A Robust Test for Normality

Transcription

1 A Robust Test for Normality Liangjun Su Guanghua School of Management, Peking University Ye Chen Guanghua School of Management, Peking University Halbert White Department of Economics, UCSD March 11, 2006 Abstract This paper proposes a simple robust test for normality based on order statistics. The test statistic has the same asymptotic null distribution and is as easy to compute as Jarque and Bera s (JB) test statistic for normality. We also show that out test is unaffected by parameter estimation error. The JB test is highly distorted in the presence of outliers, whereas ours is not. Nor does our test lose much power in the absence of outliers. We apply our test to USA salary income data and find that we cannot reject the normality of the logarithm of salary income for 1999 data. Keywords: Normality; Order Statistics; Parameter Uncertainty; Test JEL Classification: C12, C. Address Correspondence to: Liangjun Su, Department of Business Statistics and Econometrics, Guanghua School of Management, Peking University, Beijing , China; lsu@gsm.pku.edu.cn, Phone: (0086) Ye Chen, Guanghua School of Management, Peking University, Beijing , China; ye_chen@pku.edu.cn. Halbert White, Department of Economics, UCSD, La Jolla, CA ; hwhite@weber.ucsd.edu. The first author gratefully acknowledges financial support from the NSFC (Project ). The usual disclaimers apply.

2 1 Introduction Several tests for normality have been proposed in the literature, among which the Jarque and Bera s (1980, 1987, hereafter JB ) test is the most popular. This test was first given by Bowman and Shenton (1975), who only stated the expression of the statistic and noted it was asymptotically distributed as χ 2 (2) under normality but did not give it an LM interpretation. It is a special case of White s (1994) information matrix test. Of particular interest is that for a wide range of alternative distributions, the JB test outperforms such tests as the Kolmogorov- Smirnov test, the Cramer-von 0 Mises test and the Durbin test. Despite its good power properties and computational simplicity, it is well known that the JB test is highly sensitive to outliers because it is constructed from the moment-based skewness and kurtosis measures. This unsatisfactory feature can lead to quite misleading results in empirical applications where outliers are frequently present. For example, in the finance literature, while it has become a stylized fact that stock market returns have negative skewness and severe excess kurtosis (see Hwang and Satchell, 1999, Harvey and Siddique, 2000), using some robust measures of skewness and kurtosis, Kim and White (2004) reveal interesting evidence in sharp contrast to what has been previously thought. One seemingly simple solution is to eliminate the outliers from the data. But as Kim and White (2004) have remarked, this can cause problems. The decision to eliminate outliers is often taken usually after one visually inspects the data, which can invalidate subsequent statistical inference; deciding which observations are outliers can be somewhat arbitrary. Consequently, eliminating outliers manually is not necessarily simple. It is therefore desirable to have robust measures of skewness and kurtosis that are not sensitive to outliers. These measures can then serve as the basis for a robust test for normality. In this paper we propose a simple robust test for normality. The test statistic is based on certain robust measures of skewness and kurtosis and is both scale and location invariant. We will show that the test statistic is easy to compute and that it has the same asymptotic null distribution as the JB test statistic. We also prove that our test is unaffected by parameter estimation error. This facilitates the application of our test since one can conduct a normality test for the residual series from a regression model. The paper is organized as follows. We describe the test statistic and derive its asymptotic null distribution in Section 2. We prove that our test is robust against parameter uncertainty in Section 3. We report a Monte Carlo study in Section 4 and an application in Section 5. Section 6 concludes. 1

3 2 Robust Measures of Skewness and Kurtosis and Robust Tests for Normality We consider a process {y t } n t=1 and assume that the y0 ts are independent and identically distributed (i.i.d.) with cumulative distribution function F and probability density function f. The JB test for normality is based upon the statistic µ JB n = n dsk 2 /6+ dkr 3 2 /24, (2.1) where SK d and KR d are the conventional moment-based estimates of skewness and kurtosis for {y t }. Under normality, JB shows that JB n is asymptotically distributed as χ 2 (2). Due to its reliance on moments up to the fourth order, the JB test for normality is highly sensitive to outliers. This motivates the search for robust measures of skewness and kurtosis and robust test for normality. 2.1 Robust Measures of Skewness and Kurtosis Robust measures of location and dispersion are well documented in the literature. Following this tradition, Bowley (1920) proposed a coefficient of skewness based on quantiles SK 1 = Q 3 + Q 1 2Q 2, (2.2) Q 3 Q 1 where Q i is the ith quartile of y t. Clearly, SK 1 is zero for any symmetric distributions and otherwise lies between -1 and 1. Moors (1988) showed that the conventional measure of kurtosis can be interpreted as a measure of the dispersion of a distribution around the two values µ ± σ, where µ and σ are respectively the mean and standard deviation of F. Based on this interpretation, Moors proposed a robust measure of kurtosis (E 7 E 5 )+(E 3 E 1 ), (2.3) E 6 E 2 where E i is the ith octile, i.e., E i = F 1 (i/8) for i =1, 2,...,7. It is easily seen that for the normal distribution, the above measure takes the value Kim and White (2004) center the Moors coefficient of kurtosis at the value for the normal distributions: KR 1 = (E 7 E 5 )+(E 3 E 1 ) (2.4) E 6 E 2 Several other measures of skewness and kurtosis are also available. Like SK 1 and KR 1, most of them satisfy the properties put forward by Groeneveld and Meeden (1984). Nevertheless, as Kim and White (2004) note, some of these measures depend on the first or second moments of F and thus are still sensitive to outliers. 2

4 2.2 Robust Tests for Normality We now propose a robust test for normality based upon the robust measures for skewness and kurtosis. Let SK d 1 and KR d 1 be the sample analog estimates of SK 1 and KR 1, respectively. Denote b θ 1 = dsk1, KR 1 0. d Thefirstteststatisticis T n,1 = n b θ 0 1Ω 1 1 b θ 1, where Ω 1 is the asymptotic variance-covariance matrix of b θ 1 under normality as specified in Theorem 2.1. The asymptotic null distribution of T n,1 is given in the following theorem. Theorem 2.1 Under normality, T n,1 = n b θ 0 1Ω 1b d 1 θ 1 χ 2 (2), where Ã! Ω 1 = That is, T n,1 = n dsk 2 1/ KR d 1/ d χ 2 (2). A sketch of the proof. By definition, SK d 1 = bq3 + Q b 1 2Q 2 b / bq3 Q 1 b and KR 1 = ( E b 7 E b 5 + E b 3 E b 1 )/( E b 6 E b 2 ) , where Q b i and E b i are sample analog estimates of Q i and E i, respectively. Noticing that under normality, SK 1 = KR 1 =0, by the standard asymptotic normality results on quantile estimators (e.g., Cramer, , p.349), we have Ã! dsk1 n = n = n dkr 1 d N (0, Ω 1 ), Q 3 + Q 1 2Q 2 Q 3 Q Q 3+Q 1 2Q 2 1 Q 3 Q 1 E 7 E 5 + E 3 E 1 E 6 E E 7 E 5 +E 3 E 1 2 E 6 E 2 ( Q 3 Q 3)+( Q 1 Q 1) 2( Q 2 Q 2) Q 3 Q 1 ( E 7 E 7)+( E 3 E 3) ( E 5 E 5) ( E 1 E 1) {( E 2 E 2) ( E 6 E 6)} E 6 E 2 + o P (1) with Ã! Ã! w 11 w Ω 1 = =, w 12 w where, for example, ½ 1 F (Q3 )(1 F (Q 3 )) w 11 = (Q 3 Q 1 ) 2 f 2 + F (Q 1)(1 F (Q 1 )) (Q 3 ) f 2 + 4F (Q 2)(1 F (Q 2 )) (Q 1 ) f 2 (Q 2 ) + 2F (Q 1)(1 F (Q 3 )) 4F (Q 2)(1 F (Q 3 )) 4F (Q ¾ 1)(1 F (Q 2 )) f (Q 3 ) f (Q 1 ) f (Q 3 ) f (Q 2 ) f (Q 1 ) f (Q 2 ) = when F is a normal distribution function. 3

