The inference in probit models relies on the assumption of normality. However, tests of this assumption are not implemented in standard econometric software. Therefore, the paper presents a simple representation of the Bera-Jarque-Lee test, that does not require any matrix algebra. Furthermore, the representation is used to compare the Bera-Jarque- Lee test with the RESET-type test proposed by Papke and Wooldridge (1996).probit model, Lagrange multiplier test, normality assumption, artificial regression

Joachim Wilde

English

Research Papers in Economics

Institut für Halle Institute for Economic Research
Wirtschaftsforschung Halle
IWH-Diskussionspapiere
IWH-Discussion Papers
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
A Simple Representation of the Bera-Jarque-Lee Test  
for Probit Models 
Joachim Wilde 
 
 
 
Dezember 2007                                            No. 13 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
A Simple Representation of the Bera-Jarque-Lee Test  
for Probit Models 
Joachim Wilde 
 
 
 
Dezember 2007                                            No. 13 
 
 
 
 
 
  
IWH  __________________________________________________________________ 
 
IWH-Diskussionspapiere 13/2007  2
Author: Dr Joachim Wilde 
  Chief Econometrician 
  Email: Joachim.Wilde@iwh-halle.de 
  Tel.: ++49 345 77 53 836 
 
 
 
 
 
The responsibility for discussion papers lies solely with the individual authors. The views ex-
pressed herein do not necessarily represent those of the IWH. The papers represent preliminary 
work and are circulated to encourage discussion with the author. Citation of the discussion pa-
pers should account for their provisional character; a revised version may be available directly 
from the author. 
Comments and suggestions on the methods and results presented are welcome. 
 
IWH-Discussion-Papers are indexed in RePEc-Econpapers. 
 
 
Herausgeber: 
INSTITUT FÜR WIRTSCHAFTSFORSCHUNG HALLE – IWH 
Prof. Dr. Ulrich Blum (Präsident), Dr. Hubert Gabrisch (Forschungsdirektor) 
Das IWH ist Mitglied der Leibniz-Gemeinschaft 
Hausanschrift: Kleine Märkerstraße 8, 06108 Halle (Saale) 
Postanschrift: Postfach 11 03 61, 06017 Halle (Saale) 
Telefon:    (0345) 77 53-60 
Telefax:    (0345) 77 53-8 20 
Internetadresse:  http://www.iwh-halle.de 
  
__________________________________________________________________  IWH 
 
IWH-Diskussionspapiere 13/2007  3
A Simple Representation of the Bera-Jarque-Lee Test 
for Probit Models 
Abstract 
The inference in probit models relies on the assumption of normality. However, tests of 
this assumption are not implemented in standard econometric software. Therefore, the 
paper presents a simple representation of the Bera-Jarque-Lee test, that does not require 
any matrix algebra. Furthermore, the representation is used to compare the Bera-Jarque-
Lee test with the RESET-type test proposed by Papke and Wooldridge (1996). 
Keywords:  probit  model,  Lagrange  multiplier  test,  normality  assumption,  artificial 
regression 
JEL Classification: C25   
Zusammenfassung 
Die statistische Inferenz in Probitmodellen beruht maßgeblich auf der Gültigkeit der 
Normalverteilungsannahme für die Störgrößen. Dennoch sind Tests auf Normalvertei-
lung in vielen ökonometrischen Paketen nicht implementiert und werden daher oft auch 
nicht durchgeführt. Das Papier präsentiert deshalb eine einfache Darstellung des Bera-
Jarque-Lee Tests, der die Normalverteilung gegen die vergleichsweise allgemeine Al-
ternative der Pearson-Verteilungen testet. Die Darstellung beruht auf einer Hilfsregres-
sion, wie sie von Davidson und MacKinnon (1984) für einen Test gegen Heteroskeda-
stie vorgeschlagen wurde. Sie kann mit jeder Statistik- oder Ökonometriesoftware, die 
die Schätzung eines Probitmodells ermöglicht, problemlos berechnet werden.  
Darüber hinaus zeigt die Darstellung, dass der Test auf Normalverteilung asymptotisch 
identisch  ist  mit  einem  von  Papke  und  Wooldridge  (1996)  vorgeschlagenen RESET 
Test, der die Linearität des Erwartungswertes der latenten Variablen überprüft. Folglich 
kann  ohne  zusätzliche  Informationen  nicht  zwischen  der  funktionalen  Form  des 
Erwartungswertes und dem Verteilungstyp der Störgrößen unterschieden werden. Sofern 
allerdings die Linearität des Erwartungswertes unterstellt werden kann, ist die künstliche 
Regression des Bera-Jarque-Lee Tests informativer, da sie Hinweise darauf gibt, ob die 
Ablehnung  der  Nullhypothese  durch  Schiefe  oder  eine  von  der  Normalverteilung 
abweichende Kurtosis verursacht wurde.  
Schlagwörter: Probitmodell, Lagrange Multiplier Test, Normalverteilungsannahme 
JEL-Codes: C25  
IWH  __________________________________________________________________ 
 
