Journal of Econometrics 62 (1994) 265276. North-Holland

Simple tests of distributional form

Gordon Anderson* University of Toronio, Toronto, Ont. MS.5 lA1, Can&

Received December 1990, final version received April 1993

This paper discusses a class of tests, based upon modifications to the goodness-of-fit test, for examining the accord between distributional assumptions and the data-generating process. They are easy to compute, flexible and applicable in a wide range of circumstances, and have intuitive appeal. A Monte Carlo study comparing them to a widely used test for normality is presented.

Key words: Goodness-of-fit test; Distributional assumptions; Data-generating process; Monte Carlo; Normality JEL classijication: Cl

1. Introduction

This note discusses a class of tests for examining the accord between distribu- tional assumptions and the data-generating process. The class of tests proposed is based upon a rearrangement of the well-known Pearson goodness-of-fit test [which is essentially the same as Rao’s score test, see Rao (1973) for a discussion] to focus upon particular features that may be of interest.

Information Matrix (IM) or Lagrange Multiplier (LM) based tests [White (1982), Domowitz and White (1982) and Jarque and Bera (1980)] are the main alternative instruments of investigation of this issue. Much concern has been expressed over the size problems associated with LM tests [see Chesher and Spady (1991) Kennan and Neumann (1988) and Orme (1990)] though size corrections are always a possibility as are specialized tables [see Bera and Jarque (1987)]. Similar problems are likely with IM tests given their relationship to LM tests [see Chesher (1984)] though it will depend upon which form of the

Correspondence to: Gordon Anderson, Department of Economics, University of Toronto, 150 St.

George Street, Toronto, Ont. M5S lA1, Canada.

* I am indebted to two anonymous referees for their helpful comments. This work, supported by the SSHRC under grant number 410-85-0476, has benefited from discussions with Trevor Breusch, Andy Tremain, Angelo Melino, and Dale Poirier. Responsibility for any errors is however entirely

my own.

0304-4076/94/$07.00 cc) 1994--Elsevier Science B.V. All rights reserved

266 G. Anderson, Simple tests qf distributional form

test is used [see Hall (1989)]. On the other hand the folklore associated with the Pearson test is that it lacks power.

The class of tests proposed are applicable in a wide range of circumstances (for example, there is no reason why these tests cannot be employed on residuals from fitted equations), they are easy to compute and flexible (in that they may be directed at specific or general alternatives), and have intuitive appeal. A Monte Carlo study comparing tests for normality from this class with those in Jarque and Bera (1980) is reported which, aside from confirming the size problems associated with statistics based upon estimates of higher moments, indicates good size and reasonable power properties for the Pearson-based tests.

2. The test statistics

The Pearson test is based upon partitioning the range of a random variable y into w mutually exclusive and exhaustive categories (where the sample size n is sufficiently large for the expected frequencies in the categories to be nontrivial based upon some hypothesized distribution). Then Xi, the number of sample elements falling in each category, is distributed multinomially with probabilities pi, i = l,..., w, such that

A multivariate form of the Central Limit Theorem implies that the w x 1 vector x, with typical element xi, is asymptotically distributed N(p, 8) [Kendall and Stewart (1979)], with p and 52 satisfying

Pl Pl(l -P*) -P1P2 ... -P1 Pw,

n-l@= 42 ) -PZP1 II 1: C’L?=

Pz(l - P2) ... -P2Pw

! PM’ -PwP1 - PwP2 i 1.

(1)

PWU - Pw)

There exists a collection of vectors dimensioned w, denoted ij = 1, . . . . w - 1, such that

(a) iji,=O,j#r,

(b) Uj = ij(.x - /L) is asymptotically distributed N(0, ngf) with gj’ a known function of the category probabilities.

Concomitant with (a) and (b) z~f/nof are w - 1 independent asymptotically distributed x2( 1) random variables. When the p;s have been based upon u estimated

G. Anderson, Simple tests of distributional form 267

parameters, there will be (w - 1 - u) independent random variables [see Rao (1973, p. 391) for a proof predicated upon first-order efficient estimates of the parameters, continuous partial derivatives of the cell probabilities with respect to the parameters, and an information matrix of full rank]. Thus, when estim- ated parameters are involved care must be taken to allow sufficient categories for the number of test statistics required. Noting that the choice of categories (with the exception of avoiding the triviality issue) is arbitrary the pi’s and hence the vis can be chosen to focus upon particular moments or characteristics of the distribution of interest.

For example, let w = 8 and let

i,= (-1, -1, -1, -l,l,l, l,l},

i, = (-1, - 1, 1, 1, 1, 1, - 1, - l},

is= c-1, - 1, 1, 1, - 1, - 1, 1, l},

i, = (1, - 1, - 1, 1, 1, - 1, - 1, l},

then (a) and (b) follow immediately with 03’s being of the form:

2 cm = 1 - (-PI - P2 - P3 - p4 + p5 + p6 + p7 + pd2,

d = ’ - (-PI - p2 + p3 + p4 + p5 + p6 - p7 - p,j)‘,

g: = ’ - (-PI - p2 + p3 + p4 - PS - p6 + p7 + PS)‘,

By suitable choice of pi’s, v,, v,, v,, and v, can be focussed on the mean, variance, skewness, and kurtosis of the null respectively, and the tests may be employed jointly or separately. (Indeed other aspects of the distribution such as the median may equally be the subject of attention.) Since v, and u, are of interest in the Monte Carlo study, discussion is confined to them whilst noting that an analogous discussion applies to v, and v,. Letting h(y) be the hypothesised distribution of y and g(y) its empirical distribution (corresponding to a potential alternative distribution), it follows that

[S a1

v, = {g(y) - h(y)) dy - s @* {g(y) - My)} dy ~aj a,

+ s a; {g(Y) - h(y)) dy - je; {g(y) - WY)} dy ] n,

268

and, similarly,

- {g(y) - h(y)ldy j”’ {g(y) - 4y)) dy B3

idr) - Wy))dy n, 1 where the choice of X’S and p’s is governed by the choices of the pi’s. Thus, the two statistics correspond to functions of sum and differences of areas between the hypothetical and empirical distribution curves of the random variable of interest.

Whilst the nominal size of the tests (which are consistent under the alternative hypothesis) is unaffected by the choice of the p;s, their power is. For null and relatively skewed alternative distribution functions centred on the same mean with at most three points of intersection other than at the tails, a necessary condition for the skewness test to have maximal local asymptotic power (see appendix) is satisfied when Zi’s are chosen such that

s(zi) = k(ri)> i= 1,2,3. (2)

Similarly, for null and relatively kurtotic alternative distributions centred on the same mean with at most four points of intersection other than at the tails, the same algebra indicates that the /$‘s satisfy

g(Pi) = k(Bi), i = 1, . ...4. (3)

Note that it may not be possible to examine different moments with maximal power jointly since the optimal choices of categories may preclude independence of the corresponding individual tests. On the other hand if a priori there exists a well specified alternative distribution, its intersection points with the null provide natural choices for the above.

It is easy to show that a test of m characteristics of a distribution as suggested above and a standard Pearson test with m degrees of freedom based upon the same partition are observationally equivalent. However, when there exists a ‘suspected’ alternative distribution, gains in power over the general Pearson test can be achieved in two ways, either by suggesting what an appropriate partition would be if all m degrees of freedom are to be exploited or by focussing

G. Anderson, Simple tests o~distriburionol,f~rm 269

on a limited number of characteristics and hence reducing the degrees of freedom. Two examples will illustrate these possibilities.

Firstly consider Ho: h(y) - 1, 0 < y < 1, against H,: g(y) - rry + 1 - 7r/2, 0 < y < 1, 17~1 < 2 (as rt + 0, Ho 4 H,). Partitioning the range of variation of y into four equal segments yields a Pearson test statistic of 0.07812mr2 and a modified test statistic directed at the mean (the point of intersection of the two distributions) of 0.0625nrt2. Noting that the former is x2(3) and the latter is x2(1), the conclusions with regard to power are obvious since the former test requires 100/r? observations to secure rejection under the alternative and the latter requires 61/n2 at a 95% critical value.

Secondly consider He: h(y) - (l/A)e-““, y, i > 0, against H,:

g(y) - (l/~.r)e~y’“‘, y,, A1 > 0, where A, = (1 + s)A. The intersection point for two distributions is readily shown to be at y = A((1 + 6)/6)ln(l + 6). Partition- ing the range of variation of y arbitrarily at the point ‘a will yield a test statistic of

ne-u/l W Ob’a(l +@ - 1J2 (1 _ e=“/A) ’

which at its maximum is a function only of 6, the proportionate difference between the parameter under the null and under the alternative. For 6 = 0.2, choosing a as the intersection of the two distributions yields a test statistic of n(0.01399) whereas choosing it arbitrarily at the mean yields a statistic of n(0.01372), a 2% increase in the size of the statistic.

3. The Monte Carlo study

To investigate more comprehensively their power and size characteristics, tests from this class directed at skewness and kurtosis have been subject to a Monte Carlo study. For comparison the widely used normality test [Bera and Ja.rque (1980, 1981)], which can be decomposed into skewness and kurtosis tests, are included in the study. The experimental design is based on employing response surface techniques as discussed in Hendry (1984) [see Maasoumi and Phillips (1982) for a critique of this approach]. Moments of the statistics calculated for each experiment are treated as estimates of the true moments of the corresponding statistic under the conditions of that experiment. The ‘esti- mated moments’ are regressed on functions of variables in the experimental design so that approximations to their behaviour in response to experimental design variation (i.e., response surfaces) can be made. Functional forms are chosen which permit coincidence with known values of the true moments in appropriate circumstances and which generate suitably distributed residuals for the regressions.

Given z N N(0, 1). symmetry tests were studied via the generation of n inde- pendent random variables y, of the form

y, = z(1 + d) + g when z < 0,

= (z/(1 + d)) + g otherwise.

Here g plays the role of a mean shifting nuisance parameter. For g = 0, the intersection of the null and alternative distributions is at

ys = -J{l + d(4 - d)/(2 + tt,>, y,Y < 0,

= Jc1 - a2 + 4), Ys > 0.

Kurtosis tests were examined via the generation of N independent random variables of the form

y, = -7(121”)(1 + f).

Here t can be thought of as a variance-shifting nuisance parameter. For t = 0 intersection of the null and alternative distributions is at yc = f solutions to .$-(I - y;-2q(‘-q)) = -{q/(1 - q))ln/yci - In(1 + 4).

A thousand replications per experiment were generated for each combination of d = {O 0.05 0.10 0.15 0.2), g = { -0.02 0.0 0.02 0.041, and n = {SO 75 100 125) for the asymmetric variable and for each combination of q = (0 0.2 0.4 O.(S), f = { -0.5 -0.25 0 0.25 OS}, and n = (50 7.5 100 125) for the kurtotic variable, generating X0 experiments in each case. Setting the values of d and g or q and t to zero in their respective models results in a standard normal random variable, thus deviations from zero of these parameters correspond to deviations from the null of no skewness or kurtosis. Categories were chosen such that pi = 0.125, i=l , . . ., 8, implying a value of one for both of and CT,“. This was partly for convenience, but also because choice of categories to equahse probabilities is recommended for the Pearson test. This results in intersection points being fairly close to optimal for the skewness test (at least for small d) but not for the kurtosis test.

Response surfaces are developed for the mean and variance of the statistics rather than their rejection frequencies since for accuracy reasons the latter requires many more replications per experiment and linear approximations are easier to derive and interpret for the former. Under the null the x’(j) statistic has a mean and variance ofj and 2j respectively, forming the basis for the response surface equations. Denoting the mean and variance of the estimated test statistic in an experiment as chi and v&i respectively, the estimated response surfaces for

G. Anderson, Simple tests of distributional form 271

the skewness tests are of the form

ln(Chii/ji) or In(Whii/2ji)

= 90 + 01gi + e,gf + 8,ngf + 04di + 05dF + Osnd.? + O_rg/ni

+ Qgd/ni + Vgni + ei,

and for the kurtosis tests they are of the form

ln(Chii/ji) or h(llChii/2ji)

The motivation for this is that under the null all regressors except the intercept and n are zero and, since the expected value of the dependent variable is zero in this case, a correctly sized test would yield zero coefficients with respect to the intercept and n regressors. Power in the test is reflected in the magnitude of the coefficients subscripted 1 through 6 (with some adjustments for sign in the case of the nonsquared regressors) with consistency of the test [see Engle (1984)] indicated by the magnitude of coefficients subscripted 3 and 6. Coefficients subscripted 7 and 8 are intended to reflect small sample effects.

Results for these experiments are reported in tables 1 and 2. All response surfaces are well characterised in the data with satisfactory diagnostics. In general both types of test have power in the right directions; however, whilst the BJ tests appear to have more power than the corresponding MP tests, they consistently have the wrong size unlike their nonparametric counterparts. Since the size issue is largely reflected by the intercept term a note of caution is appropriate. Response surfaces, when likened to a Taylor series approximation to the truth, embody in the intercept term the average value of the remainder which (as with the Taylor series), whilst hopefully small, will nevertheless depend upon the alternatives being considered. Of particular note in this regard is the absence of any heteroskedasticity in the residuals (see footnotes to the tables) which is encouraging. However, the significance of the intercept could in part reflect a misspecified functional form for the response surface. The size problem associated with the BJ tests is probably related to their relying on estimating higher moments.

More specifically, whilst the BJ skewness test tends to overreject substantially, with its mean and variance both significantly larger than appropriate under the null, the BJ kurtosis test will underreject to a lesser extent so that the joint test tends to overreject significantly, but not by the same magnitude as the BJ

272 G. Anderson, Simple tests of distributionaljorm

Table 1

Skewed distribution, means and variances of statistics.”

Mean Variance

BJS MPS BJS MPS

Constant

9 [O.OOSO]

2 [0.0005]

ns? [0.0004]

d [O. 1 0001

dZ [0.0150]

;,“.:128]

sin [O.OOO 1 ]

d/n [O.OO 13

n [85.0000]

R2 Forecastb Normality’

[1.5173]

0.7011* (0.0641)

-6.883* (1.8552)

43.3290 (89.3886)

-31.0832 (105.7880)

8.9265* (0.9284)

1.1661 (4.9210)

8.0062 (5.0769)

146.2556 (136.0894)

- 239.0306* (59.7852)

0.0016* (0.0007)

0..9815 1.78 2.80

CO.54361

0.1285* (0.0427)

- 2.4103* (1.2351)

11.2395 (59.5057)

-21.8994 (70.4228)

5.8338* (0.6180)

4.9902 (3.2759)

8.0954* (3.3797)

127.9398 (90.5943)

- 149.3134* (39.7988)

0.0014* (0.0005)

0.9874 1.16 0.48

12.49501

1.2701* (0.0879)

- 5.1238* (2.5428)

76.9075 (122.5135)

-98.5870 (144.9922)

14.2727* (1.2725)

0.8823 (6.7447)

- 1.7821 (6.9583)

46.8 188 (186.5230)

-385.3018 (81.9411)

0.0041* (0.0009)

0.9111 1.30 0.83

[0.7858]

- 0.2382* (0.0971)

- 2.1953 (2.8105)

- 23.8099 (135.4114)

24.0961 (160.2543)

10.1247* (I .4064)

10.8548 (7.4547)

0.2716 (7.6908)

122.3082 (206.1567)

- 263.7301* (90.5663)

0.0024* (0.00 10)

0.9677 0.90 0.35

Constant

Y [O.OOSO]

d [0.0005]

ng2 [0.0004]

d [O. 1 0001

d2 co.01 501

nd’ CO.01 281

Mean Variance

BJSK MPSK BJSK MPSK

[1.0491] CO.33381 [2.3364] CO.51481

0.3438* - 0.0393 1.4317* PO.1 142 (0.0564) (0.0268) (0.1578) (0.0724)

~ 6.0695* ~ 1.6797* - 6.1444 - 1.1581 (1.6317) (0.7760) (4.5648) (2.0941)

5.8850 15.1288 - 232.0365 35.9023 (78.6156) (37.3873) (219.9331) (100.8966)

14.6616 -24.4145 306.578 1 - 55.6570 (93.0385) (44.2464) (206.2825) (119.4073)

7.0239* 2.8867* 11.5116* 5.9465* (0.8165) (0.3883) (2.2843) (1.0479)

2.8324 1.784 4.8201 8.0328 (4.3280 (2.0583) (12.1078) (5.5546)

9.7413* 10.7552* 1.9994 6.1717 (4.4650) (2.1234) (12.4912) (5.7305)

G. Anderson, Simple tests of distributional form 213

Table 1 (continued)

Mean Variance

BJSK MPSK BJSK MPSK

sin [0.0001]

din [0.0013]

;85.0000,

R2 Foreca& Normality’

144.9599 (119.6881)

-194.1615 (52.5799)

0.0013* (0.0006)

0.9822 1.57 1.21

83.6236 (56.9202)

- 79.3143* (25.0055)

0.0004 (0.0003)

0.9884 2.19 2.12

258.9781 (334.8365)

-295.3578* (147.0963)

- 0.0003 (0.0017)

0.9349 1.53

32.95

26.2801 (153.6097)

- 182.5906 (67.4819)

0.0010 (0.0008)

0.9643 1.34 0.33

“The nomenclature for the tests is as follows. The prefix BJ denotes a Bera-Jarque test and a prefix MP denotes a Modified Pearson test; the suffix S denotes a test for skewness, the suffix K denotes a test for kurtosis, and the suffix SK denotes a joint test. Sample means of the controlled variates and the test statistic outcomes are reported in brackets; OLS standard errors are reported in parentheses, with significance at 95% indicated by asterisks. All equations were subjected to heteroskedasticity tests, and since in every case the null of no heteroskedasticity was not rejected, standard OLS coefficient standard errors are reported.

‘The forecast test, which is distributed as x’(k), is based upon the forecast erros e{, i = 1, . . . . k, and is of the form

where 0: is the error variance calculated over the estimation period.

‘The normality test, distributed x’(2), follows Jarque and Bera (1980)

skewness test. Though the MP skewness test has a significantly larger mean than appropriate under the null, it is of a much smaller magnitude than its BJ counterpart and is compensated for by a significantly smaller variance. The kurtosis test and the joint tests appear to have the correct size. These observa- tions are corroborated by table 3, where means and variances of rejection frequencies based upon a 5% critical region and sample sizes ranging from 50 to 300 are reported. The MP tests appear less responsive to sample size under the null, with similar responsiveness to the BJ test under the alternative in the case of testing for skewness.

Results not reported here (but available from the author on request) show that BJS is sensitive to nonnormal kurtosis as is BJK to nonnormal skewness, which is to be expected given the specificity of the BJ tests. On the other hand, a similar phenomenon does not occur with MPS and MPK, which maintain their appropriate size under kurtosis and skewness respectively, which is no surprise given their demonstrable orthogonality.

214 G. Anderson. Simple tests qf distributional form

Table 2

Kurtotic distribution, means and variances of statistics.”

BJS

Mean Variance

MPK BJS MPK

Constant

4 [0.3000]

q2 [O. 14001

4 112.251

[ 1 .OOOO]

;f.,*,o,

nt’ [98.4375]

0 [0.0039]

tin [0.0128]

n [85.0000]

R2 Forecast” Normality’

Cl.85661 [I.31911 [4.5254] Cl.71251

-0.4704* 0.0499 0.2446 0.1636 (0.0583) (0.1830) (0.3023) (0.2709)

11.3951* 3.2505* 20.8541* 4.6558* (0.34241 (1.0739) (1.7747) (1.5901)

-3.6413* -0.8323 - 12.5253* - 1.8327 (0.6156) (1.9310) (3.1910) (2.8591)

-0.8829 1.1717 3.3803 0.8741 (0.7630) (2.3933) (3.9551) (3.5436)

0.0514 1.7354* 0.462 1 2.0798* (0.0778) (0.2441) (0.4034) (0.3615)

0.0846 1.0054 0.7926 1.0646 (0.28 19) (0.8842) (1.4612) (1.3092)

- 0.0466 0.7461 -0.7087 0.9060 (0.3160) (0.9912) (1.6380) (1.4676)

- 158.7323* - 4 1.6229 -143.8158 -51.8476 (22.1357) (69.4334) (114.7438) (102.8073)

- 4.2415 -43.2352 - 34.5285 - 55.6015* (2.5944) (17.5479) (28.9992) (25.9825)

0.0033* 0.0035 0.0023 0.0042 (0.0006) (0.0020) (0.0033) (0.0029)

0.9975 0.9181 0.9835 0.8773 0.43 1.82 1.45 1.44 0.35 4.33 3.38 4.757

Mean Variance

BJSK MPSK BJSK MPSK

Constant

4 [0.2800]

q2 CO.12531

nq2 [0.1007]

t [0.0000] t2 [0.1250]

nt’ [O. 10631

11.83121 CO.79681 [4.5767] Cl.24141

0.2779* -0.0140 1.3851* 0.0088 (0.0465) (0.1160) (0.23 11) (0.1872)

6.6049* 2.1569* 14.9204* 3.41X3* (0.2729) (0.6807) (1.3566) (1.0989)

- 1.7339* -0.4571 - 7.7773: -0.9274 (0.4907) (1.2239) (2.4392) (1.9758)

1.0580 1.3305 4.6483 0.9604 (0.6083) (1.5170) (3.0231) (2.4488)

0.0066 1.3933* 0.1990 1.7448: (0.0620) (0.1547) (0.3084) (0.2498)

0.1887 0.6247 0.6027 0.7928 (0.2245) (0;.5605) (1.1169) (0.9047)

-0.1833 0.8291 -0.2350 0.8966 (0.2519) (0.6283) (1.2520) (1.0142)

G. Anderson. Simple tests of distributional form 215

Table 2 (continued)

Mean Variance

BJSK MPSK BJSK MPSK

4/n [0.0038]

tin [O.OOOO]

n [SWOOO]

RZ Forecastb Normalitv’

-90.3809* (17.6463)

- 1.2939 (4.4598)

0.0018* (0.0005)

0.9970 3.88 1.94

-31.3753 (44.0104)

- 33.7206* (11.1228)

0.0024 (0.0013)

0.9430 2.34 3.56

- 118.1875 -50.1371 (87.7059) (7 1.0449)

- 18.7058 -48.7831 (22.1659) (17.9552)

-0.0010 0.0034 (0.0025) (0.0020)

0.9858 0.9080 3.32 1.87 5.11 5.20

a.b,cSee footnotes of table 1.

Table 3

Means and variances of tvue 1 error frequencies (5% critical regionha

Means Variances

BJS

[0.2046] [0.0242]

BJK

[0.0325] [0.0004]

MPS

[0.0502] [0.0001]

MPK

[0.0490] [O.OOl]

‘See footnote a of table 1.

4. Conclusions

The class of tests proposed are simply computed, easily directed at specified alternatives, have simple intuition and desirable properties, and are applicable in a wide range of circumstances. The application considered in the Monte Carlo study suggests that the tests could overcome size problems associated with frequently employed alternatives and yet still maintain power in appropriate directions.

Appendix

In general the test statistic lj may be written as

where Ai - Ni = vi/n and where Ni is of the form

s OLI a?. a3 m

Ni = Ny)dy - s W)dy + s W)dy - s h(y)dy for i = s, -cc

216

Noting that

G. Anderson. Simple tests of distributional jorm

azyaz = 2n(Ai - Ni) [aA,/aZ

- (1 - (Ai - Ni)Ni/(l - N2)}&Vi/az]/(l - IV?),

(2) and (3) will satisfy their respective first-order conditions asymptotically when

Ai = Nj + ~/(n”‘5+E),

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