Application of Equality Test of Coefficients of Variation to the Heteroskedasticity Test

The presence of heteroskedasticity in a considered regression model may bias the standard deviations of parameters obtained by the Ordinary Least Square (OLS) method. In this case, several hypothesis tests on the model under consideration may be biased, for example, CHOW’s coefficient stability test (or structural change test), Student’s t-test and Fisher’s F-test. Most of the heteroscedasticity tests in the literature are based on the comparison of variances. Despite the multiplication of equality tests of coefficients of variation (CVs) that have appeared in the literature, to our knowledge, the first and only use of the coefficient of variation in the detection of heteroskedasticity was offered by Li and Yao in 2017. Thus, this paper offers an approach to determine the existence of heteroskedasticity by a test of equality of coefficients of variation. We verify by a Monte Carlo robustness and performance test that our method seems even better than some tests in the literature. The results of this study contribute to the exploitation of the statistical measurement of CV dis-persion. They help technicians economists to better verify their hypotheses before making a scientific decision when making a necessary forecast, in order to contribute effectively to the economic and sustainable development of a company or enterprise.


Introduction
Gauss-Markov's theorem states that the least squares estimator is called BLUE, because it is the Best linear Unbiased Estimator, in the sense that it provides the lowest variances for estimators ( [1], p. 53). However, the presence of heteroskedasticity in a considered regression model may bias the standard deviations of parameters obtained by the Ordinary Least Square (OLS) method ( [2] [4], the application of the Rényi divergence proposed by Pardo (1999) [5], the test based on a numerical approach by Gokpinar (2015) [6], the Forkman test [7], McKay and Miller's statistics [8].
To our knowledge, the first use of the coefficient of variation in the detection of heteroskedasticity was offered by Li and Yao (2017) [9]. Thus, the question is: "is it possible to find an application of these CV equality tests to detect the existence of heteroskedasticity?" The rest of this article is organized as follows: Section 2 will discuss the position of our problem; Section 3 will present a state of the art on heteroskedasticity test; Section 4 will propose an approach to using a CV equality test when detecting heteroskedasticity; and finally, a conclusion is given at the end.

Position of Problem
We have a simple linear regression model 0 1 , 1, t t t y a a x t n = + + =  (1) such that the t  are the errors made when applying the model. We want to check if the variance of the errors is constant for t ranging from 1 to n. That is, we want to test if the model is homoscedastic or heteroscedastic. Figure 1 shows an example of homoscedastic model, and    heteroscedastic model. We note that these four models all have the same regression line equation:

State of the Art on the Homoskedasticity Test
We consider the general linear regression model Y Xa = +  . The various tests, which we will mention below, consist in testing the following hypothesis: ( )

Breusch-Pagan Test
The Breusch-Pagan Test assumes that the squares of the errors 2 i  are related to the dependent variable Y. According to Leblond (2003) where k is the number of explanatory variables i x , n is the sample size and 2 R is the coefficient of determination of 2  and Y.
Decision-making: We accept the null hypothesis 0 H at the confidence level is the critical value of F-distribution at risk α , at k and 1 n k − − degrees of freedom.

Goldfeld-Quandt Test
The Goldfeld-Quandt test assumes that there is an explanatory variable i X that influences the variance of errors, such as ( ) 2) Divide the observations into two groups: where 1 3 n n = and 2 2 3 n n = .
3) Calculate the error variance estimators for each sub-sample: Y X a n n k e n n k where â is the estimator of the parameter a by the least squares method, The GQ statistic follows the F-distribution at 1 1 n k − − and

Gleisjer's Test
The Gleisjer test can detect both heteroskedasticity and the form that this heteroskedasticity takes ([1], p. 150). The Gleisjer test assumes that there is a relationship between the error  of the model and the variable i X assumed to be the cause of heteroskedasticity. The steps of the test are summarized as follows: Step 1: Determination of the residues generated by the suspected variable i X .
1) Regress Y to X. This gives the simple regression model , 1, 2) Calculate the estimators of a and b using the Ordinary Least Squares method: â and b .
3) Estimate the model's residues k ε by its estimators: Thus, the vector of residues k e is known.
Step 2: Proposal of possible forms of existing heteroskedasticity.
Gleisjer suggests testing different forms of possible relationships between e and i X , for example: 1) Type 1: where k v is the residue of this model. This relationship generates the type of 2) Type 2: This relationship generates the type of heteroskedasticity 2 This relationship leads to heteroskedasticity of type Step 3: Detection of heteroskedasticity Significance test of the regression coefficient 1 a : , for a type 1 relation ship; , for a type 2 relation ship; 1 , for a type 3 relation ship.
Decision-making: The null hypothesis If the existence of heteroskedasticity is validated, then the relationship with the highest * t represents the form of existing heteroskedasticity.

White's Test
White's test consists in testing the existence of a relationship between the square of the residue and one or more explanatory variables or its squares. The test procedures can be summarized as follows: Step 1: Determination of model's residues.
1) When the parameters of the model Y Xa = +  are estimated, then we have the estimation of the residues: ( ) x and validation. 3) We consider the model: what can be written in matrix form: The estimator of u is: Calculate the variance of the errors: ) Calculate the variance-covariance matrix of parameters i a and i b : In this case, the variance of i-th element of the vector u is: ˆi The statistics * Decision-making: The null hypothesis 0 H is rejected at the confidence level

ANOVA Methods
In order to determine the existence of heteroskedasticity, researchers proposed the method of analysis of variances, commonly said ANOVA. According to the application example presented in ( 3) Group the values of the variable to be explained Y according to their corresponding classes ( i y in the class corresponding to i x ). Thus, we obtain z samples of Y. 4) Apply the ANOVA test to the z samples of Y, then draw a conclusion.
In the following subsections, we will present some ANOVA tests that can be done in step 4.

Bartlett's Test
Bartlett's statistic 1 is defined as follows: H is rejected at con-

Levene's Test
The Howard Levene's statistic proposed in 1960 ( [13], p. 4) is defined as follows: where,  z is the number of groups or value categories obtained,

Zhaoyuan Li and Jianfeng Yao Test
Zhaoyuan Li and Jianfeng Yao [9] proposed two measures to detect heteroskedasticity in a multivariate linear model.

1) Test based on the likelihood ratio:
2) Coefficient of variation test: ( ) This last test shows a trend in the use of coefficient of variation in the detection of heteroskedasticity.

Our Approach
In this section, we will show that the test of equality of coefficients of variation allows us to detect the existence of heteroskedasticity. The steps of our approach can be summarized as follows: 1) Estimate the parameter a of the regression model of Y to X, noted as â .

4)
As the Goldfeld-Quandt method, divide the residue squares into two groups:

Monte Carlo Simulation
Now, we will test the robustness of these measures proposed in the literature and the one in which we have proposed.

Methodology
Like the Gleisjer method, our simulation consists of generating two variables X Moreover, in order to enrich the forms of heteroskedasticity studied, we also propose to take the other three forms considered by Li and Yao: 4) , p. 15). In this simulation, we consider only the simple regression model. We repeat 100 m = times this test, and we count the number k of times the test rejects the 0 H hypothesis at the 95% confidence level. Then, the probability p k m = is calculated.
As p is a random variable, then we repeat these procedures several times (1000 times), then we calculate ( ) , then the test is considered robust. In addition, the measure with the highest p is the measure considered most sensitive to the type of error i considered ( 1,6 i = ). As we want to test the robustness of the test, then it would be better to check whether the test in question detects small variations or not. During the simulations we did, we took 3 a = , 2 b = , 0 2 a = and 1 1 a c = = . We took 1 1 a = , because it is already different from 0, but judged subjectively low value.
In Table 1, the probabilities 1 p , 2 p , 3 p , 4 p , 5 p , 6 p , 7 p and 8 p correspond respectively to the rejection probabilities of the null hypothesis 0 H of the Breush, Goldfeld-Quandt, Gleisjer, White, Bartlette, Levene, Li and Yao tests, and our proposal.

Simulation Results
From Table 1, we obtain the classifications in Tables 2-6.

Discussion
First of all, from these simulations, it is indisputable that the Levene test is the most robust and sensitive of all the tests considered in this study.
However, these results show that, among the 06 forms of heteroskedasticity proposed, our proposal can detect 04 for 50 n < , and 05 for 50 n ≥ . In general, our proposal fails to detect the only form of heteroskedasticity . In addition, our proposal seems better than the Li and Yao test, which is, to our knowledge, the first tendency to use the coefficient of variation to detect heteroskedasticity.          tlette's test. Indeed, we see from these results that this test is less robust than our proposal.

Conclusions
In this paper, we proposed a technique to detect the existence of heteroskedasticity by an equality test of the coefficients of variation. Next, we also presented the heteroskedasticity test of Zhaoyuan Li and Jianfeng Yao. To the best of our knowledge, the Zhaoyuan Li and Jianfeng Yao test was the first tendency to use coefficients of variation to determine the existence of heteroskedasticity.
Among the equality tests of coefficients of variation available in the literature, we have considered Forkman's test to illustrate our approach, as it is a robust and stable test for a sample with size 3 n ≥ . The results of our performance tests have shown that our approach can detect 5 types of heteroskedasticity among the 6 types considered in this paper.
At the end of this analysis, we affirm that the equality test of coefficients of variation allows us to detect the existence of possible heteroskedasticity in a simple regression model. Thus, our study contributes to the reapplication of several equality tests of coefficients of variation that have already appeared in the literature.