Investigating Performances of Some Statistical Tests for Heteroscedasticity Assumption in Generalized Linear Model: A Monte Carlo Simulations Study

In a linear regression model, testing for uniformity of the variance of the residuals is a significant integral part of statistical analysis. This is a crucial assumption that requires statistical confirmation via the use of some statistical tests mostly before carrying out the Analysis of Variance (ANOVA) technique. Many academic researchers have published series of papers (articles) on some tests for detecting variance heterogeneity assumption in multiple linear regression models. So many comparisons on these tests have been made using various statistical techniques like biases, error rates as well as powers. Aside comparisons, modifications of some of these statistical tests for detecting variance heterogeneity have been reported in some literatures in recent years. In a multiple linear regression situation, much work has not been done on comparing some selected statistical tests for homoscedasticity assumption when linear, quadratic, square root, and exponential forms of heteroscedasticity are injected into the residuals. As a result of this fact, the present study intends to work extensively on all these areas of interest with a view to filling the gap. The paper aims at providing a comprehensive comparative analysis of asymptotic behaviour of some selected statistical tests for homoscedasticity assumption in order to hunt for the best statistical test for detecting heteroscedasticity in a multiple linear regression scenario with varying variances and levels of significance. In the literature, several tests for homoscedasticity are available but only nine: Breusch-Godfrey test, studentized Breusch-Pagan test, White’s test, Nonconstant Variance Score test, Park test, structures: exponential and linear (generalize of square-root and quadratic structures) were injected into the residual part of the multiple linear regression models at different categories of sample sizes: 30, 50, 100, 200, 500 and 1000. Evaluations of the performances were done within R environment. Among other findings, our investigations revealed that Glejser and Park tests returned the best test to employ to check for heteroscedasticity in EHS and LHS respectively also White and Harrison-McCabe tests returned the best test to employ to check for homoscedasticity in EHS and LHS respectively for sample size less than 50.


Introduction
One of the crucial assumptions in the multiple linear regression models is that the variance of the errors should be constant [1]. The Ordinary Least Squares (OLS) method is very popular with statistics practitioners as it provides efficient and unbiased estimates of the parameters when the assumptions, especially the assumption of homoscedastic error variances, are met. But in many real-life applications, variances of the errors vary across observations. Since homoscedasticity is often unrealistic assumption, researchers should consider how the results are affected by heteroscedasticity. Even though the OLS estimates retain unbiasedness in the presence of heteroscedasticity, its estimates become inefficient [2] [3].
However, heteroscedasticity yields hypothesis tests that fail to keep false rejections at the nominal level, or estimated standard errors as well as confidence intervals that are either too narrow or too large [4]. Every statistical procedure carries with it certain assumptions that must be at least approximately true before the procedure can produce reliable and accurate results [3]. Researchers often apply a statistical procedure to their data without checking on the validity of the assumptions of the procedure. If one or more of the assumptions of a given statistical procedure are violated, most especially in multiple linear regression analysis, then misleading results will be produced by the procedure [5].
In short, a number of assumptions are associated with the analysis of data using OLS in multiple linear regression but the current study deals with only one of them, that is, homogeneity of variance assumption. Literarily, assumptions refer to basic principles that are accepted on faith, or assumed to be true, without proof or verification. It is frequent and common that a researcher applies a statistical method to a set of data without thoroughly checking that the assumptions of the methods are valid [6]. This may be especially true in multiple linear modelling. Notwithstanding, it is a known fact that all statistical procedures should have underlying assumptions; some are more stringent than others. In some cases, violation of these assumptions will not change substantive research conclusions. In other cases, violation of assumptions will undermine meaningful research. Establishing that one's data meet the assumptions of the procedure one is using is an expected component of all quantitatively-based theses, journal articles, and dissertations. This assumption practically and usually exists in regression and experimental design but this research discusses its relation with regression analysis.
The homogeneity of variance assumption is one of the critical assumptions underlying most parametric statistical procedures such as the analysis of variance and it is very important for a good researcher to be able to test this assumption before the application of ANOVA technique. Simply, the term "homo" means "the same" while "hetero" means different, therefore variance homogeneity assumption, which is equivalently called "homoscedasticity", means that the variance of each residual should be the same throughout the experiment. If the errors (residuals) fail to possess equal (but sometimes unknown) variances, the reliability of application of analysis of variance technique may be badly affected [3]. Direct opposite in meaning to "homogeneity assumption" is "heterogeneity of error variances", which simply refers to a situation where the variance of the residuals is affected by at least one predictor variable leading to unequal magnitude in spread. Thus, heterogeneity problem may arise in most of the economic (econometric), experimental and agricultural modelling where specifically analysis of variance technique is applied. Thus, homogeneity of variance is a major assumption underlying the validity of many parametric tests. More importantly, it serves as the null hypothesis in substantive studies that focus on cross-or within-group dispersion.
In addition, showing that several samples do not come from populations with the same variance is sometimes of importance per se. The statistical validity of many commonly used tests such as the t-test and ANOVA depend on the extent to which the data conform to the assumption of homogeneity of variance. When a research design, however, involves groups that have very different variances, the p-value accompanying the test statistic, such as t and F may be too lenient or too harsh. Thus, substantive research often requires investigation of cross-or within-group fluctuation in dispersion. For example, in quality control research, homogeneity of variance tests is often "a useful endpoint in an analysis" [7]. In human performance studies, an increase or decrease in the dispersion of performance scores within the same group of subjects may shed light on how changing condition affect human behaviour. Recent studies on gender-related differences in the dispersion of academic performance have provoked substantive as well as methodological interest in homogeneity of variance [8] [9].
It has been reported in some literatures that the assumption of the error term is such that its probability distribution remains the same over all observations of for all values of the predictor variables [10]. This assumption is also known as Assumption of Homogeneity of Variances or the Assumption of Constant Variance of the error term. If it is not satisfied in any particular case, we say that the error term ( i e ) is heteroscedastic. The meaning of the assumption of homoscedasticity is that the variation of each error ( i e ) around its zero mean does not depend on the values of predictor variables. The variance of each i e remains the same irrespective of small or large values of the explanatory variables.
Apparently, the present research intends to investigate the best statistical test, through the computation of the number of time (frequency) each test commits type II error (when sigma = 0) and type I error (sigma ≠ 0) for confirming homoscedasticity assumption when different levels of heteroscedasticity are injected into the multiple linear regression models at 30, 50, 100, 200, 500 and 1000 sample sizes.

