity security of market effectively. In short, it has the very vital significance to focus on the research of the MEM both in theory and application field.The main research contents and conclusions of this paper are as follows:Firstly, this paper puts forward the partially linear MEM based on the parametric MEM and the partly linear model and gives the corresponding estimation algorithm and the proof of the uniform convergence of the two phase estimation algorithm of the partial linear MEM form①. It makes the MEM expand from the parametric field to the semiparametric field and provides a powerful research tool to deal with the nonnegative financial time series variables. And it is also the theoretical innovation on the model and the estimation method.Secondly, in order to simulate the nonnegative time series variables and leverage effect in the financial markets better and strengthen the effectiveness and scientificity of the argument, this paper generates the nonnegative variable sequence and its real conditional mean sequence through three different kinds of data generation process and two different kinds of residual distribution whose length is500. Then calculate the data generation process500times. Finally evaluate the estimated effects of the parametric MEM(1,1) and the partial linear MEM(1,1) through five different prediction ability evaluation indexes and find that no matter by what kind of data generation process and residual distribution, after500times loop calculation, the fitting ability of the partial linear MEM(1,1) is better than the parametric MEM(1,1). When the residual obey exp(1) distribution, the parametric MEM(1,1) is better than the partial linear MEM(1,1) and when the residual obey Gamma(1.5,1.5) distribution, the partial linear MEM(1,1) is better than the parametric MEM(1,1). Thus, this paper proves that the partial linear MEM(1,1) is effective from the perspective of the Monte Carlo simulation.Thirdly, this paper deals with a specific empirical analysis in the parametric MEM(1,1) and the partial linear MEM(1,1) based on the real sample data of market depth of China Unicom. First of all, this paper chooses the market depth sequence of China Unicom as the research object and divides the whole sample lnto two parts:estimate sample and forecast sample. Then it conducts the descriptive statistic analysis and model estimation for the sample. Finally, compare the fitted ability and the prediction ability of the parametric MEM(1,1) and the partial linear MEM(1,1) by five different predictive ability evaluation indexes. The empirical research shows that the market depth sequence of China Unicom isn’t excessive dispersion, but has the characteristics of obvious sustained correlation, positively biased and thick tail. The parametric MEM(1,1) has not completely extracted the relevant features of the estimate sample and can’t fit the estimate sample well. The partial linear MEM(1,1) can extract more relevant features of the estimate sample than the parametric MEM(1,1) and is more suitable for fitting the estimate sample. From the view of evaluation index, both the parametric MEM(1,1) and two kinds of model forms of the partial linear MEM(1,1) have better prediction ability than fitting ability. Compared with the parametric MEM(1,1), both kinds of model forms of the partial linear MEM(1,1) are more able to depict characteristics of the market depth of China Unicom and more suitable for depicting the liquidity of China Unicom shares. And the partial linear MEM(1,1) form①has better performance and has a better ability to predict the liquidity of China unicom. It also shows that the liquidity of China unicom is affected by itself in a linear fashion and is affected by its average in a nonlinear fashion. This part proves that the partial linear MEM(1,1) has better fitting ability than the parametric MEM(1,1) from the perspective of empirical analysis, confirming with the simulation result, and makes the result more convincing.Compared with the other papers, the innovation of this paper mainly has the following two points:Firstly, this paper puts forward two forms of the partial linear MEM and uses a test statistic which can be used to automatically choose what kind of model should be used for the first time. It also gives two estimation algorithms with uniform convergence for each form of the partial linear MEM and the proof of the consistency of the iterative algorithm for form①under the weak condition. By combing the domestic and foreign literature related to MEM, this paper finds that the scholars’ researches of MEM are mainly concentrated in terms of parametric model and haven’t involved in partial linear MEM. Thus, combining with the parametric MEM and the partly linear model, the paper gives the partial linear MEM and its uniformly convergent estimation algorithm. It makes up the blank of the research in this aspect and enrichs the theoretical study of the partial linear MEM. Based on the Monte Carlo simulation data and real sample data of China stock market, this paper compares the fitted ability of the parametric MEM(1,1) and the partial linear MEM(1,1).The results show that two kinds of the partial linear MEM(1,1) have better fitting ability than the parametric MEM(1,1).Secondly, this paper uses the parametric MEM(1,1) and the partial linear MEM(1,1) to research the market depth of China Unicom for the first time. Empirical study shows that compared with the parametric MEM(1,1) both form of the partial linear MEM(1,1) are more able to depict the characteristics of the market depth of China Unicom and more suitable for depicting the liquidity of China Unicom, and the partial linear MEM(1,1) form①has better performance. Confirming with the simulation result it makes the result more convincing.This paper is supported by the2011National Natural Foundation of China (71101118).