A Modification of the Jarque-Bera Test. for Normality
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1 Int. J. Contemp. Math. Sciences, Vol. 8, 01, no. 17, HIKARI Lt, A Moification of the Jarque-Bera Test for Normality Moawa El-Fallah Ab El-Salam Department of Statistics & Mathematics an Insurance Faculty of Commerce, Zagazig University, Egypt Copyright 01 Moawa El-Fallah Ab El-Salam. This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. Abstract In regression analysis, many statistical tests have been propose to fin out whether the error term is normally istribute or not. These statistical tests are typically constructe using OLS resiuals. However, since iagnostic tests for normality are very sensitive to outliers, they have a zero breakown value. In this paper, we attempt to investigate the effects of using resiuals from robust regression replacing OLS resiuals in test statistics for normality. We stuy a moifie Jarque-Bera test statistic for normality base on robust regression resiuals. The asymptotic istribution of this robustifie normality test is erive, an its breakown property is iscusse. Keywors: Normality; Jarque-Bera test; Robust regression; Least Trimme Squares; Breakown point
2 844 Moawa El-Fallah Ab El-Salam 1. Introuction Significant eviations from normality of the regression resiuals can substantially affect the performance of usual inference techniques. Thus, iagnostic tests for normality are important for valiating inferences mae from regression moels. Several such tests have been propose (see, Pearson et al., [10]; White an Maconal, [15]; Pierce an Gray, [11]; an Jarque an Bera, [6]). In general, these tests are conucte by testing the orinary least squares (OLS) resiuals for normality. However, since iagnostic tests for normality are very sensitive to outliers, these tests fail to etect any lack of normality in the OLS analysis. Consier we have inepenently sample n observations X n =(x 1, x,, x n ) with mean μ an stanar eviation σ. The coefficients of skewness an kurtosis are efine as: b 1 = μ σ E [(x μ) ] = / [E (x μ) ] (1) μ 4 E [( x μ) 4 ] b = = 4 σ [E ( x μ) ] () where μ r = E[(x-μ) r ] be the rth centeral moment of X n with μ =σ. Davison an Mackinnon [] pointe out that, if X n is i.i.. an normally istribute, then: nb1 N(0,6) an n(b ) N(0,4) () Regaring univariate normality tests amongst the several alternatives, we have selecte Jarque an Bera (JB) as being one of the popular tests for testing normality in regression applications. It is well known that the JB test is base on a weighte average of the skewness an kurtosis as:
3 Moification of the Jarque-Bera test 845 JB = n ( b 1 ) (b ) +, (4) 6 4 where n is the sample size, b is the sample skewness an b 1 is the sample kurtosis. Jarque an Bera [6] prove that, if the alternatives to the normal istribution belong to the Pearson family, JB is a score test for normality. However, because the classical skewness an kurtosis coefficients b an b 1 are base on OLS resiuals, they are very sensitive to outliers in the ata. Therefore, the main iea of this paper is to use the same JB test using resiuals from a robust regression estimator instea of OLS resiuals, an enote this by JB*. The particular estimator we stuy is calle Least Trimme Squares (LTS), introuce by Rousseeuw [1]. The paper is organize as follows. Section (), introuces the basic iea of the least trimme squares robust proceure. In section (), we erive the asymptotic istribution of the robustifie normality test, an the high breakown property of the test statistic is iscusse. Section (4) inclues simulation results, while section (5) conclues.. The Least Trimme Squares Metho The least trimme squares (LTS) metho has the property of being highly resistant to a relatively large proportion of outliers. Thus LTS has a high breakown value. For the etails of this technique an its properties, see Rousseeuw an Leroy [1]), an also chapter 5 of Zaman [18]. Consier the following regression moel: y i = x i β + u i, i = 1,,., n, (5) where x i is a (1xk) vector of regressors, β is a (kx1) vector of regression coefficients, y i are observe responses, an u i represent the error terms which are
4 846 Moawa El-Fallah Ab El-Salam assume to be inepenent an ientically istribute with mean 0 an stanar eviation σ. The LTS estimate is given by: h Min e (i), (6) i= 1 where e (1) e (),, e (n) are the orere square resiuals e i =(y i x i βˆ ), i=1,,n, an the value of h must be etermine base on trimming the ata values. For example, the 0% trimme LTS estimator is efine to be the value of (4n/5) βˆ minimizing e ( i) ( β ). Chen [] efine h in the range [(n/) + 1] h i= 1 [(n+k+1)/4], an recommene that, the breakown value for the LTS estimator is ( n-h/n), when h=[(n+k+1)/4]. Although, there exist other high breakown estimators, the LTS has many avantages to recommen itself. A recently evelope algorithm has permitte fast computation of the LTS (See Rousseeuw an Van Driessen, [14]). The main avantages of the LTS metho is as follows. Firstly, LTS is simple to unerstan an easy to motivate. Also, it is more efficient than the LMS (Least Meian of Squares) introuce by Rousseeuw [1] with which it shares these avantages. Many estimators commonly regare as robust in the econometrics literature have low breakown values an cannot eal with any significant number of outliers. For example, the boune influence estimator of Krasker an Welsch [7], an the least absolute eviation metho, both suffer heavily from the presence of a small subgroup of outliers; see Yohai [16].. Asymptotic an Robustness properties In this section, we iscuss the asymptotic istribution an the breakown property of the robustifie normality test statistics.
5 Moification of the Jarque-Bera test 847 Theorem 1: uner the null hypothesis of normality of the error terms, JB* is istribute asymptotically chi-square with egrees of freeom. Proof: As can be seen from Hossjer [5] that. n(ˆ β) β = O P (1), (7) then using (7) an Davison an Mackinnon ( [], p ), we obtain: 1/ n ui 1/ n ei (6n) (6n) 0, (8) i 1 i 1 ˆ p = σ = σ 1/ n ui (6n) N(0, 1), (9) σ i= 1 where e i are the resiuals from LTS regression an σˆ is the LTS estimate of the stanar eviation. So we have: 1/ n ei (6n) N(0, 1) (10) i= 1σˆ Similarly for 4 n e 1/ (4) i N(0, 1) (11) 4 i 1 ˆ = σ Then, combining (10) an (11) in the test statistic, we have: JB* x () (1) The breakown point of a statistical function is the smallest fraction of ata contamination that can prouce arbitrarily extreme results. So, a power breakown point is the least amount of ata contamination that can rive the test statistic to its null value regarless of the true alternative value (see He et al., [4] an Markatou an He, [8], for a etaile iscussion). On the other han the weighte likelihoo estimator (Markatou et al., [10]) may reuce to the LTS estimator with the following weights:
6 848 Moawa El-Fallah Ab El-Salam w i = 1 if i h, (1) 0 if i > h where h is the trimming point. Following Agostinelli an Markatou [1], we efine the score type test function as: 1 JB* = / n Sw( θ) wii( θ) Sw( θ), (14) i= 1 where θ is the vector of parameters, the superscript / enotes transpose an I(θ) is n the Fisher information matrix evaluate at θ,s w (θ) = w i u(y i, θ) is the score i= 1 function evaluate at θ, an u(y i, θ) = L(yi, θ), where L(y i, θ) is the loglikelihoo function. Following the same technique as Agostinelli an Markatou θ [1], P.507, but making minor moifications require to allow for the fact that we have a Pearson family alternative, it may be possible to establish that the power breakown point of the JB* test is the same as the breakown point of LTS estimator, if the alternative belongs to the Pearson family. 4. Numerical Results Rousseeuw an Leroy [1] use several ata sets for their analysis relate to the robust regression. As a preliminary test for the valiity of our basic iea, we conucte a stuy of tests for normality using the same five ata sets. The results, which are presente in Table (1), are highly encouraging. In each of the five ata sets stuie, we can say that, the JB test fails to etect any lack of normality in the OLS resiuals, while JB* picks up the lack of normality of the errors very clearly.
7 Moification of the Jarque-Bera test 849 Table (1): The values of the normality test statistics from Rousseeuw an Leroy ata sets: Series N JB JB* Brain Clou Salinity Aircraft Delivery Rousseeuw an Leroy (1987) from pp. 57, 96, 8, 154, 155. In aition, since, the presence of many outliers (corresponing to lack of normality) in these series is well known, we observe that the JB test, conucte to the basis of OLS resiuals, can lea to misleaing outcomes. On the other han, in orer to unerstan the reasons for the superiority of JB* test on the Rousseeuw an Leroy ata sets, we carry out a Monte Carlo simulation to obtain the power comparisons of JB an JB* using the GAUSS program. To generate ranom variates from normal an other istributions consiere throughout the stuy, we use the ranom number generator of GAUSS.1.1. We consiere a linear moel with a constant term of one an one aitional regressor. Every experiment in this simulation allows some alternative istributions than just i.i.. with some non-normal ensity. To accomplish this, we starte by using the mixture of two normal istributions (αn (0, 1) + (1 - α) N(μ, σ)) as an alternative to the null of normality. In this way, α allowe us to choose the percentage of outliers an μ an σ allowe us to control the behavior of the outliers. In the examples relate to the failure of OLS series, we observe that the outliers lie in clusters (see, for example, Rousseeuw an Leroy, [1], p.58). Therefore, we stuy the effects of clustere outliers (referre to as case A below) as oppose to ranomly generate outliers (referre to as case B below). The nature of the outliers also turns out to be crucial. High variance outliers from
8 850 Moawa El-Fallah Ab El-Salam N (0, 9) an N (0, 16) compare to the base istribution N(0, 1) were stuie, as well as mean-shift outliers generate from N(5, 1) an N(10, 1) istributions. In orer to make a comparison, we also generate outliers at ranom places in samples. The first case, where the outliers are clustere, is labele case A, an the secon case, where they are at ranom places, is labele case B. The results for 0 an 50 observations are presente in Table (). In all cases, 80% of the true errors are generate from N (0, 1) an 0% of them from normal istributions with ifferent means an variances. So, 4 outliers are generate for the case of 0 observations an 10 outliers for the case of 50 observations. We compute the values of JB an JB* an we fin out whether null hypothesis (normality) is rejecte by each iniviual test. The level of significance use was 0.10 throughout the simulations. We carrie out 1000 replications an estimate the power of each test by computing the percentage when the null hypothesis was rejecte. The results for 0 an 50 observations are presente in Table (). Table () Power comparisons of mixture of normal alternatives with 0% outliers uner ifferent conitions. N = 0 Case A Case B N ( 5, 1) JB JB* N (10, 1) JB JB* N ( 0, 9) JB JB* N ( 0,16) JB JB* N = 50 Case A Case B N ( 5, 1) JB JB* N (10, 1) JB JB* N ( 0, 9) JB JB* N ( 0,16) JB JB*
9 Moification of the Jarque-Bera test 851 The results of Table () show that, when the outliers are clustere (case A) an generate by shifting the mean of the normal, the robust version of the test has substantially greater power than the test base on OLS resiuals. In all other cases, both tests have similar performance. This can be explaine as follows. With balance outliers on both sies of the regression line, the OLS estimator is not much affecte by the outliers, an hence OLS resiuals are similar to robust resiuals. In aition, outliers generate by high variance are equally likely effect an i not lea to improve performance for JB* as we ha expecte. While, outliers generate by a mean shift all lie on one sie of the regression line an hence are much more effective in systematically istorting OLS estimates. Also, clustering of the outliers