Aim and Objectives of the Study
This study aims at providing a comprehensive comparative analysis of asymptotic behaviour of some selected statistical tests for homoscedasticity assumption in order to hunt for the best statistical test for detecting heteroscedasticity in a multiple linear regression scenario with varying variances. In order to achieve this aim, the following objectives are pursued: 1) To compare nine different statistical tests under different heteroscedastic conditions such as Exponential and Linear (generalized structure for Quadratic and Square-root) Forms; 2) To evaluate, through the computation of the number of time (frequency) each test commits type II error (when sigma = 0) and type I error (sigma ≠ 0) as the case may be; 3) To investigate the asymptotic behaviour of the selected tests when the variances are varied across all simulations.

Theoretical Framework and Literature Review
The analysis of variance (ANOVA) is one of the most important and useful techniques for variety of fields such as economics, agriculture, biology and so on with a view to comparing different groups or treatments with respect to their means [3]. Let us consider testing equality of means of k populations given samples Hence, a set of assumptions such as normality, homogeneity and independence of observations has to be made in order to employ an F test for (2.1). As [1] has pointed out, in practice, the assumption of homogeneity of variances is the one most often unmet in ANOVA. In the absence of homogeneity of variances, Unfortunately, these tests (Bartlett and Levene) are sensitive to the assumption of normality [20]. Specifically, the probability of a Type I error ( α ) is dependent upon the kurtosis of the distribution. However, these alternative tests are affected adversely by non-normality [21]. As [22] pointed out, none of the procedures directly handle the problem of skewness bias. [23] has also argued that failure to consider the impact of the combined violations of variance equality and distribution normality is an important omission of a statistical procedure. Under these circumstances, the development of new alternative tests, namely, trimming, transforming statistics, bootstrapping [24] to deal with unequal variances, and non-normality is worthwhile. [25] employed four homogeneity tests, named SNHT, Buishand Range test, Pettitt test and Von Neumann Ratio (VNR) test to the European Climate. The results are categorized into three classes, which are useful, doubtful and suspect according to the number of tests rejecting the null hypothesis. Three testing variables were used, each consists of annual values. For temperature, the two testing variables are annual mean of diurnal temperature range and annual mean of the absolute day-to-day differences. Meanwhile for precipitation, the annual number of wet days (threshold 1 mm) is employed.
Tests for equality of variances are of interest in many situations such as analysis of variance or quality control [26]. The classical approach to hypothesis testing begins with the likelihood ratio test under the assumption of normal distributions given by [27].  [28]. This problem was also studied by [7], who performed an increasing study of many of the existing parametric and non-parametric tests. In his last paper Monte Carlo simulations of some distributions for several sample sizes show a few tests that are robust and have good power.

Forms of Heteroscedasticity
The study considers four different heteroscedastic structures coined from [29] additive and multiplicative heteroscedastic model but in our model, we assume that the variance of the error varies as the mean of the responses. The two general forms are: 3) Square-rooted Form:

Procedure for the Monte Carlo Simulation Experiment
To investigate the finite sample properties of the test statistics of the presence of heteroscedasticity in any given dataset, we use a Monte Carlo experiment. We simulate a linear multiple regression model with three explanatory variables model using a simple Least square function of the form: where e is a normal error variable. Following [30], we then simulate the independent variables X i as follows: X i 's are a set of independent variables that are fixed following 1 1, , We also generate the two error terms as follows Then the model for the heteroscedasticity function was formed as follows: The parameters were then set at:

Tests for Detecting Heteroscedasticity
This study considers nine (9)

Park Test
Park (1966) formalizes the graphical method by suggesting that 2 i σ is some function of the explanatory variable i X . The functional form he suggested was: where i v is the stochastic disturbance term. Since 2 i σ is generally not known, Park suggests using 2 i u as a proxy and running the following regression: 2 2 ln ln ln ln If β turns out to be statistically significant, it would suggest that heteroscedasticity is present in the data. If it turns out to be insignificant, we may accept the assumption of variance homogeneity. The Park test is thus a two-stage procedure. In the first stage, we run the OLS regression disregarding the heteroscedasticity question. We obtain ˆi u from this regression, and then in the second stage, we run the regression (3.5).
Although empirically appealing, the Park test has some problems. [31] has argued that the error term i v entering into (3.5) may not satisfy the OLS assumptions and may itself be heteroscedastic. Nonetheless, as a strictly exploratory method, one may still use the Park test.

Glejser Test
The [32] is similar in spirit to the Park test. After obtaining the residuals ˆi u from the OLS regression, Glejser suggests regressing the absolute values of ˆi u on the X variable that is thought to be closely associated with 2 i σ In his experiments, Glejser used the following functional forms: Onifade where i v is the error term.
Again, as an empirical or practical matter, one may use the Glejser approach.
But Goldfeld and Quandt pointed out that the error term i v has some problems in that its expected value is nonzero, it is serially correlated. Above all, we shall choose the best form of regression which gives the best fit from the viewpoint of correla-

Goldfeld-Quandt Test
This is a simple and intuitive test. One orders the observations according to i X and omits c central observations. Next, two regressions are run on the two sepa- The only remaining question for performing this test is the magnitude of c. Obviously, the larger c is, the more central observations are being omitted and the more confident we feel that the two samples are distant from each other. The loss of c observations should lead to loss of power. However, separating the two samples should give us more confidence that the two variances are in fact the same if we do not reject homoscedasticity. This trade off in power was studied by [32] using Monte Carlo experiments. Their results recommend the use of c = 8 for n = 30 and c = 16 for n = 60. This is a popular test, but assumes that we know how to order the heteroscedasticity. In this case, we use i X . But what if there are more than one regressor on the right-hand side? In that case one can order the observations using ˆi Y .

Breusch-Pagan Test
The success of the Goldfeld-Quandt test depends not only on the value of c (the Assuming that the error variance 2 i σ is described as follows: that is, 2 i σ is some function of the non-stochastic variables z's; some or all of the x's can serve as z's. Specifically, assume that: that is, 4) Obtain the regression such that

White Test
Another general test for homoscedasticity where nothing is known about the form of this heteroscedasticity is suggested by [5]. This test is based on the difference between the variance of the OLS estimates under homoscedasticity and that under heteroscedasticity. This test does not rely on normality assumption making it very easy to implement. Consider the following three-variable regres- The White test is tailored as follows: 1) Given a set of data, obtain the residual, i e from (3.16); 2) Run the following auxiliary regression: R obtained from the auxiliary regression asymptotically follows the chi-squared distribution with degree of freedom equal to the number of regressors (excluding the constant term) in the auxiliary regression. Mathematically, we have: 5) It is expected that the null hypothesis will be rejected when 2 nR exceeds the critical value obtained from chi-square table at a given level of significance.
It is observed that if a model has several regressors, then introducing all the regressors, their squared (or higher-powered) terms, and their product can quickly consume degreed of freedom. Therefore, one must be very cautious of using the test; this is one of the demerits of this test.

Spearman's Rank Correlation Test
This test ranks the i x 's and the absolute value of OLS residuals, the i e 's. Then it computes the difference between these rankings, that is, For this simple linear regression model, we obtain the Spearman's Rank Correlation Coefficient as follows: Having obtained (3.19), the next step is to test for the significance of the coefficient using t-test as follows: The statistic is t-distributed with (n − 2) degree of freedom under any level of significance.
In a situation where the number of regressors are more than one (multiple linear regression case), that is,

Breusch-Godfrey Serial Correlation
The Breusch-Godfrey serial correlation LM test is a test for heteroscedasticity in the errors in a regression model. It makes use of the residuals from the model being considered in a regression, and a test statistic is derived from these.
This test is valid with lagged dependent variables and can be used to test for heteroscedasticity Procedure Step 1. Estimate.
obtain the residuals (e t ).
Step 2. Estimate the following auxiliary regression. model: Step 3. For large sample sizes, the test statistic is: Step 4. If the test statistic exceeds the critical chi-square value we can reject the null hypothesis of no serial correlation in any of the ρ terms.
Other tests are "Non-constant variance score test" and Harrison-McCabe test.

Heteroscedasticity Correction
From the section above, a general linear regression model with the assumption of heteroscedasticity can be expressed as follows: Noting that the t subscript attached to sigma squared indicates that the disturbance for each of the n-units is drawn from a probability distribution that has a different variance.
Given such a non-constant variance function where α is the unknown parameter in the model.

Taking the natural logarithm
Then taking exponential of equation If the variance depends on more than one explanatory variable (a multiple regression case) Taking the exponential function is best because it gives non-negative Using the OLS technique to estimate the coefficients 1 2 , , , s and so on.
We then took the square root of the exponential of the fitted estimate ( ) Then ˆi σ is the weight required to transform the data set by dividing through. But; ( ) Using the estimate of our variance function 2 We then defined the transformed variable as Therefore; which is the Weighted Least Squares model with homoscedasticity.

Comparative Analyses of Some Statistical Tests for Homoscedasticity Assumption
As earlier mentioned, nine statistical tests are compared in this study with the use of the number of time (frequency) each test commits type II error and type I error as the case may be, such that the one with the least frequency type II error) when sigma = 0 shall be considered as the best among others and the test with the highest frequency (type I error) when sigma ≠ 0 shall be considered as the best among others. The null hypothesis is such that homoscedasticity assumption is upheld.  Table 1 and Figure Table 1 and Figure 4) riance Score tests are also best at sigma 0.7 and 0.9 while Park test is also best at sigma = 0.7.   Table 2 and Figure   6). Hence, the celebrated White test at sigma = 0 was outperformed by Glejser and so on (see Table 2 and Figure 7). and so on (see Table 2 and Figure 8) while Park test is also best at sigma 0.1, 0.3 and 0.7. Table 3 and Table 4 present the frequency of tests of significance at 1% and 5% levels respectively after 1000 replications for errors that follow LHS. As observed from the simulation results in Table 4 and Figure 9,     Table 3 and Figure 10). Hence, Har-  Table 3 and Figure 11) Table 3 and Figure 12). Hence, the earlier celebrated  Table 3 and Figure 13). Hence, Non-constant Variance Score and  Table 3 and Figure 14). Hence, Spearman rank test is the best in terms of de-   Table 4 and Figure 16).  Table 4 and Figure 17) Table 4 and Figure 18).

Summary
This study focuses on comparative analysis that determines the asymptotic behaviour of some selected statistical tests for homoscedasticity assumption by Monte Carlo simulations, and seeks to recommend the best statistical test for detecting heteroscedasticity in a multiple linear regression scenario with varying variances.

Conclusions
The  Following our findings, Table 5 and Table 6 present the summary of the best tests across all board.
As observed from Table 5 and Table 6: