Bayesian prediction of the stock price rate in the Iraq stock market based on the symmetric heavy tails regression model

prediction


Introduction:
The concept of the Iraq stock market is economic market enjoy financial and administrative independence that is not related to a set party.It is managed a council consisting of (9) members representing the different economic segments the investment sector, named a council of governors.The market had placed where investors meet, as stock markets dealt with in buying and selling.It constitutes one of the channels through which money flows between individuals, institutions and different sectors, which helps in mobilizing and developing and mobilizing and preparing savings for different investment range.Trading in the market takes place according to the manual method and based on the public bidding system written on boards.A plastic board has been allocated to each listed joint stock company, arranged according to the following sectors: banking sector, insurance sector, investment sector, services sector, industrial sector, hotels sector, tourism, and agriculture sector.Sometimes a multiple regression models are estimated when random errors follow a normal distribution (ND).However, there are cases in which the (ND) of errors is not suitable.Therefore, mixture distributions are more suitable, one of these distributions use such as Generalised Multivariate Modified Bessel (GMMB) distribution.This distribution has applications in variety regions that included displaying stock market data .
The researchers [1] estimated parameters of the multivariate semi-parametric regression model when the error follows a matrix-variate (GMB) distribution using the Bayes method when previous information is not available, in addition to testing hypotheses for the model parameters through the Bayes factor criterion.The theoretical results were applied to experimental data and they concluded that the sample that was used was drawn from a population does not belong to the (GMB) community.The researchers [2] tested statistical hypotheses about the population means of a (GMMB), as well as statistical tests related to the equality of the population means, and they concluded the probability distribution of the statistics (Hotelling-T2) and (Scheffe-T2) for the two tests above respectively and when they have the same covariance matrix.The researchers [3] studied Bayes prediction of a multiple linear regression model in case of equal correlations, as it was assumed that the prior probability function of the variance parameter of the general regression model follows a Generalised Inverse Gaussian (GIG) distribution, predictive distribution was obtained that follows a (GMMB) distribution.This result differs inversely with the t-distribution obtained using the inverse chi-square distribution the prior probability function of the variance parameter, where the predictive distribution was based on interclass correlation.The research covered several sections, where the first section was a general introduction to stock market, and the (GMBR) model was described in the second section, as well as Bayesian estimation and Bayesian prediction.The third section included the application of real data related to the Iraq stock market for the purpose of Bayesian prediction based on the best Bayesian estimator.In the fourth and fifth sections, the most prominent conclusions reached by the research and future studies were presented.edings.pdf.

Description of the Generalized Modified Bessel Regression (GMBR) model:
The multiple linear regression models are described by the following linear equation: Where: : Response variable with dimension (n× 1).
: Non-random matrix of explanatory variables with dimension (n× ), where  is number of explanatory variables.

𝛽: A vector of regression model parameters with dimension (p× 1).
: Vector of random errors has amplitude (n×1), which has a (GMMB) distribution, the probability function for  is defined to the following equation: As (λ,  , ) it represents the shape parameters, and the field of its parameters is defined as (): Modified Bessel function of the third type to order (v), and is defined according to the following equation: The probability distribution of  is expressed descriptively by:  ~  (0 ,  2   ,  ,  , ) Equation (1) represents a combination in vector , which follows a (GMMB) distribution.
Therefore, [7] deduced the probability distribution of the random vector Y, which follows a (GMMB) distribution.The probability function for the random response Y, is as follows: Descriptively expresses this distribution ~   ( ,  Location parameter of the (GMBR) model was estimated based on informative prior.In this section, we also find on the predictive distribution when an additional sample is available for the response variable Y and the explanatory variables that sample have no relationship to the sample that was used in the estimation process.The Bayesian prediction properties of the future response variable were also studied.
2.2.1 Estimating the location parameter using informative prior: In this study, [7] concluded the estimation of the location parameter for a (GMBR) model and it was as follows: This estimator is biased and the amount of bias is equal to: Therefore, the mean square error matrix for  ̂ is as follows: After merging the complete posterior distribution of  conditional on  2 , , determined by the following equation: With the probability density function of   conditional by the random variable τ and defined according to the following equation: We obtain that the predictive distribution of (  |, ) is as follows: Since  ̂ was previously defined in equation ( 7), and integrating equation ( 14) relative to the random variable τ, the predictive distribution of   as follows: Where ℜ = (  −    ̂) ´(  −    ̂).
Since the predictive distribution of   is not a common probability distribution, the Bayesian prediction for the response variable Y is found by the following formula:

