APPLICATION OF THE COPULA METHOD TO ANALYZE THE RELATIONSHIPS OF MACROECONOMIC FACTORS AFFECTING THE CSPI

ABSTRACT


INTRODUCTION
The Composite Stock Price Index (CSPI) is used to measure the performance of stocks combined in the capital market, allowing investors to look at market conditions and determine the investment strategy [1]. For the national economy, the rise and fall of the CSPI can describe macroeconomic conditions, which, if economic conditions weaken, can impact company performance in general so that business also cuts, eventually impacting company profits which decline. This also; this investor confidence to decrease.
The factors that cause the rise and fall of the CSPI are macroeconomic factors in the form of exchange or exchange rates, interest rates, inflation rates, and the number of regulations or economic deregulation. In addition, [2] describe several factors that affect the CSPI of the Indonesia Stock Exchange. Inflation is a wrong signal for investors, indicating that company earnings are declining due to weak purchasing power. The same thing can also happen if there is a change in interest rates, where when the economic situation worsens, the Central Bank will reduce interest rates to boost the economy because if there is a decline in the economy, this can weaken investor confidence so that the CSPI will also experience a fall. The same will also be experienced if the rupiah exchange rate continues to change; this will affect the company's balance sheet and impact price adjustments in the capital market. The exchange rate decline could reflect weaknesses, especially in exports and international finance [3].
Based on this, it is necessary to examine how much the above factors can affect the rise and fall of the CSPI. Bearing in mind that the CSPI can describe the economic condition of a country. In statistics, correlation analysis can be used to analyze the influence or relationship between variables. Correlation analysis is one of the methods used to detect relationships between associations that assume that the data is assumedly distributed [4] and [5]. However, sometimes this assumption cannot be fulfilled properly. Therefore is needed to overcome this so that the Copula can be an alternative.
The Copula is a method with flexibility because it does not have a data normality assumption requirement, so this method is considered appropriate for use in cases in the financial sector where in general, it is often found that the data has an abnormal distribution. Besides that, the Copula can also combine several marginal distributions into a joint distribution The Copula technique has been widely used for financial and actuarial applications. Cases in the financial sector are generally found to have data that is not normally distributed. Research that uses Copula in the financial industry includes [9], [10] which analyses the relationship between macroeconomic factors and the world gold price level. This study used the Copula Archimedean approach, in which the parameters were estimated using Tau Kendall. Then research by [10] has also researched to estimate claims reserves for several lines of business using the Archimedean Copula and the Generalized Linear Model. Apart from that, another study using the Ellips Copula Family is researched by [12] in calculating the value of risk in a stock portfolio.
The difference between this research and the research above is that this study uses two Copula families, namely Archimedean Copula and Elliptical Copula, in its analysis. So the purpose of this study is to apply the Copula approach in identifying the relationship of macroeconomic factors in the form of inflation, interest rates, and exchange rates to the Stock Price Index (CSPI) and to choose the best Copula model using the Maximum Likelihood Estimation (MLE) method by previously estimating parameters with Tau Kendall as part of the procedure.

Research Variables
The variables used in this research are independent variables and dependent variables. The dependent variable is CSPI (Y) and the independent variables are macroeconomic factors in the form of inflation ( 1 ), interest rates ( 2 ) and exchange rates ( 3 ).

Data Sources
This study uses secondary data obtained from the official website of the Indonesia Stock Exchange for CSPI data, while inflation and interest rate data are obtained from the Central Statistics Agency (BPS) website via www.bps.go.id, and data on the Rupiah exchange rate against the USD obtained through the website satudata.kemendag.co.id. The data used is monthly data from 2018 to 2021.