5 The result follows. The above theorem says that SK d 1 and KR d 1 are asymptotically independent under normality, analogous to the asymptotic independence between SK d and KR d under normality. In addition, T n,1 is both scale and location invariant. Even though T n,1 is robust to outliers, it is not necessarily powerful in detecting deviations from normality since it depends on the data only through seven observations: Ei b,i=1,, 7. Also, the Q 0 i s in (2.2) and the E0 is in (2.4) can be far from the tails, so a normality test based on SK 1 and KR 1 may not have much power to detect asymmetry and fat-tailedness far in the tails. For these reasons, we propose another two sets of robust measures of skewness and kurtosis: SK 2 = S 15 + S 1 2S 8,KR 2 = (S 15 S 11 )+(S 5 S 1 ) , (2.5) S 15 S 1 S 12 S 4 and SK 3 = S 15 + S 11 + S 5 + S 1 4S 8,KR 3 = KR 2, (2.6) S 15 S 1 where S i = F 1 (i/) for i =1, 2,...,15. Let SK d i and KR d i,i=2, 3, be the sample analog estimates of SK i and KR i, respectively. Denote b θ 2 = dsk2, KR d 2 0 and b θ3 = dsk3, KR d 3 0. Analogous to Theorem 2.1, we have Theorem 2.2 Under normality, T n,i = n b θ 0 iω 1 b d i θi χ 2 (2),i=2, 3, where Ã! Ã! Ω 2 = and Ω 3 = Clearly SK d i and KR d i,i=2, 3, are asymptotically independent under normality. In Section 4, we will evaluate the finite sample performance of the tests T n,i,i=1, 2, 3. 3 Testing with Parameter Uncertainty In many empirical applications, the normality assumption is made for an unobservable random variable. This is the case for a regression model, where the error term is often assumed to be normally distributed. Thus one must first estimate the parameters in the model and get the residuals. Then one tests for the normality of the residuals. It is well known that the asymptotic distribution of a test statistic that depends on an unknown parameter β 0, can be quite different from the asymptotic distribution of the same test statistic with β 0 replaced by its consistent estimator β. b This is the parameter uncertainty problem. Fortunately, this is not the case for our test statistic for the reason that will shortly become apparent. 4

6 Consider a class of nonlinear regression models of the form y t = m(x t,β)+σ (x t,β) t,t=1, 2,...,n, (3.1) where y t is the dependent variable, x t is a vector of independent variables, β is a vector of parameters, and the 0 ts are iid disturbance terms with mean 0 and variance 1. The functional forms of m and σ are assumed to be known. Let t (β) = y t m(x t,β). σ (x t,β) When the parameter β = β 0, the true parameter value, we write t β 0 = t. We are interested in testing the null hypothesis H 0 : 1 N (0, 1). The alternative hypothesis is: H 1 : 1 is not distributed according to N (0, 1). Since the 0 ts are not observed, we cannot base our test for normality directly on the 0 ts. Denoting the estimator for β as β, b we obtain residuals b t = t bβ = y t m(x t, β) b σ x t, β b,t=1, 2,...,n. (3.2) Our test statistics are then based upon {b t }. For example, we define T n,2 ( β)=n b b θ bβ 0 2 Ω 1b 2 θ 2 bβ, (3.3) where b θ 2 bβ = dsk2 bβ, KR d bβ 0 2, SK d 2 bβ and KR d 2 bβ are defined as SK d 2 and KR d 2 using the b t series. b θ 1 bβ, b θ 3 bβ,t n,1 ( β), b and T n,3 ( β) b are similarly defined. Let f (z; β) be the probability density function (pdf) of t (β). We make the following assumptions. Assumption A1 {x t, t } are iid and { t } is independent of {x t }. A2 n bβ β 0 = O p (1). A3 For each z, f (z; β) is a probability density differentiable in β, and there exists a function g (z; β) such that (i) f (z; β) / g z; β 0 for all β in a neighborhood of β 0, and (ii) R g z; β 0 dz <. Note that Condition (i) in Assumption A3 imposes a uniformity requirement on the boundedness of f (z; β) /. In conjunction with A3(ii), this ensures that we can take differentiation under the integral for f (z; β), i.e., Z z f ( ; β) d = Z z f ( ; β) d. 5

7 Theorem 3.1 Suppose Assumptions A1-A3 hold. Under H 0 : 1 N (0, 1),T n,i ( β)=n b b θ 0 i Ω 1 b i θi bβ has the same asymptotic distribution as T n,i (β 0 ), i =1, 2, 3. Proof. We only prove the case for T n,2 since the proof of other cases are similar. It suffices to show à dsk2 ( β) b d! SK 2 (β 0 ) n dkr 2 ( β) b KR d 2 (β 0 = o p (1). (3.4) ) First, by the Taylor expansion, we have à dsk2 ( β) b d! SK 2 (β 0 ) n dkr 2 ( β) b KR d 2 (β 0 ) bβ à = dsk2 (β 0 ) 0 dkr 2 (β 0 ) à dsk2 (β 0 ) SK 2 (β 0 ) = 0 dkr 2 (β 0 ) KR 2 (β 0 ) A 1n + A 2n + o P (1).! n( b β β 0 )+o P (1)! à n( β b β 0 )+ 0 SK 2 (β 0 ) KR 2 (β 0 )! n( b β β 0 )+o P (1) (3.5) By the consistency of the quantile estimators and Assumptions A2-A3, A 1n = o p (1) O p (1) = o p (1). (3.6) Write t (β) = y t m(x t,β) σ (x t,β) σ x t,β 0 = t = y t m(x t,β 0 )+m(x t,β 0 ) m(x t,β) σ σ xt,β 0 x t,β 0 σ (x t,β) σ (x t,β) + m(x t,β 0 ) m(x t,β) σ (x t,β) = t s (x t,β)+µ(x t,β), where s (x t,β)=σ x t,β 0 /σ (x t,β), and µ (x t,β)= m(x t,β 0 ) m(x t,β) /σ (x t,β). Under the null hypothesis, it is easy to show that the cumulative distribution function (cdf) of t (β) is given by µ z µ (xt,β) F (z; β) = P ( t (β) z) =EΦ s (x t,β) = Z z f ( ; β) d where Φ (. ) and ϕ (. ) are the cdf and pdf for the standard normal distribution, respectively, and Z µ µ (x, β) 1 f ( ; β) = ϕ dg (x) s (x, β) s (x, β) 6

8 with G (x) being the cdf of X. Fix τ (0, 1), and define a function H τ such that H τ (β,z) = Z z f ( ; β) d τ. (3.7) By the implicit function theorem, there is a function z = z (β) such that H τ (β,z) =0, and Let and z = H τ (β,z) / H τ (β,z) / z = c 1 β 0 = c 2 β 0 = Z Z It is easy to verify that under the null, m x, β 0 σ x, β 0 R z f ( ; β) /d. (3.8) f (z; β) 1 σ x, β0 dg (x), 1 σ x, β 0 dg (x). f ; β 0 = ϕ ( ), (3.9) and f ; β 0 = c 1 β 0 ϕ 0 ( ) c 2 β 0 ϕ 0 ( ) + ϕ ( ). (3.10) Denote the inverse function of F (z; β) by F 1 (τ; β), i.e., F (z; β) =τ for z = F 1 (τ; β). Clearly, when β = β 0,F z; β 0 = Φ (z), and F 1 τ; β 0 = Φ 1 (τ). By (3.7) through (3.10), we have F 1 τ; β 0 R Φ 1 (τ) c1 β 0 ϕ 0 ( )+c 2 β 0 [ϕ 0 ( ) + ϕ ( )] ª d = ϕ (Φ 1 (τ)) 1 = c1 β 0 ϕ (Φ 1 ϕ Φ 1 (τ) + c 2 β 0 ϕ Φ 1 (τ) Φ 1 (τ) ª (τ)) = c 1 β 0 + c 2 β 0 Φ 1 (τ). Now, let S i (β) =F 1 (i/; β),i=1,...,15. Then SK 2 β 0 = ½ S15 (β)+s 1 (β) 2S 8 (β) S 15 (β) S 1 (β) ¾ β=β 0 S 15 (β 0 ) + S 1(β 0 ) 2 S 8(β 0 ) S15 (β 0 )+S 1 (β 0 ) 2S 8 (β 0 ) h S 15 (β 0 ) = S 15 (β 0 ) S 1 (β 0 ) S15 (β 0 ) S 1 (β 0 ) 2 = c 2 β 0 Φ Φ 1 1 2Φ Φ 1 15 Φ 1 1 0=0. 7 S 1(β 0 ) i

9 and KR 2 β 0 ½ (S15 (β) S 11 (β)) + (S 5 (β) S 1 (β)) = = S 12 (β) S 4 (β) S 15 (β 0 ) S 11(β 0 ) + S 5(β 0 ) 2 S 1(β 0 ) S 12 (β 0 ) S 4 (β 0 ) ¾ β=β 0 S15 (β 0 ) S 11 (β 0 )+S 5 (β 0 ) S 1 (β 0 ) h S 12 (β 0 ) S12 (β 0 ) S 4 (β 0 ) 2 = c 2 β 0 Φ 1 15 Φ 1 ( 11 )+Φ 1 5 Φ 1 1 Φ 1 12 Φ 1 4 = 0. Φ 1 15 S 4(β 0 ) Φ 1 ( 11 )+Φ 1 5 Φ 1 1 c2 β 0 Φ 1 12 Φ 1 12 Φ i Φ 1 4 Consequently, we have A 2n =0, (3.11) which, in conjunction with (3.5) and (3.6), implies (3.4). REMARK. From the proof of the above theorem, we see that the properties Φ 1 (1 τ) = Φ 1 (τ),τ (0, 1), and Φ 1 (0.5) = 0 guarantee that Ã! SK 2 (β 0 ) 0 KR 2 (β 0 =0. (3.12) ) Accordingly, the asymptotic distributions of T n,i ( β) b and T n,i (β 0 ) (i =1, 2, 3) coincide,which implies that our test statistics are robust against parameter uncertainty. 4 A Monte Carlo Study We now conduct a Monte Carlo experiment to illustrate the finite sample performance of our test. We also compare our tests with the Jarque-Bera (JB) test. 4.1 Test without Parameter Uncertainty We first consider tests without the parameter uncertainty problem. In the absence of outliers, consider data drawn from the 7 symmetric and 8 skewed distributions used in Bai and Ng (2005): S1: N(0, 1); 8

10 Table 1: Empirical rejection frequency of the normality tests n = 100 n = 200 n = 500 JB n T n,1 T n,2 T n,3 JB n T n,1 T n,2 T n,3 JB n T n,1 T n,2 T n,3 S S S S S S S A A A A A A A A S2: t 5 ; S3: e 1 1(z 0.5) + e 2 1(z>0.5) h, where zi U (0, 1),e 1 N ( 1, 1), and e 2 N (1, 1) ; S4-S7: F 1 (u) =λ 1 + u λ 3 (1 u) λ 4 /λ 2, where λ 1 =0, and (λ 2,λ 3,λ 4 )=( , , ), ( 1, 0.08, 0.08), ( , 0., 0.), ( 1, 0.24, 0.24) in S4-S7 respectively; A1: lognormal: exp(e), where e N (0, 1) ; A2: χ 2 (2) ; A3: exponential(1) ; h i A4-A8: F 1 (u) =λ 1 + u λ 3 (1 u) λ 4 /λ 2, where λ 1 =0, and (λ 2,λ 3,λ 4 )=(1, 1.4, 0.25), ( 1, , 0.03), ( 1, 0.1, 0.18), ( 1, 0.001, 0.13), ( 1, , 0.17) in A4-A8 respectively. Table 1 reports the empirical rejection frequency for the tests JB n and T n,i,i=1, 2, 3, where the nominal significance level is α =5%and the number of replications is throughout. Under normality, all tests have the correct size even though T n,3 is a little oversized. When normality does not hold true, JB n is most powerful in detecting non-normality except for S4. As expected, T n,1 does not have much power; both T n,2 and T n,3 are more powerful. For asymmetric distributions A1-A8 and some symmetric distributions (S3, S7 for n = 500), T n,2 and T n,3 are almost as powerful as JB n. Next, we consider testing for normality in the presence of outliers: O1-O7: Y t is generated according to N (0, 1) but with the first observation replaced by a 9

11 Table 2: Empirical rejection frequency of the normality tests in the presence of outliers n =100 n = 200 n =500 JB n T n,1 T n,2 T n,3 JB n T n,1 T n,2 T n,3 JB n T n,1 T n,2 T n,3 O O O O O O O O O O O O O O single outlier o, whichis3,4,5,6,8,10,15ino1-o7respectively. O8-O14: Y t is generated according to N (0, 1) but with probability Y t is replaced by an outlier o and with probability replaced by an outlier o, where o is 3, 4, 5, 6, 8, 10, 15 in O8-O14 respectively. Table 2 reports the empirical rejection frequency for the tests JB n,t n,i,i =1, 2, 3, where the nominal significance level is α =5%. Clearly,inthepresenceofoutliers,thesizeofJB n is highly distorted; this is true even if we have only one outlier in 500 observations and the outlier is only 4 or 5 times as large as the standard deviation of the other observations. In contrast, the levels of T n,1 or T n,2 are still well behaved whereas that of T n,3 is slightly distorted. Given the almost equal power performance of T n,2 and T n,3, we recommend using T n,2 in practice. 4.2 Test with Parameter Uncertainty We now consider tests when the parameter uncertainty problem exists. To save space, we only compare T n,2 with the JB test. We start by simulating data from a linear regression model: y t = β 0 + β 1 x 1t + β 2 x 2t + t,t=1,...,n, (4.1) where x 1t and x 2t are independent Uniform(0,1) random variables, and 0 ts are first generated according to S1-S7 and A1-A8 and then centered around their sample means. Set (β 0,β 1,β 2 ) 0 = (1, 1, 1) 0. 10

12 Table 3: Empirical rejection frequency of the normality tests with parameter uncertainty n =100 n = 200 n = 500 JB n T n,2 JB n T n,2 JB n T n,2 S S S S S S S A A A A A A A A We estimate the model (4.1) by fitting a linear regression model and denote the resulting parameter estimator as β b = bβ0, β b 1, β b 2 0. Let b t = t bβ = y t β b 0 β b 1 x 1t β b 2 x 2t. We calculate T n,2 bβ and JB n based on {b t }. Table 3 reports the empirical rejection frequency for the tests JB n and T n,2 bβ, where the nominal significance level is α =5%. Under normality, both tests have the correct size like the case of no parameter uncertainty. When normality does not hold true, both the JB n and T n,2 bβ test behave as if there were no parameter estimation error. Next, we consider testing for normality in the presence of outliers, parameter uncertainty, and heteroskedasticity. We simulate data from the following model y t = β 0 + β 1 x 1t + β 2 x 2t + p exp (β 3 + β 4 x 1t ) t,t=1,...,n, (4.2) where x 1t and x 2t are independent Uniform(0,1) random variables, and 0 ts are generated according to O1-14. Set (β 0,β 1,β 2,β 3,β 4 ) 0 =(1, 1, 1, 1, 1) 0. We estimate the model (4.2) in two steps. First, we estimate (β 0,β 1,β 2 ) 0 by regressing y t on a constant, x 1t, and x 2t. Denote the estimator of (β 0,β 1,β 2 ) 0 by bβ0, β b 1, β b 2 0 and define bu t = y t β b 0 β b 1 x 1t β b 2 x 2t. In the second step we regress log bu 2 t on a constant and x 1t to estimate (β 3,β 4 ) 0, and denote the resulting estimator by bβ3, β b 4 0. Let b t = t bβ = 11

13 Table 4: Empirical rejection frequency of the normality tests in the presence of outliers and parameter uncertainty n = 100 n = 200 n = 500 JB n T n,2 JB n T n,2 JB n T n,2 O O O O O O O O O O O O O O r y t β b 0 β b 1 x 1t β b 2 x 2t / exp bβ3 + β b 4 x 1t. We calculate T n,2 bβ and JB n based on {b t }. Table 4 reports the empirical rejection frequency for the tests JB n and T n,2 bβ, where the nominal significance level is α =5%. Similar to the case where we have the presence of outliers but not parameter estimation error, the JB n test is highly distorted when both outliers and parameter estimation error are present. In sharp contrast, our test T n,2 bβ is still well behaved and robust against both outliers and parameter estimation error. 5 Application to the Distribution of Salary Income In this section we apply our test to the distribution of individual salary income. Ever since Pareto, it has been recognized that the shape of the income distribution is skewed and heavytailed. Accordingly, the logarithm of income is usually analyzed in the literature (e.g., Jones, 1997). To apply our test we obtain the annual salary income of household heads from a recent PSID dataset. After eliminating missing values, we have 1350, 1510 and 1445 valid observations for the years 1999, 2001, and 2003, respectively. Table 5 gives summary statistics. From Table 5 we see that the maximum lies far from the average or median income for each year. For example, in 1999 the maximum is about 19 standard deviations from the mean or 12

14 Table 5: Summary statistics of annual salary income Year Obs Mean Std. dev Skewness Kurtosis Median Minimum Maximum Table 6: Test statistics for the normality of log salary income (p-values in brackets) Year JB n T n,1 T n,2 T n, (0.000) 5.30 (0.071) 6.35 (0.040) 8.42 (0.015) (0.000) 4.95 (0.084) (0.001).32 (0.000) (0.000) (0.003) (0.003) (0.000) median. We thus suspect the presence of outliers in the right tail of the income distribution. Figure 1 plots kernel density estimates for the logarithm of annual salary income in 1999, 2001, and We use the standard normal kernel and choose the bandwidth by Silverman s rule of thumb. Also plotted in Figure 1 are the normal densities with mean and variance equal to those of the logarithm of annual salary income in the corresponding year. From Figure 1 we see that the kernel estimate is pretty close to the normal density for the year 1999, whereas the two densities are quite different in 2001 and Since extreme values are quite common in income distributions, we expect the JB test and our robust test to perform quite differently. This is indeed the case as reported in Table 6. We see that the JB test strongly rejects the normality of log salary for all cases. In contrast, our test statistics are much smaller, although they share the same asymptotic null distribution as the JB statistic. This suggests the presence of outliers that seriously distort the JB test. Based on our recommended T n,2 test, we cannot reject the normality of log salary in 1999 at the 1% significance level. 6 Conclusion We propose a simple robust test for normality based on order statistics and show that our test statistics have the same asymptotic null distributions as the JB test statistic for normality. We demonstrate that our test is robust against parameter uncertainty. The JB test is highly distorted in the presence of outliers, whereas ours is not. Nor does our robust test lose much power in the absence of outliers. The joint use of JB and our new test can thus be highly informative. We also apply our test to salary income data and demonstrate the differing conclusions that can be drawn from our test and the traditional JB test. 13

15 Density Density Density Logarithm of salary income, Logarithm of salary income, Logarithm of salary income, 2003 Figure 1: Kernel density estimates of the annual salary income (dotted line: estimates, solid line: normal density) 14

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