Applied side:
The Iraqi market during the period between 1992 and 2003 was known as the Baghdad stock market.It was instituted according to Law (24) for the year 1991.This market was a government exchange that was able to list 113 companies (Iraqi and mixed stock), and was able to attract in the last year It has an annual trading rate of over (17,500,000) dollars.The Iraq Stock market was established on (18 April 2004).Temporary Law (74) was issued to establish two important institutions in the capital sector, which Iraq Stock market and Iraq stock Commission. [10]

Determine basic variables and preparing data:
In this side, practical application will be made on data related to the Iraq Stock market, as monthly data related to the services sector, represented by the Iraq Baghdad for General Transportation sector for the year 2018, will be studied, as the effect of both (Stock turnover rate X1, Closing price X2) which represents the explanatory variables on (Stock price rate Y) which represents the response variable, and Table 1 shows the measurements for the study variables measured in Iraqi dinars.Source: http://www.isx-iq.net/isxportal/portal/homePage.htmlBefore using the (GMBR) model to represent the relationship between variables, the data is time series or not.This was done by drawing Auto-correlation function for stock price rate; it became clear from the drawing that the data is white noise.Figure 1 shows a plot autocorrelation function for stock price rate variable:  In order to determine the suitability of the data to the model used, the study data was tested through goodness of fit and based on the default shape parameters (λ = 5, ψ = 2, v = 3).The value of the Kolmogorov-Smirnov test extracted (0.2140) for this model was lowerhan the tabulate value   (0.05,12) = 0.2420, which indicates that the data fit to model used.
Multicollinearity between explanatory (independent) variables was tested based on criterion called the Variance Inflation Factor (VIF).Where both values were less than 10, which indicates that there is no problem of multicollinearity, as shown in the following Table 2: The sample data was divided into two parts, the first consisting of a random sample of size (n=10).This data was used for estimation and the last two observations were used for the purpose of prediction.The parameter vector β was estimated for the sample data with a size of (n=10) by the classical method to choose the initial values, these values were as follows: In this aspect, the location parameter for the (GMBR) model will be estimated based on the shape parameters (λ=5, ψ=2, v=3) and future values will be predicted.The following Table 3 shows the location parameter estimation using the Bayesian method.4. We note from the above figure that the estimated values of the response variable vector have the same pattern as the real values, which indicates that the estimated model was appropriate to the study data.

Bayesian prediction:
In this study, future values will be predicted based on the best Bayesian estimator.The predictive value represents the predictive mean defined in equation (16).

Figure 1 :
Figure 1: Plot of Auto-correlation function for stock price rate variable significance limits for the autocorrelations)

Figure 2 :
Figure 2: Behavior of the real and estimated values of the stock price rate

Figure 2
Figure2shows the real estimated values for Table4.We note from the above figure that

Table 1 :
Measurements of the variables of closing price and stock turnover rate affecting the stock price rate for the year 2018

Table 2 :
Variance inflation factor values for explanatory variables

Table 3 :
Location parameter estimation for the generalized

Table 4
shows the real and estimated values of the stock price rate by using the Bayesian estimator based on informative prior.

Table 4 :
Real and estimated values for the stock price rate

Table 5 :
Real and Predictive Values