Logarithmic Multiplicative Error Model And Its Application

Since Engle (1982) proposed the autoregressive conditional heteroskedasticity (ARCH) model, modeling of non-negative financial time series variables has been numerous. Especially when the generalized autoregressive conditional heteroskedasticity (GARCH) model and the autoregressive conditional duration (ACD) model came out, modeling studies on financial volatility and financial duration had got more and more attention. And we know that there has a large number of non-negative financial time series. Such as volume, volatility, absolute yield, financial duration, bid-ask spreads, transaction size and other variables. More and more research in financial markets also depends on dynamic analysis of these non-negative financial time series variables. To solve this problem, Engle is no longer tied to a specific non-negative financial time series variable, but rather construct a new model that is suitable for all non-negative financial time series variables——multiplicative error model (MEM). GARCH model、ACD model and CARR model are all special cases of MEM.The study of volatility and duration is very fruitful and successful. In this paper, the multiplicative error model (MEM) on the basis of the bid-ask spreads variable in China’s securities market were studied and discussed. We found that the ACD model and the GARCH model have a non-negative restriction on their parameters, which is a big inconvenience, especially when you want to add some more other variables, non-negative limits of parameters will bring you a lot of trouble. To solve this problem, we can logarithm the model, thereby Log-GARCH model (EGARCH model) and the Log-ACD model emerged. The same, Engle (2002) proposed a new model applies to all non-negative financial time series variables——multiplicative error model (MEM) also has a non-negative restriction on its parameters, we naturally come up with the logarithmic model technology, to build the Log-MEM. Thus, the theoretical part of this paper which starts from the building of the Log-MEM separates to three parts.1. According to the previous model development direction, the logarithm of the multiplicative error model is constructed on the classical multiplicative error model. Similar to the construction of the Log-ACD model, Log-MEM1and Log-MEM2are built due to two different specification of the error. Meanwhile, we deduced the conditions of the existence and analytical expressions of their unconditional moments.2. Stock market events are often characterized by overdispersion, meaning that the standard deviation of the data is larger than their mean. Another important stylized fact is the shape of the ACF, which usually decreases slowly from a relatively low positive first-order autocorrelation. It is therefore essential that Log-MEM be able to fit such stylized facts, for some parameter values. We deduced from the analysis of the presence and expression of the difference between the index and the autocorrelation function. According to the theoretical results derived, we gives the corresponding multiple sets of tables, with a more clear perspective presented.3. More systematic review of the classic multiplicative error model. The unconditional moments previously mentioned before and its existing conditions, dispersion index and the autocorrelation function, apply it to the classic multiplicative error model, we found a linear set of classical models of these formulas are quite more simple, but because of the stability and non-negative parameter setting is set so that the limitations of these formulas are also increased a lot.In the empirical part, we choose Shanghai Composite Index and the CSI300Index. First, the basic statistical analysis of the data of Shanghai Composite Index and the CSI300Index appears that the bid-ask spreads variable has an obvious characteristic peak and positive bias, a certain autocorrelation and memory. Then this paper, these two sets of data model selection problem is studied, as this article relates to the comparison of the three models, also involves three basic distributions of residuals, a total of nine possible models for these data, the maximum likelihood estimates for nine times for each one. Finally, we compare the mean, the dispersion index and the standard deviation with the real data. We found that the Log-MEM1(Exponential distribution) is suitable for the Shanghai Composite Index and the classic MEM (Gamma distribution) is for the CSI300Index.