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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, Spearman Rank, Glejser test, Goldfeld-Quandt test, Harrison-McCabe test were considered for this study; this is with a view to examining, by Monte Carlo simulations, their asymptotic behaviours. However, four different forms of heteroscedastic 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.

One of the crucial assumptions in the multiple linear regression models is that the variance of the errors should be constant [

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 [

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 [

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 [

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” [

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 the explanatory variables, and in particular the variance of each e i is the same for all values of the predictor 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.

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.

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 [^{th} population with mean μ i , variance σ i 2 and distribution function F { σ i − 1 ( x − μ i ) } . While doing ANOVA, the null hypothesis to be tested is

H 01 : μ 1 = μ 2 = ⋯ = μ k versus H 11 : μ i ≠ μ j for some i ≠ j (2.1)

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 [

Furthermore, [

Obviously, there are a good number of methods available to test for homogeneity of variances for different situations [

H 02 : σ 1 2 = σ 2 2 = ⋯ = σ k 2 against H 12 : σ i 2 ≠ σ j 2 for some i ≠ j (2.2)

Unfortunately, these tests (Bartlett and Levene) are sensitive to the assumption of normality [

[

Tests for equality of variances are of interest in many situations such as analysis of variance or quality control [

The study considers four different heteroscedastic structures coined from [

1) V a r ( e ′ e ) = σ 2 e E ( y i ) ;

2) V a r ( e ′ e ) = σ 2 E ( y i ) g ; where g ≥ 0 .

Emanating from the two above, four heteroscedastic structures were formulated as follows:

1) Exponential Form: h i 1 = σ 2 e E ( y i ) = σ 2 e ( β 1 x 1 + β 2 x 2 ) ;

2) Linear Form: h i 2 = σ 2 E ( y i ) g = σ 2 ( β 1 x 1 + β 2 x 2 ) 1 ;

3) Square-rooted Form: h i 3 = σ 2 E ( y i ) g = σ 2 ( β 1 x 1 + β 2 x 2 ) 0.5 ;

4) Quadratic Form: h i 4 = σ 2 E ( y i ) g = σ 2 ( β 1 x 1 + β 2 x 2 ) 2 .

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:

y i = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + e (3.1)

where e is a normal error variable.

Following [_{i} as follows:

X_{i}’s are a set of independent variables that are fixed following X 1 = 1 , ⋯ , N , X 2 = 1 , ⋯ , N , X 3 = rnorm ( 10 , 1 ) and Z = i , for i = 1 , ⋯ , N .

We also generate the two error terms as follows v ~ N ( 0 , σ 2 ) and u ~ | N ( 0 , σ 2 ) |

Then the model for the heteroscedasticity function was formed as follows:

σ v = exp ( α 0 + σ α 1 ln X 1 i + σ α 2 ln X 2 i ) (3.2)

The parameters were then set at:

β 0 = α 0 = γ 0 = α 1 = α 2 = γ 1 = 1

β 1 = β 2 = β 3 = 0.5 ( constantreturntoscale )

The parameter σ measures the degree of heteroscedasticity. We use several degrees of heteroscedasticity by letting σ to vary (0, 0.1, 0.3, 0.7 and 0.9). When σ = 0 , we obtain the homoscedastic case. We considered also different sample sizes: 30, 50, 100, 200, 500 and 1000 observations (Hadri and Garry, 1998). To analyze the performance of the heteroscedasticity test statistic as we estimated the OLS model and then tested with the following heteroscedastic test statistics (Breusch-Godfrey test, studentized Breusch-Pagan test, White’s Test, Non-constant Variance Score Test, Park test, Spearman Rank, GLEJSER test, Goldfeld-Quandt test, Harrison-McCabe test) for the true presence of the problem using the test frequency of significance at 1%, 5% and 10% respectively. We also set the number of replications to 1000. Refer to Appendix 1 for the R simulation code.

This study considers nine (9) popularly used heteroscedastic test as evident in past works of literature. These tests include: Breusch-Godfrey test, studentized Breusch-Pagan test, White’s Test, Non-constant Variance Score Test, Park test, Spearman Rank, GLEJSER test, Goldfeld-Quandt test, Harrison-McCabe test.

Park (1966) formalizes the graphical method by suggesting that σ i 2 is some function of the explanatory variable X i . The functional form he suggested was:

σ i 2 = σ 2 X i β e v i (3.3)

or ln σ i 2 = ln σ 2 + β ln X i + v i (3.4)

where v i is the stochastic disturbance term. Since σ i 2 is generally not known, Park suggests using u ^ i 2 as a proxy and running the following regression:

ln u ^ i 2 = ln σ 2 + β ln X i + v i = α + β ln X i + v i (3.5)

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 u ^ i from this regression, and then in the second stage, we run the regression (3.5).

Although empirically appealing, the Park test has some problems. [

The [

| u ^ i | = β 0 + β 1 X i + v i (3.6)

| u ^ i | = β 0 + β 1 X i + v i (3.7)

| u ^ i | = β 0 + β 1 1 X i + v i (3.8)

| u ^ i | = β 0 + β 1 1 X i + v i (3.9)

| u ^ i | = β 0 + β 1 X i + v i (3.10)

| u ^ i | = β 0 + β 1 X i 2 + v i (3.11)

where v i 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 v i 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 correlation coefficient and standard error of the coefficients. It’s reported that if β 0 = 0 and β 1 ≠ 0 , pure heteroscedasticity is suggested and, also if both β 0 and β 1 differ from zero, it is an indication of mixed heteroscedasticity. This could be achieved by conducting statistical significance of both β 0 and β 1 .

This is a simple and intuitive test. One orders the observations according to X i and omits c central observations. Next, two regressions are run on the two separated sets of observations with ( n − c ) / 2 observations in each. The c omitted observations separate the low-value X’s from the high-value X’s, and if heteroscedasticity exists and is related to X i , the estimates of σ 2 reported from the two regressions should be different. Hence, the test statistic is s 2 2 / s 1 2 , where s 1 2 and s 2 2 are the Mean Square Error of the two regressions, respectively. Their ratio would be the same as that of the two residual sums of squares because the degrees of freedom of the two regressions are the same. This statistic is F-distributed with [ ( n − c ) / 2 − k ] degrees of freedom in the numerator as well as the denominator.

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 [

The success of the Goldfeld-Quandt test depends not only on the value of c (the number of central observations to be omitted) but also on identifying the correct X variable with which to order the observations. This limitation of this test can be avoided if we consider the Breusch-Pagan (BP) test (1979). Consider the k-variable linear regression model:

y i = β 0 + β 1 x 1 i + ⋯ + β k x k i + e i (3.12)

Assuming that the error variance σ i 2 is described as follows:

σ i 2 = f ( α 0 + α 1 z 1 i + ⋯ + α m z m i ) (3.13)

that is, σ i 2 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:

σ i 2 = α 0 + α 1 z 1 i + ⋯ + α m z m i (3.14)

that is, σ i 2 is a linear function of the z’s. If α 1 = α 2 = ⋯ = α m = 0 , σ i 2 = α 0 , which is a constant. Therefore, to test whether σ i 2 is homoscedastic, one can test the hypothesis that α 1 = α 2 = ⋯ = α m = 0 . This is the basic idea behind the Breusch-Pagan test. The actual procedure is tailored as follows:

1) Obtain y i = β 0 + β 1 x 1 i + ⋯ + β k x k i + e i by OLS and compute the residuals;

2) Obtain σ ^ 2 = 1 n ∑ e i 2 , which would be MLE of σ 2 under homoscedasticity;

3) Obtain another variable such that p i = e i 2 σ ^ 2 ;

4) Obtain the regression such that p i = α 0 + α 1 z 1 i + ⋯ + α m z m i + v i , where v i is the residual term of this regression;

5) Obtain the statistic: B P G = 1 2 ( S S R ) , where SSR is the Regression Sum of Squares;

6) Assuming e i are normally distributed, one can show that if there is homoscedasticity and if the sample size n increases indefinitely, then:

B P ~ χ m − 1 2 . (3.15)

Another general test for homoscedasticity where nothing is known about the form of this heteroscedasticity is suggested by [

y i = β 0 + β 1 x 1 + β 2 x 2 + e i (3.16)

The White test is tailored as follows:

1) Given a set of data, obtain the residual, e i from (3.16);

2) Run the following auxiliary regression:

e i 2 = α 0 + α 1 x 1 + α 2 x 2 + α 3 x 1 2 + α 4 x 2 2 + α 5 x 1 x 2 + v i (3.17)

3) Obtain R 2 from this auxiliary regression;

4) Under H 0 that there is no heteroscedasticity, it can be shown that the sample size (n) times R 2 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:

n R 2 ~ χ k ; α 2 (3.18)

5) It is expected that the null hypothesis will be rejected when n R 2 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.

This test ranks the x i ’s and the absolute value of OLS residuals, the e i ’s. Then it computes the difference between these rankings, that is, d i = rank ( | e i | ) − rank ( x i ) . For this simple linear regression model, we obtain the Spearman’s Rank Correlation Coefficient as follows:

r | e i | . x i = 1 − [ 6 ∑ i = 1 n d i 2 n ( n 2 − 1 ) ] (3.19)

Having obtained (3.19), the next step is to test for the significance of the coefficient using t-test as follows:

T = r | e i | . x i n − 2 1 − r | e i | . x i 2 (3.20)

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, y i = β 0 + β 1 x 1 i + ⋯ + β k x k i + e i , it’s suggested that r | e i | . x 1 , r | e i | . x 2 , ⋯ , r | e i | . x k should be computed separately and the same t-test should be used for testing the significance of each of the correlation coefficient. This is the test intended to modify in the present research.

The choice of statistical tests for this study is in connection with the existing literature, most especially in papers entitled “Heteroscedasticity as Basis of Direction Dependence in Reversible Linear Regression Models” authored by [

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.

Y t = β 1 + β 2 X 2 t + β 3 X 3 t + β 4 Y t − 1 + U t (3.21)

obtain the residuals (e_{t}).

Step 2. Estimate the following auxiliary regression.

model:

e t = b 1 + b 2 X 2 + b 3 X 3 + b 4 Y t − 1 + c 1 e t − 1 + c 2 e t − 2 + c 3 e t − 3 + w t (3.22)

Step 3. For large sample sizes, the test statistic is:

( n − p ) R 2 ~ χ p 2 (3.23)

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.

From the section above, a general linear regression model with the assumption of heteroscedasticity can be expressed as follows:

y i = β 0 + β 1 x 1 + β 2 x 2 + ⋯ + e (3.24)

Letting e = μ t

y i = β 0 + β 1 x 1 + β 2 x 2 + ⋯ + μ t (3.25)

V a r ( μ t ) = E ( μ t 2 ) = σ t 2 for t = 1 , 2 , ⋯ , n

where:

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

V a r ( e i ) = σ i 2 = σ i 2 x i α (3.26)

where α is the unknown parameter in the model.

Taking the natural logarithm

ln ( σ i 2 ) = ln ( σ i 2 ) + α ln ( x i ) (3.27)

Then taking exponential of equation

σ i 2 = exp [ ln ( σ o 2 ) + α ln ( x i ) ] (3.28)

Letting β 1 = ln ( σ i 2 ) , β 2 = α , Z i = ln ( x i )

σ i 2 = exp [ β 1 + β 2 Z i ] (3.29)

σ i 2 = exp [ β 1 + β 2 Z i 2 + ⋯ + β 3 Z i s ] (3.29*)

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 value of variance σ i 2 .

From Equation (3.27) with β 1 = ln ( σ i 2 ) , β 2 = α , Z i = ln ( x i )

Using the OLS technique to estimate the coefficients β 1 , β 2 , ⋯ , β s of the variance function

ln ( σ i 2 ) = β 1 + β 2 Z i 2 + ⋯ + β s Z i s (3.30)

where Z i 2 = ln ( x 2 ) , Z i 3 = ln ( x 3 ) and so on.

We then took the square root of the exponential of the fitted estimate

σ ^ i = exp ( β ^ 1 + β ^ 2 Z i 2 + ⋯ + β ^ s Z i s ) (3.31)

Then σ ^ i is the weight required to transform the data set by dividing through.

But;

V a r ( e i σ i ) = 1 σ i 2 V a r ( e i ) = 1 σ i 2 × σ i 2 = 1 (3.32)

Using the estimate of our variance function σ ^ i 2 in place of σ i 2 in Equation (3.30) to obtain the Generalized Least Square Estimator of β 1 , β 2 , ⋯ , β s .

We then defined the transformed variable as

y i ∗ = y i σ ^ i , x i 1 ∗ = 1 σ ^ i , x i 2 ∗ = x i σ ^ i , ⋯ , x i s ∗ = x s σ ^ i (3.33)

Therefore;

y i ∗ = β i x i 1 ∗ + β 2 x i 2 ∗ + ⋯ + β s x i s ∗ + e i ∗ (3.34)

which is the Weighted Least Squares model with homoscedasticity.

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.

significance level | sample size (n) | level of hetero | Breusch-Godfrey test | studentized Breusch-Pagan Test | White’s Test | Non-constant Variance Score Test | Park test | Spearman Rank | GLEJSER test | Goldfeld- Quandt test | Harrison- McCabe test |
---|---|---|---|---|---|---|---|---|---|---|---|

1% | n = 30 | sigma = 0 | 18 | 3 | 0 | 10 | 34 | 22 | 59 | 6 | 9 |

1% | n = 30 | sigma = 0.1 | 13 | 6 | 0 | 27 | 42 | 20 | 417 | 23 | 17 |

1% | n = 30 | sigma = 0.3 | 11 | 8 | 1 | 33 | 74 | 54 | 727 | 48 | 20 |

1% | n = 30 | sigma = 0.5 | 18 | 3 | 1 | 6 | 34 | 22 | 59 | 6 | 9 |

1% | n = 30 | sigma = 0.7 | 6 | 37 | 1 | 178 | 90 | 84 | 413 | 69 | 12 |

1% | n = 30 | sigma = 0.9 | 6 | 70 | 2 | 314 | 70 | 70 | 284 | 92 | 11 |

1% | n = 50 | sigma = 0 | 16 | 6 | 3 | 6 | 37 | 40 | 68 | 13 | 12 |

1% | n = 50 | sigma = 0.1 | 6 | 12 | 5 | 34 | 55 | 22 | 656 | 30 | 29 |

1% | n = 50 | sigma = 0.3 | 11 | 12 | 9 | 53 | 129 | 89 | 950 | 49 | 33 |

1% | n = 50 | sigma = 0.5 | 16 | 6 | 3 | 3 | 37 | 40 | 68 | 13 | 12 |

1% | n = 50 | sigma = 0.7 | 6 | 81 | 4 | 348 | 100 | 117 | 707 | 67 | 32 |

1% | n = 50 | sigma = 0.9 | 6 | 185 | 7 | 611 | 85 | 87 | 472 | 116 | 52 |

1% | n = 100 | sigma = 0 | 9 | 9 | 12 | 25 | 9 | 30 | 6 | 73 | 11 |

1% | n = 100 | sigma = 0.1 | 8 | 19 | 12 | 46 | 151 | 28 | 924 | 40 | 34 |

1% | n = 100 | sigma = 0.3 | 11 | 14 | 16 | 74 | 350 | 74 | 1000 | 58 | 52 |

1% | n = 100 | sigma = 0.5 | 11 | 9 | 11 | 8 | 30 | 25 | 73 | 9 | 10 |

1% | n = 100 | sigma = 0.7 | 8 | 332 | 16 | 728 | 136 | 96 | 984 | 102 | 74 |

1% | n = 100 | sigma = 0.9 | 11 | 643 | 16 | 929 | 93 | 55 | 903 | 146 | 119 |

1% | n = 200 | sigma = 0 | 12 | 8 | 19 | 11 | 42 | 36 | 98 | 17 | 16 |

1% | n = 200 | sigma = 0.1 | 14 | 46 | 19 | 69 | 460 | 18 | 999 | 37 | 37 |

1% | n = 200 | sigma = 0.3 | 12 | 14 | 16 | 76 | 819 | 87 | 1000 | 52 | 49 |

1% | n = 200 | sigma = 0.5 | 12 | 8 | 17 | 6 | 42 | 36 | 98 | 17 | 16 |

1% | n = 200 | sigma = 0.7 | 9 | 818 | 18 | 962 | 355 | 71 | 999 | 115 | 94 |

1% | n = 200 | sigma = 0.9 | 6 | 991 | 17 | 999 | 173 | 24 | 998 | 167 | 159 |

1% | n = 500 | sigma = 0 | 11 | 8 | 16 | 8 | 31 | 29 | 91 | 11 | 12 |

1% | n = 500 | sigma = 0.1 | 9 | 207 | 18 | 93 | 922 | 17 | 1000 | 39 | 38 |

1% | n = 500 | sigma = 0.3 | 6 | 21 | 13 | 85 | 999 | 83 | 1000 | 60 | 56 |

1% | n = 500 | sigma = 0.5 | 11 | 8 | 16 | 12 | 31 | 29 | 91 | 11 | 12 |

1% | n = 500 | sigma = 0.7 | 11 | 999 | 22 | 1000 | 895 | 14 | 1000 | 140 | 135 |

1% | n = 500 | sigma = 0.9 | 13 | 1000 | 23 | 1000 | 584 | 0 | 1000 | 192 | 188 |

1% | n = 1000 | sigma = 0 | 10 | 15 | 12 | 11 | 29 | 31 | 88 | 7 | 9 |

1% | n = 1000 | sigma = 0.1 | 13 | 595 | 12 | 203 | 999 | 10 | 1000 | 41 | 39 |

1% | n = 1000 | sigma = 0.3 | 12 | 35 | 21 | 103 | 1000 | 85 | 1000 | 52 | 53 |

1% | n = 1000 | sigma = 0.5 | 10 | 15 | 11 | 16 | 29 | 31 | 88 | 7 | 9 |

1% | n = 1000 | sigma = 0.7 | 9 | 1000 | 19 | 1000 | 1000 | 0 | 1000 | 141 | 134 |

1% | n = 1000 | sigma = 0.9 | 10 | 1000 | 26 | 1000 | 963 | 0 | 1000 | 194 | 189 |

*Frequency of test significance after 1000 replications.

replications for errors that follow EHS. As observed from the simulation results in

Moreover, considering sample size 50, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Studentized Breusch-Pagan test 0.6%, Non-constant Variance Score test 0.6%, White test with 0.3%, and so on (see

respectively out of every 1000 replications, thus implying that the Glejser test has the highest rate of type I error of 65.6%, 95.0%, 6.8% and 70.7% which makes Glejser the best test when sigma is 0.1, 0.3, 0.5 or 0.7 at sample size of 50. However, Non-constant Variance Score test with 61.1% outperformed the celebrated Glejser test when sigma = 0.9.

In addition, considering sample size 100, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 0.9%, Studentized Breusch-Pagan test 0.9%, Park test 0.9%, White test with 1.2%, Glejser test 0.6% and so on (see

Furthermore, considering sample size 200, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two

error; Breusch-Godfrey test 1.2%, Studentized Breusch-Pagan test 0.8%, Non- constant Variance Score test 1.1%, White test with 1.9% and so on (see

Additionally, considering sample size 500, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 1.2%, Studentized Breusch-Pagan test 0.8%, Non- constant Variance Score test 0.8%, White test with 1.6% and so on (see

that the Glejser test has the highest rate of type I error of 100%, 100%, 9.1%, 100% and 100% which makes Glejser the best test when sigma is 0.1, 0.3, 0.5, 0.7 or 0.9 at sample size of 500. Interestingly, Studentized Breusch-Pagan recorded 100% performance at sigma = 0.9 also, Non-constant Variance Score test recorded 100% performance at sigma = 0.7 & 0.9. Hence, Non-constant Variance Score test is also best at sigma 0.7 and 0.9 also, Studentized Breusch-Pagan is best at sigma = 0.9.

Lastly, considering sample size 1000, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 1.0%, Studentized Breusch-Pagan test 1.5%, Non-constant Variance Score test1.1%, White test 1.2%, Goldfeld-Quandt test 0.7% and so on (see

significance level | sample size (n) | level of hetero | Breusch-Godfrey test | studentized Breusch-Pagan test | White’s Test | Non-constant Variance Score Test | Park test | Spearman Rank | GLEJSER test | Goldfeld-Quandt test | Harrison-McCabe test |
---|---|---|---|---|---|---|---|---|---|---|---|

5% | n = 30 | sigma = 0 | 66 | 34 | 7 | 35 | 137 | 128 | 241 | 41 | 50 |

5% | n = 30 | sigma = 0.1 | 71 | 49 | 11 | 88 | 175 | 102 | 614 | 72 | 60 |

5% | n = 30 | sigma = 0.3 | 56 | 67 | 15 | 97 | 276 | 149 | 884 | 105 | 79 |

5% | n = 30 | sigma = 0.5 | 66 | 34 | 8 | 40 | 137 | 128 | 241 | 41 | 50 |

5% | n = 30 | sigma = 0.7 | 37 | 161 | 15 | 288 | 246 | 190 | 783 | 131 | 70 |

5% | n = 30 | sigma = 0.9 | 34 | 263 | 11 | 476 | 201 | 163 | 649 | 161 | 87 |

5% | n = 50 | sigma = 0 | 47 | 40 | 28 | 43 | 151 | 139 | 257 | 61 | 59 |

5% | n = 50 | sigma = 0.1 | 57 | 68 | 35 | 82 | 269 | 118 | 819 | 87 | 78 |

5% | n = 50 | sigma = 0.3 | 56 | 68 | 39 | 126 | 410 | 187 | 988 | 103 | 94 |

5% | n = 50 | sigma = 0.5 | 47 | 40 | 33 | 40 | 151 | 139 | 257 | 61 | 59 |

5% | n = 50 | sigma = 0.7 | 47 | 285 | 29 | 502 | 274 | 209 | 964 | 171 | 112 |

5% | n = 50 | sigma = 0.9 | 49 | 493 | 29 | 740 | 231 | 172 | 877 | 199 | 163 |

5% | n = 100 | sigma = 0 | 57 | 48 | 55 | 128 | 48 | 139 | 40 | 273 | 58 |

5% | n = 100 | sigma = 0.1 | 48 | 94 | 58 | 154 | 455 | 76 | 973 | 103 | 101 |

5% | n = 100 | sigma = 0.3 | 51 | 78 | 61 | 171 | 720 | 177 | 1000 | 118 | 112 |

5% | n = 100 | sigma = 0.5 | 58 | 48 | 48 | 43 | 139 | 128 | 273 | 57 | 63 |

5% | n = 100 | sigma = 0.7 | 49 | 600 | 63 | 834 | 410 | 156 | 1000 | 178 | 157 |

5% | n = 100 | sigma = 0.9 | 47 | 888 | 50 | 968 | 269 | 83 | 998 | 230 | 206 |

5% | n = 200 | sigma = 0 | 55 | 63 | 68 | 63 | 150 | 123 | 301 | 59 | 55 |

5% | n = 200 | sigma = 0.1 | 62 | 183 | 65 | 146 | 732 | 77 | 1000 | 128 | 123 |

5% | n = 200 | sigma = 0.3 | 51 | 66 | 61 | 181 | 965 | 180 | 1000 | 137 | 133 |

5% | n = 200 | sigma = 0.5 | 55 | 63 | 63 | 69 | 150 | 123 | 301 | 59 | 55 |

5% | n = 200 | sigma = 0.7 | 49 | 940 | 49 | 986 | 713 | 84 | 1000 | 198 | 191 |

5% | n = 200 | sigma = 0.9 | 47 | 999 | 48 | 1000 | 463 | 40 | 1000 | 254 | 251 |

5% | n = 500 | sigma = 0 | 50 | 55 | 57 | 55 | 143 | 136 | 293 | 45 | 44 |

5% | n = 500 | sigma = 0.1 | 42 | 484 | 55 | 229 | 976 | 52 | 1000 | 101 | 98 |

5% | n = 500 | sigma = 0.3 | 50 | 83 | 51 | 184 | 1000 | 185 | 1000 | 141 | 136 |

5% | n = 500 | sigma = 0.5 | 50 | 55 | 56 | 49 | 143 | 136 | 293 | 45 | 44 |

5% | n = 500 | sigma = 0.7 | 65 | 999 | 72 | 1000 | 991 | 15 | 1000 | 224 | 224 |

5% | n = 500 | sigma = 0.9 | 58 | 1000 | 71 | 1000 | 872 | 11 | 1000 | 281 | 278 |

5% | n = 1000 | sigma = 0 | 35 | 67 | 47 | 50 | 142 | 116 | 283 | 49 | 49 |

5% | n = 1000 | sigma = 0.1 | 40 | 801 | 60 | 342 | 1000 | 30 | 1000 | 107 | 108 |

5% | n = 1000 | sigma = 0.3 | 51 | 124 | 62 | 214 | 1000 | 181 | 1000 | 129 | 128 |

5% | n = 1000 | sigma = 0.5 | 35 | 67 | 50 | 52 | 142 | 116 | 283 | 49 | 49 |

5% | n = 1000 | sigma = 0.7 | 38 | 1000 | 66 | 1000 | 1000 | 0 | 1000 | 229 | 223 |

*Frequency of test significance after 1000 replications.

in

Moreover, considering sample size 50, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Studentized Breusch-Pagan test 4.7%, Non-constant Variance Score test 4.3%, White test 2.8% and so on (see

respectively out of every 1000 replications, thus implying that the Glejser test has the highest rate of type I error of 81.9%, 98.8%, 25.7%, 96.4% and 87.7% which makes Glejser the best test when sigma is 0.1, 0.3, 0.5, 0.7 or 0.9 at sample size of 50.

In addition, considering sample size 100, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 5.7%, Studentized Breusch-Pagan test 4.8%, Park test 4.8%, White test 5.5%, Glejser test 4.0% and so on (see

Furthermore, considering sample size 200, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 5.5%, Studentized Breusch-Pagan test 6.3%, Non- constant Variance Score test 6.3%, White test 6.8%, Harrison-McCabe test 5.5% and so on (see

and 100% which makes Glejser the best test when sigma is 0.1, 0.3, 0.5, 0.7 or 0.9 at sample size of 200. Interestingly, Non-constant Variance Score test recorded 100% performance at sigma 0.9. Hence, Non-constant Variance Score test is also best at sigma 0.9.

Additionally, considering sample size 500, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 5.0%, Studentized Breusch-Pagan test 5.5%, Non- constant Variance Score test 5.8%, White test with 5.7%, Harrison-McCabe 4.4% and so on (see

Lastly, considering sample size 1000, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 3.5%, Studentized Breusch-Pagan test 6.7%, White test 4.7% and so on (see

283, 1000 and 1000 corrected tests results at 0.1, 0.3, 0.5, 0.7 & 0.9 sigma levels respectively out of every 1000 replications, thus implying that the Glejser test has the highest rate of type I error of 100%, 100%, 28.3%, 100% and 100% which makes Glejser the best test when sigma is 0.1, 0.3, 0.5, 0.7 or 0.9 at sample size of 1000. Interestingly, Studentized Breusch-Pagan and Non-constant Variance Score tests recorded 100% performances at sigma = 0.7 & 0.9 also, Park test recorded 100% performance at sigma 0.1, 0.3 and 0.7. Hence, Studentized Breusch- Paganand Non-constant Variance Score tests are also best at sigma 0.7 and 0.9 while Park test is also best at sigma 0.1, 0.3 and 0.7.

significance level | sample size (n) | Heteroscedasticity Test | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

level of hetero | Breusch- Godfrey test | Stu1dentized Breusch-Pagan test | White’s Test | Non-constant Variance Score Test | Park test | Spearman Rank | GLEJSER test | Goldfeld- Quandt test | Harrison- McCabe test | ||

1% | n = 30 | sigma = 0 | 3 | 34 | 22 | 10 | 59 | 6 | 18 | 9 | 0 |

1% | n = 30 | sigma = 0.1 | 22 | 52 | 71 | 115 | 295 | 86 | 8 | 40 | 0 |

1% | n = 30 | sigma = 0.3 | 21 | 123 | 106 | 141 | 353 | 116 | 2 | 53 | 1 |

1% | n = 30 | sigma = 0.5 | 21 | 125 | 108 | 143 | 353 | 117 | 2 | 56 | 1 |

1% | n = 30 | sigma = 0.7 | 23 | 128 | 107 | 141 | 353 | 119 | 2 | 56 | 1 |

1% | n = 30 | sigma = 0.9 | 23 | 125 | 104 | 142 | 354 | 118 | 2 | 56 | 1 |

1% | n = 50 | sigma = 0 | 6 | 37 | 40 | 6 | 68 | 13 | 16 | 12 | 3 |

1% | n = 50 | sigma = 0.1 | 28 | 87 | 87 | 217 | 352 | 108 | 11 | 71 | 0 |

1% | n = 50 | sigma = 0.3 | 30 | 153 | 195 | 259 | 389 | 123 | 8 | 84 | 4 |

1% | n = 50 | sigma = 0.5 | 30 | 144 | 187 | 266 | 391 | 123 | 7 | 83 | 4 |

1% | n = 50 | sigma = 0.7 | 30 | 146 | 191 | 266 | 392 | 123 | 7 | 83 | 4 |

1% | n = 50 | sigma = 0.9 | 30 | 142 | 190 | 266 | 393 | 124 | 7 | 84 | 3 |

1% | n = 100 | sigma = 0 | 9 | 30 | 25 | 6 | 73 | 9 | 11 | 10 | 12 |

1% | n = 100 | sigma = 0.1 | 64 | 132 | 140 | 326 | 474 | 114 | 13 | 105 | 25 |

1% | n = 100 | sigma = 0.3 | 63 | 193 | 300 | 388 | 522 | 138 | 10 | 114 | 26 |

1% | n = 100 | sigma = 0.5 | 64 | 191 | 302 | 391 | 522 | 138 | 10 | 116 | 26 |

1% | n = 100 | sigma = 0.7 | 64 | 190 | 304 | 393 | 520 | 138 | 10 | 116 | 26 |

1% | n = 100 | sigma = 0.9 | 64 | 191 | 304 | 394 | 520 | 138 | 10 | 116 | 27 |

1% | n = 200 | sigma = 0 | 8 | 42 | 36 | 11 | 98 | 17 | 12 | 16 | 19 |

1% | n = 200 | sigma = 0.1 | 117 | 218 | 231 | 461 | 643 | 137 | 9 | 128 | 28 |

1% | n = 200 | sigma = 0.3 | 122 | 283 | 467 | 538 | 687 | 154 | 11 | 143 | 29 |

1% | n = 200 | sigma = 0.5 | 120 | 288 | 470 | 540 | 690 | 154 | 11 | 144 | 29 |

1% | n = 200 | sigma = 0.7 | 120 | 280 | 473 | 542 | 690 | 155 | 11 | 144 | 29 |

1% | n = 200 | sigma = 0.9 | 120 | 272 | 474 | 543 | 689 | 155 | 11 | 144 | 29 |

1% | n = 500 | sigma = 0 | 8 | 31 | 29 | 8 | 91 | 11 | 11 | 12 | 16 |

1% | n = 500 | sigma = 0.1 | 379 | 434 | 400 | 676 | 887 | 154 | 8 | 154 | 22 |

1% | n = 500 | sigma = 0.3 | 378 | 515 | 709 | 745 | 914 | 167 | 9 | 166 | 28 |

1% | n = 500 | sigma = 0.5 | 375 | 514 | 722 | 751 | 915 | 166 | 9 | 167 | 29 |

1% | n = 500 | sigma = 0.7 | 375 | 524 | 724 | 753 | 914 | 166 | 9 | 166 | 29 |

1% | n = 500 | sigma = 0.9 | 375 | 525 | 724 | 753 | 915 | 165 | 9 | 166 | 29 |

1% | n = 1000 | sigma = 0 | 15 | 29 | 31 | 11 | 88 | 7 | 10 | 9 | 12 |

1% | n = 1000 | sigma = 0.1 | 676 | 732 | 618 | 870 | 977 | 165 | 6 | 162 | 23 |

1% | n = 1000 | sigma = 0.3 | 691 | 742 | 901 | 912 | 984 | 173 | 5 | 174 | 25 |

1% | n = 1000 | sigma = 0.5 | 691 | 749 | 911 | 914 | 984 | 173 | 5 | 171 | 26 |

1% | n = 1000 | sigma = 0.7 | 691 | 745 | 912 | 914 | 983 | 173 | 5 | 171 | 26 |

1% | n = 1000 | sigma = 0.9 | 691 | 747 | 913 | 915 | 983 | 173 | 5 | 171 | 26 |

*Frequency of test significance after 1000 replications.

significance level | sample size (n) | Heteroscedasticity Test | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

level of hetero | Breusch- Godfrey test | studentized Breusch-Pagan test | White’s Test | Non-constant Variance Score Test | Park test | Spearman Rank | GLEJSER test | Goldfeld- Quandt test | Harrison- McCabe test | ||

5% | n = 30 | sigma = 0 | 34 | 137 | 128 | 35 | 241 | 41 | 66 | 50 | 7 |

5% | n = 30 | sigma = 0.1 | 89 | 228 | 184 | 211 | 516 | 168 | 48 | 122 | 14 |

5% | n = 30 | sigma = 0.3 | 93 | 335 | 264 | 253 | 602 | 195 | 46 | 139 | 12 |

5% | n = 30 | sigma = 0.5 | 93 | 327 | 255 | 257 | 605 | 196 | 46 | 139 | 10 |

5% | n = 30 | sigma = 0.7 | 93 | 335 | 249 | 257 | 608 | 198 | 47 | 139 | 10 |

5% | n = 30 | sigma = 0.9 | 93 | 333 | 251 | 260 | 608 | 198 | 47 | 138 | 10 |

5% | n = 50 | sigma = 0 | 40 | 151 | 139 | 43 | 257 | 61 | 47 | 59 | 28 |

5% | n = 50 | sigma = 0.1 | 100 | 287 | 235 | 326 | 593 | 187 | 48 | 166 | 33 |

5% | n = 50 | sigma = 0.3 | 104 | 374 | 359 | 383 | 662 | 224 | 43 | 183 | 40 |

5% | n = 50 | sigma = 0.5 | 103 | 367 | 365 | 382 | 659 | 223 | 43 | 186 | 40 |

5% | n = 50 | sigma = 0.7 | 103 | 359 | 368 | 382 | 656 | 222 | 45 | 186 | 40 |

5% | n = 50 | sigma = 0.9 | 103 | 363 | 367 | 383 | 655 | 222 | 45 | 187 | 40 |

5% | n = 100 | sigma = 0 | 48 | 139 | 128 | 40 | 273 | 57 | 58 | 63 | 55 |

5% | n = 100 | sigma = 0.1 | 164 | 354 | 305 | 446 | 706 | 194 | 55 | 187 | 69 |

5% | n = 100 | sigma = 0.3 | 174 | 429 | 471 | 498 | 747 | 218 | 45 | 204 | 69 |

5% | n = 100 | sigma = 0.5 | 172 | 436 | 482 | 498 | 748 | 220 | 46 | 206 | 68 |

5% | n = 100 | sigma = 0.7 | 172 | 439 | 483 | 498 | 750 | 219 | 47 | 206 | 69 |

5% | n = 100 | sigma = 0.9 | 172 | 436 | 479 | 499 | 749 | 219 | 47 | 206 | 68 |

5% | n = 200 | sigma = 0 | 63 | 150 | 123 | 63 | 301 | 59 | 55 | 55 | 68 |

5% | n = 200 | sigma = 0.1 | 266 | 491 | 405 | 580 | 858 | 216 | 45 | 212 | 78 |

5% | n = 200 | sigma = 0.3 | 278 | 543 | 611 | 635 | 878 | 244 | 45 | 236 | 80 |

5% | n = 200 | sigma = 0.5 | 277 | 552 | 622 | 638 | 873 | 248 | 45 | 239 | 80 |

5% | n = 200 | sigma = 0.7 | 277 | 547 | 625 | 639 | 873 | 249 | 45 | 243 | 83 |

5% | n = 200 | sigma = 0.9 | 278 | 543 | 627 | 639 | 874 | 249 | 45 | 243 | 83 |

5% | n = 500 | sigma = 0 | 55 | 143 | 136 | 55 | 293 | 45 | 50 | 44 | 57 |

5% | n = 500 | sigma = 0.1 | 551 | 719 | 595 | 764 | 969 | 215 | 43 | 215 | 66 |

5% | n = 500 | sigma = 0.3 | 562 | 750 | 809 | 816 | 980 | 232 | 46 | 231 | 67 |

5% | n = 500 | sigma = 0.5 | 564 | 751 | 820 | 825 | 980 | 235 | 47 | 233 | 66 |

5% | n = 500 | sigma = 0.7 | 566 | 748 | 822 | 826 | 980 | 235 | 46 | 233 | 65 |

5% | n = 500 | sigma = 0.9 | 565 | 752 | 822 | 827 | 980 | 235 | 46 | 233 | 66 |

5% | n = 1000 | sigma = 0 | 67 | 142 | 116 | 50 | 283 | 49 | 35 | 49 | 47 |

5% | n = 1000 | sigma = 0.1 | 821 | 916 | 782 | 915 | 997 | 253 | 36 | 251 | 66 |

5% | n = 1000 | sigma = 0.3 | 831 | 921 | 946 | 939 | 999 | 263 | 43 | 256 | 70 |

5% | n = 1000 | sigma = 0.5 | 830 | 915 | 951 | 943 | 999 | 261 | 45 | 257 | 70 |

5% | n = 1000 | sigma = 0.7 | 830 | 915 | 951 | 943 | 999 | 262 | 45 | 258 | 70 |

5% | n = 1000 | sigma = 0.9 | 830 | 910 | 952 | 944 | 999 | 262 | 45 | 259 | 71 |

*Frequency of test significance after 1000 replications.

and 35.4% which makes Park the best test when sigma is 0.1, 0.3, 0.5, 0.7 or 0.9 at sample size of 30.

Moreover, considering sample size 50, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 0.6%, Spearman Rank tests 1.3%, White test 4.0%, Harrison-McCabe test 0.3% and so on (see

In addition, considering sample size 100, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 0.9%, Non-constant Variance Score test 0.6%, Spearman Rank test 1.3%, White test 2.5%, Harrison-McCabe test 1.2% and so on (see

Park test has the highest rate of type I error of 47.4%, 52.2%, 52.2%, 52.0% and 52.0% which makes Park the best test when sigma is 0.1, 0.3, 0.5, 0.7 or 0.9 at sample size of 100.

Furthermore, considering sample size 200, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 0.8%, Non-constant Variance Score test 1.1%, Spearman Rank test 1.7%, White test 2.9%, Harrison-McCabe test 1.9% and so on (see

Additionally, considering sample size 500, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 0.8%, Non-constant Variance Score test 0.8%, Spearman Rank test 1.11%, White test 2.9%, Harrison-McCabe test 1.6% and so on (see

Lastly, considering sample size 1000, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two

error; Breusch-Godfrey test 1.5%, Non-constant Variance Score test 1.1%, Spearman Rank test 0.7%, White test 3.1%, Harrison-McCabe test 1.2% and so on (see

sigma levels respectively out of every 1000 replications, thus implying that the Park test has the highest rate of type I error of 51.6%, 60.2%, 60.5%, 60.8% and 60.8% which makes Park the best test when sigma is 0.1, 0.3, 0.5, 0.7 or 0.9 at sample size of 30.

Moreover, considering sample size 50, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 4.0%, Spearman Rank tests 6.1%, White test 13.9%, Harrison-McCabe test 2.8% and so on (see

In addition, considering sample size 100, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 4.8%, Non-constant Variance Score test 4.0%, Spearman Rank test 5.7%, White test 13.9%, Harrison-McCabe test 5.5% and so on (see

and 74.9% which makes Park the best test when sigma is 0.1, 0.3, 0.5, 0.7 or 0.9 at sample size of 100.

Additionally, considering sample size 500, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 5.5%, Non-constant Variance Score test 5.5%, Spearman Rank test 4.5%, White test 13.6%, Harrison-McCabe test 5.7% and so on (see

Lastly, considering sample size 1000, at no presence of heteroscedasticity (sigma = 0), the following tests returned the following rate in percent of type two error; Breusch-Godfrey test 6.7%, Non-constant Variance Score test 5.0%, Spearman Rank test 4.9%, White test 11.6%, Harrison-McCabe test 4.7% and so on (see

Park test has the highest rate of type I error of 99.7%, 99.9%, 99.9%, 99.9% and 99.9% which makes Park the best test when sigma is 0.1, 0.3, 0.5, 0.7 or 0.9 at sample size of 1000.

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.

The aim and objectives of the study have been principally accomplished. The analyses show the comparative results of the null hypothesis such that homoscedasticity assumption is upheld under four error structures for sample sizes; 30, 50, 100, 200, 500 and 1000 when σ 2 was varied as follows; 0, 0.1, 0.3, 0.5, 0.7 and 0.9.

In the analyses, the homoscedasticity assumption was tested under four different error distributions namely; Exponential Heteroscedastic Structure (EHS), Linear Heteroscedastic Structure (LHS), Square-Root Heteroscedastic Structure (SHS) and Quadratic Heteroscedastic Structure (QHS), for different sample sizes as the σ 2 was varied. However, two major error structures results were reported in course of our study namely Exponential Heteroscedastic Structure (EHS) and Linear Heteroscedastic Structure (LHS). The Linear Heteroscedastic Structure (LHS) results were adopted as it explains the results in Square-Root Heteroscedastic Structure (SHS) and Quadratic Heteroscedastic Structure (QHS). Following our findings,

As observed from

- when the OLS model was not contaminated with level heteroscedasticity (i.e. sigma = 0) White test returned the best test at sample size 30 and 50 for errors following EHS while;

- Harrison-McCabe test returned the best for errors following LHS. Still on sample size 30 and 50;

- when the model was infused with the level of heteroscedasticity (i.e. sigma = 0.1, 0.3, 0.5, 0.7 & 0.9), the Glejser test and Park test returned the best test for EHS and LHS respectively at sigma = 0.1, 0.3, 0.5, 0.7 & 0.9 except for EHS at sigma = 0.9 Non-constant Variance Score test returned best (0.01 level only);

N | EHS | LHS | |||||
---|---|---|---|---|---|---|---|

Sigma | 0.01 | 0.05 | 0.1 | 0.01 | 0.05 | 0.1 | |

30 | 0 | White | White | White | Harrison-McCabe | Harrison-McCabe | Harrison-McCabe |

0.1 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.3 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.5 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.7 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.9 | Non-constant Variance Score | Glejser | Glejser | Park | Park | Park | |

50 | 0 | White | White | White | Harrison-McCabe | Harrison-McCabe | Harrison-McCabe |

0.1 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.3 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.5 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.7 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.9 | Non-constant Variance Score | Glejser | Glejser | Park | Park | Park | |

100 | 0 | Glejser | Glejser | Glejser | Non-constant Variance Score | Non-constant Variance Score | Non-constant Variance Score |

0.1 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.3 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.5 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.7 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.9 | Non-constant Variance Score | Glejser | Glejser | Park | Park | Park |

n | Sigma | EHS | LHS | ||||
---|---|---|---|---|---|---|---|

0.01 | 0.05 | 0.1 | 0.01 | 0.05 | 0.1 | ||

200 | 0 | Studentized Breusch-Pagan | Harrison-McCabe/ Breusch-Godfrey | Breusch-Godfrey | Breusch-Godfrey | Spearman rank | Spearman rank |

0.1 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.3 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.5 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.7 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.9 | Non-constant Variance Score | Glejser/Non-constant Variance Score | Glejser/Non-constant Variance Score | Park | Park | Park | |

