Introduction                                                                        

The many investigations in economic and financial mathematics focused on what makes an investor profitable in the stock market. These studies can aid the researchers to decrease the investment risks and increase opportunities for a high return of gaining. One of the important questions in the stock market is when the investors can buy the stocks and when they can sell their stocks. In research economic papers, there are two aspects of analysis: fundamental and technical analysis. In fundamental aspects, the researchers find the reasons for changing stock prices, in response to reasons of changing prices caused by exogenous geopolitical events, supply disruptions or financial operation of the companies, etc. But technical analysis noted more the statistical and probabilities rules governed by processes of the data. In an aspect of technical analysis, there are a lot of models in time series to capture the stock prices. The Markov Switching models are the popular models in time series that are applied most widely in financial and economic data. These models exhibit abrupt changes in the behavior of time series data, called switches of regimes, where the switching between the regimes is controlled by a hidden Markov Chain process. (See Chang, Yongok, & Joon 2017; Von Ganske, 2016; Billio, Casarin, Ravazzolo, & Van Dijk, 2016; Di Persio,  & Frigo, 2016; Neale, Clark, Dolan, & Hunter, 2016; Nademi, & Nademi, 2019).

Recently, Markov switching AR–ARCH[1] models have repeatedly applied for making switching regimes processes and every one of them offers an algorithm for estimating the parameters. The finding of the best algorithms for parameters estimation of these models has been the object of many expansions and usages over the last decade. Numerous algorithms based on parametric and nonparametric methods have been offered to capture the parameters of the modeling process. In this respect, the combination of parametric and nonparametric methods, called semi-parametric algorithms, are popular and most broadly applied. (See Chan, & Wang, 2017; Chang, Tang, & Wu, 2016; Chen, Shen, Wei, & Lin, 2017; Gupta, Cobre, Polpo, & Sinha, 2016; Gu, Ma, & Balasubramanian, 2015; Nademi, & Farnoosh, 2014).

In a semi-parametric class of algorithms, a special function, called kernel function, is used. The selection of proper kernel function is an important item for estimating the parameters. Such that, if we apply the proper type, we can have a fast and unbiased estimating process. So, offering the best kernel functions for estimation algorithms can be essential for the modelling process. In this paper, we first focus on selecting the best kernel function in a special class of Markov switching models called semi-parametric Markov switching offered by Nademi & Farnoosh (2014) for modeling the time series data and then offer the new strategy of buying and selling stocks by the best-selected kernel function of this model on real data.

In the next section, the model and its algorithm will be introduced. Section 3 discuss the best selection of kernel function by a simulation study and offer the buying and selling strategy by comparing the different kernel functions and the feasibility of these kernels will be shown. Finally, section 4 conclude and introduce the opportunities for future studies.

The model and EM algorithm

This section consists of two subsections. In the first subsection, we introduce the Markov switching model introduced by Nademi & Farnoosh (2014) and in the second subsection, their algorithm for estimating the parameters will be reviewed. Note that, their semi-parametric algorithm is a part of a more general algorithm as EM algorithm that applies to the class of Markov switching models.

where switching between the regimes is controlled by a hidden Markov chain , with values in , and the residuals are i.i.d. random variables with mean  and variance 1.  are random variables which assume as values of the unit vectors , i.e. exactly one of the  is 1, and the others are . The stationary distribution of the hidden regime process is given by the  transition probability matrix A, i.e, . We get the stationary probabilities  by equation .

 is considered as a semi-parametric function such that , where  is a nonparametric adjustment factor and  is a known function of  and  and  is the parametric space. So, we can refer to mean function (2) and rewrite it by the following form:

2. The EM algorithm based on the semi-parametric method

Supposing the definition of  and , Nademi & Farnoosh (2014) applied a special class of log-likelihood function, called "complete" log-likelihood function, by the following form

 Where  and  is the normal density with mean  and standard deviation  The word "complete" refers to this definition that if we would have observed the complete data , instead of just , we could maximize the complete data log-likelihood (see Franke et al. [8]), instead of the ordinary log-likelihood.

Applying this method leads to the use of the Expectation and Maximization algorithm known as the EM algorithm. The EM algorithm repeats between drawing the unseen variables  by conditional expectations  given the seen data  and using an elementary estimate of the parameters on the one phase (E-step), and by maximizing  to get an update of approximations of  or  on the other phase (M-step). These two phases until assuring certain stopping criteria are iterated. The algorithm offered with the EM algorithm can be summarized in the following steps.