Stages of Data Analysis
The stages in analyzing the data are as follows. 1) Describe the data to obtain an overview based on the descriptive statistics of the research data.
2) Testing the normality of the data to see whether the data distribution is normal. If any of the var are not normal, then the analysis is continued to Copula analysis with all normal variables still used. 3) Calculate the Tau Kendall correlation coefficient used in the Copula parameter estimation. 4) Identify the relationship between variables with Copula.
• Transform random variables into a uniform distribution [0,1] using the Van der Waerden equation.
• Make a Scatterplot of the transformation results.
• Selecting the best Copula using the Maximum Likelihood Estimation (MLE) method based on the largest log-likelihood value. • Make the selected Copula Scatterplot by first generating n=5000 data based on the selected Copula parameters so that the shape and tail of the copula can be seen clearly. 5) Create a Selected Copula Model based on the MLE fitting result Parameters.

Copula Family
Copula has two families, namely the archimedean Copula family and the elliptical Copula family. The Archimedean Copula family has a distribution with tail dependencies which indicates the possibility of a relationship between the two variables in a range of extreme regions. Archimedean Copula is divided into three classes, namely Copula Frank, Copula Clayton, and Copula Gumbel, each of which has a different tail. Copula Frank, who has no tail, Copula Clayton, who has a tail at the bottom, and Copula Gumbel, who has a tail at the top.
The Copula archimedean family generator can be presented in Table 1 below [13].

Table 1. Generators of the Archimedean Copula Family
Archimedean Copula Generator The Archimedean Copula generator is obtained from Equation (6). Meanwhile for the Elliptical Copula Generator it can be done using the following Equation

Uniform Distribution [0.1]
In the Copula analysis the initial stage is to change the random variable to form a uniform [0.1], If the marginal distribution of the random variable is unknown, then the equation becomes: In the process of transforming these random variables, it is necessary to rank each random variable, namely 1 ( ) , 2 ( ) , … , ( ) which is the rank of 1 , 2 , … , where previously formed matrices each divided by + 1, for example as follows.
with r is the rank and n is the amount of data [15] .

Copula Parameter Estimation and Model Selection
The Tau Kendall correlation value can be used to estimate the parameters of Copula. The parameter estimation procedure uses the following equation [14]. ..

Description of Variable
Obtain an overview of all the variables in the study. Description of variables from the research data are presented in the following table. Based on Table 2. The descriptive statistics of each variable used in the study show N = 48, which means that the research data for each variable is 48. The value in the variable indicates the level of the CSPI, while the value in the variable shows 1 the 2 percentage from inflation and interest rates, and the value in the variable 3 shows the price of the Rupiah exchange rate against the USD.

Normality Testing
Data normality testing was carried out to test the distribution of research data meets the normal distribution or not. For the case of the Copula approach method, it does not require normally distributed data. If all variables or some variables are not normally distributed, then Copula is the right method to use. Below is presented the p-value of normality test results with the Shapiro-Wilk test.  Table 3 shows that the variable 1 with a significant level p-value (0.3981) > 0.05 fulfills the normal distribution. While the other variables do not meet the normal distribution. The above causes the correlation cannot be used properly because most of the variables do not follow a normal distribution. So that the analysis of the dependency of the CSPI on macroeconomic factors in the form of inflation, interest rates, and exchange rates can be carried out using the Copula approach.

Tau Kendall Correlation Coefficient
The value of the correlation coefficient is useful for estimating Copula parameters. The largest positive correlation coefficient is 1, and the largest negative correlation coefficient is -1, while the smallest is 0. The relationship between variables is said to be perfect if the correlation coefficient is 1 or -1 because the smaller the correlation coefficient, the greater the error. The results of the analysis of the coefficients are as follows.  Table 4 shows the p-value of the three correlation coefficients. It can be seen that the CSPI variable and the exchange rate have a significant (p-value < 0.05), meaning that the variables have a close relationship. While the CSPI variable with inflation and interest rates is not significant (p-value > 0.05), meaning that the two variables have a weak close relationship.
The Tau Kendall correlation coefficient obtained in Table 4 is used to estimate the Archimedean Copula and Elliptical Copula parameters.