500 | 0 | Studentized Breusch- Pagan/Non-constant Variance Score | Harrison-McCabe | Harrison-McCabe | Non-constant Variance Score/ Breusch-Godfrey | Spearman rank | Spearman rank |

0.1 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.3 | Glejser | Glejser | Glejser/Park | Park | Park | Park | |

0.5 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.7 | Glejser/Non-constant Variance Score | Glejser/Non-constant Variance Score | Glejser/Non-constant Variance Score | Park | Park | Park | |

0.9 | Glejser/Non-constant Variance Score/Studentized Breusch-Pagan | Glejser/Non-constant Variance Score/Studentized Breusch-Pagan | Glejser/Non-constant Variance Score/Studentized Breusch-Pagan | Park | Park | Park | |

1000 | 0 | Goldfeld-Quandt | Breusch-Godfrey | Breusch-Godfrey | Spearman rank | Spearman rank | Spearman rank |

0.1 | Glejser | Glejser/Park | Glejser/Park | Park | Park | Park | |

0.3 | Glejser/Park | Glejser/Park | Glejser/Park | Park | Park | Park | |

0.5 | Glejser | Glejser | Glejser | Park | Park | Park | |

0.7 | Glejser/Non-constant Variance Score/Studentized Breusch-Pagan | Glejser/Non-constant Variance Score/Studentized Breusch-Pagan/Park | Glejser/Non-constant Variance Score/Studentized Breusch-Pagan/Park | Park | Park | Park | |

0.9 | Glejser/Non-constant Variance Score/Studentized Breusch-Pagan | Glejser/Non-constant Variance Score/Studentized Breusch-Pagan | Glejser/Non-constant Variance Score/Studentized Breusch-Pagan/Park | Park | Park | Park |

- Furthermore, at sample size 100 Glejser test returned the best test when sigma is 0, 0.1, 0.3, 0.5 & 0.7 and Non-constant Variance Score test returned best when sigma is 0.9 for EHS. While;

- Park test returned the best test when sigma is 0.1, 0.3, 0.5, 0.7 & 0.9 and Non-constant Variance Score test returned best when sigma is 0 for LHS;

- In addition, at sample size 200 the Glejser test and Park test returned the best test for EHS and LHS respectively when sigma = 0.1, 0.3, 0.5, 0.7 & 0.9 except for EHS at sigma = 0.9 for 0.01 level Non-constant Variance Score test returned best;

- However, when sigma = 0 (no heteroscedasticity) the following tests returned best: Studentized Breusch-Pagan (EHS at 0.01 level); Harrison-McCabe/ Breusch-Godfrey (EHS at 0.05 level); Breusch-Godfrey (EHS at 0.1 level); Breusch-Godfrey (LHS at 0.01); and Spearman rank (LHS at 0.05 & 0.1);

- Moreover, at sample size 500 the Glejser test and Park test returned the best test for EHS and LHS respectively when sigma = 0.1, 0.3, 0.5, 0.7 & 0.9 also Non-constant Variance Score at sigma = 0.7 or 0.9 and Studentized Breusch- Pagan returned best when sigma = 0.9 for EHS;

- However, when sigma = 0 (no heteroscedasticity) the following tests returned best: Studentized Breusch-Pagan/Non-constant Variance Score (EHS at 0.01 level); Harrison-McCabe (EHS at 0.05 level); Harrison-McCabe (EHS at 0.1 level); Non-constant Variance Score/Breusch-Godfrey (LHS at 0.01); and Spearman rank (LHS at 0.05 & 0.1);

- Lastly, at sample size 1000 the Glejser test and Park test returned the best test for EHS and LHS respectively when sigma = 0.1, 0.3, 0.5, 0.7 & 0.9;

- Also the following test returned best: Park (EHS at sigma 0.3 and 0.01 level); Non-constant Variance Score/Studentized Breusch-Pagan (EHS; sigma 0.7 & 0.9 at 0.01 level); Park (EHS; sigma 0.3 & 0.9 at 0.05 and 0.1 levels); and Non-constant Variance Score/Studentized Breusch-Pagan/Park (EHS; sigma 0.7 & 0.9 at 0.05 and 0.1 levels);

- However, when sigma = 0 (no heteroscedasticity) the following tests returned best: Goldfeld-Quandt (EHS at 0.01 level); Breusch-Godfrey (EHS at 0.05 and 0.1 levels); and Spearman rank (LHS at 0.01, 0.05 & 0.1 levels).

From the aforementioned, the following are recommended:

1) White and Harrison-McCabe tests should be employed to check for homoscedasticity in EHS and LHS respectively for sample size 30 and 50 (Small samples).

2) others can be employed as follows Glejser (EHS at n = 100), Non-constant Variance Score (LHS at n = 100), Studentized Breusch-Pagan (EHS at n = 200, 0.01 level), Harrison-McCabe/Breusch-Godfrey (EHS at n = 200, 0.05 level), Breusch-Godfrey (EHS at n = 200, 0.1 level), Breusch-Godfrey (LHS at n = 200, 0.01), Spearman rank (LHS at n = 200, 0.05 & 0.1), Studentized Breusch-Pagan/ Non-constant Variance Score (EHS at n = 500, 0.01 level) (Moderate samples).

3) Harrison-McCabe (EHS at n = 500, 0.05 level), Harrison-McCabe (EHS at n = 500, 0.1 level), Non-constant Variance Score/Breusch-Godfrey (LHS at n = 500, 0.01), Spearman rank (LHS at n = 500, 0.05 & 0.1), Goldfeld-Quandt (EHS at n = 1000, 0.01 level), Breusch-Godfrey (EHS at n = 1000, 0.05 and 0.1 levels), Spearman rank (LHS at n = 1000, 0.01, 0.05 & 0.1 levels) (Large samples).

4) Glejser and Park tests should be employed to check for heteroscedasticity in EHS and LHS respectively.

The authors declare no conflicts of interest regarding the publication of this paper.

Onifade, O.C. and Olanrewaju, S.O. (2020) Investigating Performances of Some Statistical Tests for Heteroscedasticity Assumption in Generalized Linear Model: A Monte Carlo Simulations Study. Open Journal of Statistics, 10, 453-493. https://doi.org/10.4236/ojs.2020.103029

Performance of the Tests when Error follows Quadratic Structure (QHS) at 1%

*Frequency of test significance after 1000 replications.

Performance of the Tests when Error follows Quadratic Structure (QHS) at 5%

*Frequency of test significance after 1000 replications.