E-step: Suppose  and  are given. S0, the conditional expectation of the unseen variables  given are calculated by

M-step: Suppose the approximations  for the unseen variables  are given. Then, the transition probabilities are calculated by

where  are the joint conditional probability of  and   given the entire sequence of observations ( ) estimated by

The probabilities  are approximated by

and the functions  are estimated by

such that, gets from  for  where  is

and  is

where  is a Gaussian Kernel function and  are estimated by

for  where   denote the sample residuals. The optimal selection of the bandwidth  are also estimated by

The estimate of the parameters are obtained by iterating these two steps (E-step and M-step) until convergence.

In relation (3), they applied the Gaussian Kernel function for estimating  . But there is a question that if other types of kernel function can improve performance of the semi-parametric algorithm. By the definition of kernel function, a function , with compact support, is a kernel function satisfying

So, by this definition, we can find other functions that have these conditions. We want to try some other types of kernel functions that are popular in the mathematics field. Table 1 shows several types of kernel functions that are commonly used. These functions consist of Uniform, Triangle, Epanechnikov, Bisquare and Triweight. We also apply the Gaussian kernel to compare this function with the candidate kernel functions. Figure 1 shows the plots of these functions. We apply these functions to compare their ability to the improvement of the EM algorithm. 

Table 1. The popular kernel functions.

Figure 1. The six popular Kernel functions

Research Background

In this section, we first carry out a simulation study to examine the finite sample performance of the proposed kernel functions, and then the Semi-parametric Markov Switching model with the selected kernel function will apply to financial observations, including the Automotive Industry Index of the Tehran Stock market data, to find the strategy of buying and selling in the stocks of industrial companies. Because there is a high correlation between the prices of stocks for automotive companies and the Automotive Industry Index, we focus on the Automotive Industry Index data to offer the strategy of buying and selling.

1. Simulation study

We intend to examine the feasibility of various kernel functions by estimating parameters in the semi-parametric model (1). We generated two sets of data with sample size N=500 based on nonlinear functions  and  by the following forms:

,

and it was supposed that   and . The transition probability matrix was considered by

.

We also chose the variance parameters as and . Figures 2(a) and (d) show the generated observations based on  and  respectively. Figure 1(b, c) and (e, f) show the corresponding scatter plot of   with  and   for two simulated data sets. These plots also indicate the degree of dependency of the data is 2. Such that, this degree of dependency is because of the structure of the model (1).

For comparing the models, we apply two indices the square Root of Mean Squared Error (RMSE) by the following form:

and classifying index "Max . The classifying index "Max is defined by the following definition:

" is belonging to regime k if and only if ."

This index is suitable for the evaluation of the models, such that the proper model is powerful in classifying the observations in the right regimes.

Table 2 shows the estimated parameters of the simulated model based on . Comparing the measures of RMSE for the six kernel functions, we can find that the Triangle kernel with RMSE 0.0710 is more efficient than the others kernel functions. After that Triweight kernel with RMSE 0.0781 can be selected as the proper kernel. Figure 3 shows Max  for six kernel functions in which for Triangle kernel, except for a few cases, almost all are at least greater than 0.8, i.e. there is a clear decision for one of the two phases in the large majority of cases and we find that high percentage of the data is correctly classified. With making change the nonlinear function  by  for the second simulated data, we founded the different order of the kernel functions (table 3). Such that, the Uniform kernel function (RMSE=0.0521) had the best performance among other kernel functions. Also, after the Uniform kernel function, The Gaussian kernel with RMSE 0.0584 is more proper than the others. Figure 4 shows  for the second simulation data set which indicates the ability of the Uniform kernel in classifying the data comparing the others. These results indicate the different efficiency of kernel functions in various problems. This demonstrates in applying the semi-parametric algorithms based on selecting kernel functions, trying different kernel functions can help to get proper estimating.

Table 2. The estimated parameters for the simulated data based on .  

Table 3. The estimated parameters for the simulated data based on .

Conclusion

We have offered the strategy of buying and selling stock in financial markets by a special class of Markov switching models based on the joint conditional probability matrix. This strategy can be capable of selecting the various kernel functions. Such that, the best selection of this function, increase the accuracy of the joint conditional probability matrix.  We have compared a set of kernel functions with the semi-parametric Markov switching models in terms of their ability to capturing the parameters in simulation studies. The estimation results of the models have indicated that the proper kernel function for estimating the parameters refer to the nature of data and there is not a firm decision in selecting these functions. Such that we found the proper performance of the Triweight kernel in the first simulation data set. While we found Uniform kernel as the best kernel function in the second simulation one. This reveals that the Gaussian kernel function is not the best selection in every data modeling. This reality shows that the researchers in estimating processes must be trial some kernel functions to find the best performance of their algorithms and the use of Gaussian kernel function in every algorithm can be deceptive. 

We also suggest to academics that they compare these kernel functions with other semi-parametric and nonparametric models to improve the current knowledge about the better model for forecasting time series data

[1] Autoregressive–Autoregressive Conditionally Heteroscedasticity models.