Transformation of Random Variables to Uniform Distribution [0,1]
In the analysis with Copula, the variable is transformed to a uniform marginal distribution [0,1]. The following shows the scatter plot of the transformed data in the form of a plot of the relationship pattern of variable Y (CSPI) with each variable X (inflation, interest rates, and exchange rates). Figure 1, a scatterplot of Transformation Results to Unifrom Distribution [0.1], it can be seen that the spread of data between Y and X1 dots has a spread but is one pattern in the form of a straight line. This means that the relationship between the two variables is quite low. Meanwhile, for the distribution of data between Y and X2, there are more patterns in the form of lines. This means that there is a relationship between variables. Meanwhile, the distribution of data between Y and X3 has a negative relationship because when the variable X3 increases, it is followed by the variable Y decreasing. To identify the relationship further, the Copula method is used.

Estimation of Archimedean Copula and Elliptic Copula Parameters Using Kendall 's Tau
Based on Table 5, if the parameter estimate is positive, it means that there is an indication of a positive relationship between the CSPI variables and macroeconomic factors, whereas if the parameter estimate is negative, there is an indication of a negative relationship between the variables. The positive relationship  in question is that variable X is only related to variable Y when variable X is very high. Furthermore, to obtain the best Copula model that can explain the dependency structure between CSPI variables and macroeconomic factors. The best copula that can explain the best dependency structure is chosen based on the largest loglikelihood value using the maximum likelihood ratio (MLE) method. Calculations for each variable with Copula types are presented in the following table. Based on Table 6, the dependency model between the CSPI and Inflation variables follows the Gumbel Copula because it has the largest log-likelihood value compared to other Copulas with a parameter of ̂= 1.264. Likewise, the dependency model between the CSPI variable and interest rates follows the Gumbel Copula with parameter ̂= 1.175. While the CSPI and exchange rate variables follow Copula Student-t with parameters ̂= −0.6037. Next, make a plot to see the pattern of relationships between variables. In this case, because the amount of data is only n = 48, it is generated as many as n = 5000 points so that the pattern of relationships between variables with Copula types can be seen clearly. The dependency plots and selected Copula types based on parameters between variables can be shown in Figure 2. Figure 2, it can be seen the pattern of each selected Copula. It is known that the pattern formed by Copula Gumbel has a tail at the top, while Copula Student-t forms an ellipse with a tail above and below.

Based on
The Gumbel Copula is part of the Archimedean Copula, which has a dependency tail at the top, which means that when the explanatory variable is very high, it will have a relationship with the response variable. When the explanatory variable is low, the relationship between the two is low or has no relationship. So it can be concluded that inflation only has a dependency or relationship with the Composite Stock Price Index (CSPI) when inflation is very high, but when inflation is low or very low, then the relationship with the CSPI is very low or has no relationship.
The same thing also happens if the interest rate set by Bank Indonesia is very high, then the interest rate will have a relationship with the Composite Stock Price Index (CSPI), otherwise, if the interest rate set by Bank Indonesia is low, then the interest rate has no relationship with the Index-Joint stock price. While the relationship between the CSPI and the exchange rate follows the Copula student-t, where this Copula is part of the Ellips Copula, which has a tail at the top and bottom, this means that the exchange rate can have a close relationship with the Composite Stock Price Index (CSPI) when the exchange rate conditions very high or very low.

CONCLUSIONS
Based on the results of the analysis that has been done, it can be concluded that analysis of the relationship between the CSPI and macroeconomic factors in the form of inflation, interest rates, and the rupiah exchange rates using the Copula approach found that the CSPI and inflation and interest rates only have a relationship when inflation and interest rates are very high, while when both are low, they have no relationship with CSPI or can not affect the CSPI. While the exchange rate has a relationship with the CSPI both when the exchange rate is very high or very low. Best Copula models: