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A Test for Normality of Observations and Regression Residuals Author(S): Carlos M

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A Test for Normality of Observations and Regression Residuals Author(S): Carlos M A Test for Normality of Observations and Regression Residuals Author(s): Carlos M. Jarque and Anil K. Bera Reviewed work(s): Source: International Statistical Review / Revue Internationale de Statistique, Vol. 55, No. 2 (Aug., 1987), pp. 163-172 Published by: International Statistical Institute (ISI) Stable URL: http://www.jstor.org/stable/1403192 . Accessed: 06/03/2013 11:23 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. International Statistical Institute (ISI) is collaborating with JSTOR to digitize, preserve and extend access to International Statistical Review / Revue Internationale de Statistique. http://www.jstor.org This content downloaded on Wed, 6 Mar 2013 11:23:52 AM All use subject to JSTOR Terms and Conditions InternationalStatistical Review (1987), 55, 2, pp. 163-172. Printedin GreatBritain ? InternationalStatistical Institute A Test for Normality of Observations and Regression Residuals Carlos M. Jarque1and Anil K. Bera2 1Direcci6nGeneral de Estadistica,Secretaria de Programaciony Presupuesto,Mexico D.F. 2Departmentof Economics, Universityof Illinois, Urbana-Champaign, Illinois 61820, USA Summary Using the Lagrangemultiplier procedure or score test on the Pearson family of distributionswe obtain tests for normality of observationsand regression disturbances.The tests suggested have optimumasymptotic power propertiesand good finite sample performance.Due to their simplicity they should prove to be useful tools in statisticalanalysis. Keywords: Lagrange multiplier test; Normality test; Regressionmodel; Score test. 1 Introduction Statisticians'interest in fitting curves to data goes a long way back. As noted by Ord (1972, p. 1), although towards the end of the nineteenth century 'not all were convinced of the need for curves other than the normal' (K. Pearson, 1905), 'by the turn of the century most informed opinion had accepted that populations might be non-normal'; some historical accounts are given by E. S. Pearson (1965). This naturally led to the development of tests for the normalityof observations.Interest in this area is still very much alive, and recent contributionsto the literature are the skewness, kurtosis and omnibustests proposedby D'Agostino & Pearson (1973), Bowman & Shenton (1975) and Pearson, D'Agostino & Bowman (1977), the analysisof variancetests of Shapiro& Wilk (1965) and Shapiro & Francia (1972), and the coordinate-dependentand invariant proceduresdescribed by Cox & Small (1978). There has also been considerable recent interest in testing the normality of (unobserved) regression disturbances. This is noted below, but first we introduce necessary notation. We consider the linear regression model, y, = xf + ui, for i = 1, ... , N, where x' is a 1 by K vector of observationson K fixed regressors,f is a K by 1 vector of unknown parameters, and ui is the ith unobservabledisturbance assumed to have zero mean and to be homoscedastic,identically distributed and seriallyindependent. An additional assumptionfrequently made in this model is that the probabilitydensity function of ui, f(u,), is the normalprobability density function. The consequences of violation of this normality assumption have been studied by various authors. In estimation, for instance, the ordinary least-squares estimator b = (X'X)-IX'y, which is known to be efficientunder normality,may be very sensitive to long-taileddistributions; for example, see Hogg (1979). Regardinginferential procedures, Box & Watson (1962) considerthe usual t and F-tests, and demonstratethat sensitivityto nonnormalityis determined by the numericalvalues of the regressors. They also show that, to obtain the desired significancelevel, some adjustmentin the degrees of freedom of these tests may be required. Similarly, Arnold (1980) studies the distribution of This content downloaded on Wed, 6 Mar 2013 11:23:52 AM All use subject to JSTOR Terms and Conditions 164 C.M. JARQUE and A.K. BERA 2 = (y - Xb)'(y - Xb)IN and shows that the significancelevel of the usual X2 test of the hypothesis a2= U2 is not even asymptoticallyvalid in the presence of nonnormality. Furthermore, it has been found that homoscedasticity and serial independence tests suggested for normal disturbances may result in incorrect conclusions under non- normality; for example, see Bera & Jarque (1982). In all, violation of the normality assumptionmay lead to the use of suboptimalestimators, invalid inferential statements and to inaccurateconclusions, highlightingthe importanceof testing the validity of the assumption. In ? 2, we present a procedure for the constructionof efficient and computationally simple statisticalspecification tests. This is used in ? 3 to obtain a test for the normalityof observations, and in ? 4 to obtain a test for the normality of (unobserved) regression disturbances.We then present in ? 5 an extensive simulationstudy to compare the power of the tests suggested with that of other existing procedures. 2 The score test The procedurewe present for the constructionof specificationtests consists of the use of the score test on a 'general family of distributions'.The score test, also known as the Lagrangemultiplier test, is fully described elsewhere, for example, see Cox & Hinkley (1974, Ch. 9), so here we only introducenotation and state requiredresults. Considera randomvariable u with probabilitydensity functionf(u). For a given set of N independent observations on u, say ul,..., UN, denote by 1(O) = 1(0) +... + lN(O) the logarithmof the likelihood function, where li(O)= logf(u,), 0 = (0', 0')' is the vector of parameters (of finite dimension), and 02 is of dimension r by 1. Assume we are interested in testing the hypothesisH: 02 = 0. = Define dj = E a1(O)/aoj and jk E[E (91(O)/aoj)(a,~O)/aOk)'], where summation = goes from i = 1 to N, for j 1, 2 and k = 1, 2. Let di and Jjk denote dj and Jjk evaluated at the restricted (obtained by imposing the restriction 02=0) maximum likelihood estimatorof 0, say 0. It may be shown that under general conditions, the statisticdefined by LM= a2(22- 2- 1~2)-1a2(1)2 is, under Ho:02 = 0, asymptoticallydistributed as a X2 with r degrees of freedom, Xnr).say A test of Ho: 02 = 0, based on (1), will be referredto as a score or LMtest. Two aspects of this test are worth nothing. First, that it is asymptoticallyequivalent to the likelihood ratio test, implying it has the same asymptoticpower characteristicsincluding maximum local asymptotic power (Cox & Hinkley, 1974). Secondly, that to compute it we only require estimation under the null hypothesis. In the inferential problems studied here, estimation under Ho is easily carried out making the test computationallyattractive, as compared to other asymptoticallyequivalent procedures(i.e., likelihood ratio and Wald tests). For these two reasons (good power and computational ease) we use the LM test, ratherthan others, in the inferentialprocedure. It should be mentioned that the Lagrangemultiplier method has been applied recently in many econometric problems; for example, see Byron (1970), Godfrey (1978) and Breusch & Pagan (1980). Here the procedure for the constructionof specificationtests also uses this principle, but has its distinctfeature in the formulationof 1(0). Rather than assuming a 'particular' probability density function for ui (or transformation of ui), we assume that the true probability density function for u, belongs to a 'general family' (for example, the Pearson family), of which the distribution under Ho is a particular member. This content downloaded on Wed, 6 Mar 2013 11:23:52 AM All use subject to JSTOR Terms and Conditions Test for Normality and Regression Residuals 165 We then apply the Lagrangemultiplier principle to test Ho within this 'general family' of distributions. Our subsequent discussion will help make this point clearer. The tests obtained are known to have optimal large sample power propertiesfor members of the 'general family' specified. Yet, this does not imply they will not have good power properties for nonmember distributions.Indeed, as shown in ? 5, the tests can perform with extremely good power even for distributionsnot belonging to the 'general family' from which the test was derived. 3 A test for normalityof observations In this section we make use of the Lagrangemultiplier method to derive an additional test for the normality of observations which is simple to compute and asymptotically efficient. Consider having a set of N independent observations on a random variable v, say vl,..., vN, and assume we are interested in testing the normality of v. Denote the unknown population mean of v, by M= E[vi] and, for convenience, write v, = M+ ui. Assume the probabilitydensity function of u,, f(u,), is a member of the Pearsonfamily. This is not very restrictive,due to the wide range of distributionsthat are encompassedin it (e.g., particular members are the normal, beta, gamma, Student's t and F distributions).This means we can write (Kendall & Stuart, 1969, p. 148) - df (u,)/dui = (c1 - u)f (u,)/(co c1u + c2) (-00 < i < oo). (2) It follows that the logarithm of the likelihood function of our N observations v1, ... , VN may be written as cc2) = -N log[7 exp Cl-- du,] 4l(, Co,C I Ui+ du] 2di ICOC-C "+C2U + '- . - (3) i=1- CO cAu + 2i dui - C2U Our interest here is to test the hypothesis of normality, which means, from our expression for f(ui), that we want to test = = 0. Let 01 = (m, 02 = Ho:Cl c2 co)', (cl, c2)' and 0 = (0', 0')'. Using these, and the definitions of ? 2, we can show that the LM test statisticis given by LM= N[(Vb1)2/6 + (b2 - 3)2/24], (4) - and 7 = with summations where Vbl1= -3I/2 , b2 =442-, = (i i')/N E vilN, going from i = 1 to N.
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