has a significant effect on the relative performance of these tests. Clustere outliers ten to systematically istort OLS estimators to the possible largest extent (outliers with high leverage have a similar effect an result in similar outcomes). Thus, these situations lea to maximum improvement for tests base on robust resiuals. When the outliers are not clustere, their impact on the OLS estimators is reuce (equivalently, they have less leverage). In such situations, both OLS an robust resiuals pick out the outliers easily, an therefore both achieve high power, as inicate in Table (). 5. Conclusions In this paper, we iscusse the Jarque-Bera test of normality, which is not able to etect normality in the presence of outliers. Therefore, we propose to use resiuals from a robust regression technique (LTS) instea of OLS resiuals as a basis for the normality test. We foun that, the iea was vali by using some examples from the literature, in which the robustifie Jarque Bera test (JB*) reveals lack of normality not etecte by the test base on OLS resiuals. We obtaine the asymptotic istribution of the moifie test as chi-square with egrees of freeom uner the null hypothesis of normality, an the breakown point of the test statistic was investigate. From the results of simulation, we
10 85 Moawa El-Fallah Ab El-Salam foun that, the robust test yiels substantially greater power for systematic an clustere outliers situations. References [1] C. Agostinelli, an M. Markatou, Tests of hypotheses base on the weighte likelihoo methoology, Statistica Sinica, 11, (001), [] C. Chen, Robust regression an outlier etection with the ROBUSTRER proceure, SUGI paper, SAS Institute (00), [] R. Davison, an J. Mackinnon, Estimation an Inference in Econometrics, Oxfor Univ. Press, New York. (199). [4] X. He, D.G. Simpson, an S.L. Portnoy, Breakown robustness of tests, J. AM. Statist.Assoc., 85, (1990), [5] O. Hossjer, Rank-base estimates in the linear moel with high breakown point, J. AM. Statist.Assoc., 89,(1994), [6] C.M. Jarque, an, A.K Bera, A test for normality of observations an regression resiuals, Int. Statistical Review, 55, (1987), [7]W.S. Krasker, an R.E. Welsch, Efficient boune influence regression estimation, J. AM. Statist.Assoc., 77, (198), [8] M. Markatou, an X. He, Boune influence an high breakown point testing proceures in linear moels, J. Amer. Statist.Assoc., 89, (1994), [9] M. Markatou, A., Basu, an B. G. Linsay, Weighte Likelihoo estimating equations with a bootstrap root search, J. Amer. Statist.Assoc., 9, (1998), [10] E.S. Pearson, R.B. D Agostina, an K.O. Bowman, Tests for eparture from normality: Comparison of Powers, Biometrika, 64, (1977), [11] D.A. Pierce, an R.J. Gray, Testing normality of errors in regression moels, Biometrika, 69 (198), -6. [1] P. J. Rousseeuw, Least meian of squares regression, J. Amer. Statist. Assoc., 79, (1984),
11 Moification of the Jarque-Bera test 85 [1] P.J. Rousseeuw, an A.M. Leroy, Robust Regression an Outlier Detection, Wiley, New York, (1987). [14] P.J. Rousseeuw, an K. Van Driessen, A fast algorithm for the minimum covariance eterminant estimator, Technometrics, 41,(1999), 1-. [15] H. White, an G.M. Maconal, Some large sample tests for nonnormality in the linear regression moel, J. Amer. Statist.Assoc.,75,(1980), [16] V.J. Yohai, High breakown-point an high efficiency robust estimates for regression, Ann. Statist.,15,(1987), [17] V.J. Yohai, an R.H. Zamar, High breakown point estimates of regression by means of the minimization of an efficient scale, J. Amer. Statist. Assoc., 8, (1988), [18] A. Zaman, Statistical Founations for Econometric Techniques, Acaemic Press, New York, (1996). [19] A.Zaman, P.J,Rousseeuw an M. Orhan, Econometric applications of High breakown robust regression techniques, Econometrics Letters,71,(001), 1-8. Receive: September 1, 01
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