IWH-Diskussionspapiere 13/2007  4
A Simple Representation of the Bera-Jarque-Lee Test 
for Probit Models* * * * 
1  Introduction 
Probit models are often applied without testing the normality assumption. This is rather 
problematic because the standard maximum likelihood estimator of the probit model is 
mostly biased for nonnormal disturbances (cf. Greene 2003, p. 673). One reason might 
be that suitable tests are not implemented in standard econometric software. Therefore, 
the paper presents a simple representation of the test of Bera, Jarque, and Lee (1984). It 
only requires the estimation of a probit model and some standard transformations. Bera, 
Jarque, and Lee test the assumption of normality against a rather general alternative, 
whereas other authors assume more specific distributions (e.g. Silva 2001). 
The representation is based on an artificial regression which was suggested by Davidson 
and  MacKinnon  (1984)  for  a  similar  test  on  homoscedastic  errors.  The  dependent 
variable is the so-called standardized residual of the probit estimation (cf. Wooldridge 
2002,  p.  462).  An  alternative  artificial  regression  is  based  on  a  regressand  of  ones. 
However, the small sample properties are rather poor (cf. Davidson and MacKinnon 
1993, p. 477). Hence, the paper does not consider this alternative. 
The artificial regression of the Bera-Jarque-Lee test can be transformed into the artificial 
regression of the RESET-type test proposed by Papke and Wooldridge (1996). Although 
two regressors are numerically changed by the transformation, the explained sums of 
squares of both regressions and thus the test statistics are identical. Therefore, without 
additional information the distributional form of the disturbances is not distinguishable 
from the functional form of the mean of the latent variable. 
Section 2 summarizes the ideas of Bera, Jarque and Lee in a slightly different notation 
which  makes  it  easy  to  present  the  results  in  Section  3.  In  Section  3  the  simple 
representation of the test will be derived and interpreted. Section 4 concludes with a 
comparison of the two tests.  
                                                 
*  I thank an anonyomous referee for rather useful comments. The usual disclaimer applies.  
__________________________________________________________________  IWH 
 
IWH-Diskussionspapiere 13/2007  5
The Bera-Jarque-Lee (BJL) test 
BJL consider the usual probit model: 
( )
* 2
i i i i
*
i
i
y x u , u iid N 0, , i 1, ..., N
1, y 0
y
0, otherwise,
¢ =b + s =
 >
= 
   (1) 
xi being a (K´1)-vector of exogenous variables, b a vector of unknown parameters, yi
* a 
latent  tendency,  yi  an  observed  indicator,  and  ui  a  stochastic  error  term,  which  is 
stochastically  independent  of  the  exogenous  variables.  They  derive  a  Lagrange-
multiplier  test  of  the  null  hypothesis  of  normally  distributed  error  terms.  Under  the 
alternative hypothesis BJL assume the Pearson family of distributions. Implying a mean 
of  zero,  the  density  function  f(u)  of  this  family  is  characterized  by  the  following 
differential equation (BJL 1984, p. 565): 
( ) 1
2
0 1 2
lnf u c u
u c c u c u
¶ -
=
¶ - +
.  (2) 
The normal distribution is a special case of (2) with c0=s
2 and c1=c2=0. Hence, the 
following hypotheses are tested: 
H0: c1=c2=0   against   H1: c1¹0 and/ or c2¹0. 
"c1¹0"  implies  a  skew  distribution,  "c2¹0"  a  kurtosis  different  from  3  (given  the 
distribution  is  symmetric)  (cf.  Johnson,  Kotz  and  Balakrishnan  1994,  p.  22).  The 
variance s
2 is normalized to 1 as usual (BJL 1984, p. 566). Therefore, the vector q of 
the remaining unknown parameters is equal to (b, c1, c2). 
The  Pearson  family  of  distributions  contains  many  common  distributions  as  special 
cases,  for  instance  the  t-distribution  or  the  gamma  distribution  (Johnson,  Kotz  and 
Balakrishnan 1994, pp. 18 and 21). Nevertheless, the alternative hypothesis is not as 
general as that of the well established Jarque-Bera test in the linear regression model 
(BJL  1984,  p.  564).  However,  it  is  the  most  general  alternative  hypothesis  in  the 
literature. 
Using  the  information  matrix  equality  the  Lagrange  multiplier  test  statistic  can  be 
denoted as   
IWH  __________________________________________________________________ 
 
IWH-Diskussionspapiere 13/2007  6
( ) ( ) ( )
( ) ( ) ( )
0
0
1 r r
H r
r r
H r y
ˆ ˆ lnL y, lnL y,
ˆ LM I ,
lnL y, lnL y,
I E ,
-
¢     ¶ q ¶ q
      = q       ¶q ¶q
   
¢ ¶ q  ¶ q 
q =    ¶q ¶q      (3) 
lnL  being  the  log-likelihood  function  under  H1,  ( ) 1 N y y ... y ¢ = ,  ( ) r 0 0 ¢ q = b , 
( ) r r ˆ ˆ 0 0 ¢ q = b , and  r ˆ b  the probit maximum likelihood estimator of b.  
Define  ( ) ( ) ( ) ( ) ( )
2 2
i i i i i X x 1/3 x 1 1/4 x 3 x
¢         ¢ ¢ ¢ ¢ = - b - b + b                
,  ( ) i i x ¢ j = j b , 
( ) i i x ¢ F = F b ,  i ˆ X ,  i ˆ j ,  i ˆ F  the analogous functions after substituting b by  r ˆ b , j and F 
being  the  density  function  and  the  distribution  function  of  a  standard  normal 
distribution. Given the independence of the sample units the test statistic (3) is obtained 
as (cf. ibid., pp. 570, 571) 
( )
( ) ( )
( )
( )
1
2 N N N
i i i i i i i
i i i i
i 1 i 1 i 1 i i i i i i
ˆ ˆ ˆ ˆ y y ˆ ˆ ˆ ˆ ˆ LM X X X X
ˆ ˆ ˆ ˆ ˆ ˆ 1 1 1
-
= = =
¢       -F j -F j j       ¢ =
      F -F F -F F -F      
∑ ∑ ∑ .  (4) 
It is asymptotically c
2(2) distributed.  
__________________________________________________________________  IWH 
 
IWH-Diskussionspapiere 13/2007  7
3  A simple representation of the test 
The statistic (4) is equal to the explained sum of squares of an artificial regression. 
Define r as vector of the standardized residuals of a probit estimation, i.e.  
( )
( )
i i
1 N i
i i
ˆ y
r r , ..., r , r , i=1, ..., N,
ˆ ˆ 1
-F ¢ = =
F -F
 and  (5) 
( )
1
i
i i
i i
N
R
ˆ ˆ R , R X ',  i = 1, ..., N.
ˆ ˆ 1 R
 
j   = =  
  F -F
 
M   (6) 
Using (5) and (6) the statistic (4) can be written as follows: 
( )
1
LM r R R R R r
-
¢ ¢ ¢ = .  (7) 
Defining the artificial regression 
r = Rd + e,  (8) 
and letting  ˆ d be the ordinary least squares estimator of d (7) is equivalent to 
( ) ˆ ˆ LM R R ¢ = d d,  (9) 
i.e. the LM statistic is the explained sum of squares of (8). 
Based on (7)-(9) LM statistic and artificial regression can be interpreted as follows: The 
test rejects the hypothesis of normally distributed errors if the variables on the right hand 
side of (8) "explain" the standardized residuals of the probit model. The products of 
those variables with the dependent variable are already equal to the gradient of the log-
likelihood function under H1 (cf. (3) and (4)). The first K components of this vector are 
equal again to the gradient of the log-likelihood function of a standard probit model (cf. 
Greene  2003,  p.  671).  Hence,  the  first  K  regressors  of  the  artificial  regression  are 
orthogonal to the dependent variable.  
Furthermore, in case the null hypothesis is valid  r ˆ q  is close to the unrestricted maximum 
likelihood estimate. Therefore, the derivates with respect to c1 and c2 are close to zero, 
i.e.  the  last  two  regressors  of  the  artificial  regression  are  almost  orthogonal  to  the 
dependent variable. Thus, no regressor explains the dependent variable, and the LM 
statistic becomes small. However, in case of skewness or a kurtosis unequal to 3  r ˆ q  
might be rather different from the unrestricted maximum likelihood estimate. Then, the  
IWH  __________________________________________________________________ 
 
IWH-Diskussionspapiere 13/2007  8
last two regressors are not orthogonal to the vector of the standardized residuals, and the 
LM statistic becomes large.  
Concerning a sample unit regression and defining 
( )
i
i
i i
ˆ ˆ f
ˆ ˆ 1
j
=
F -F
 (8) is 
( )
i i
1 i i1 K i iK
i i
ˆ y ˆ ˆ f x f x
ˆ ˆ 1
-F
= d + +d
F -F
K  
( ) ( ) ( )
2 2
K 1 i r i K 2 i r i r i i
1 1 ˆ ˆ ˆ ˆ ˆ f x 1 f x 3 x
3 4
+ +
       ¢ ¢ ¢ +d - b - +d b + b +e                
.  (10) 
Using  (10)  all  variables  of  the  artificial  regression  can  be  calculated  easily  without 
matrix algebra. Merely standard forecasts of a probit estimation are needed. Afterwards, 
the LM statistic is computed via (9) and compared with the critical value of a c
2(2) 
distribution.  The  LIMDEP  code  of  the  complete  procedure  can  be  found  in  the 
appendix.  
__________________________________________________________________  IWH 
 
IWH-Diskussionspapiere 13/2007  9
4  Comparison with a RESET-type test 
Papke  and  Wooldridge  (1996,  p.  624)  test  the  hypothesis  that ( )
*
i i E y |x   is  a  linear 
function of xi against the alternative that  ( )
*
i i E y |x  is a nonlinear function of xi. Using 
the notation above, their artificial regression is 
( ) ( ) ( )
2 3
i i
1 i i1 K i iK K 1 i r i K 2 i r i i
i i
ˆ y ˆ ˆ ˆ ˆ ˆ ˆ f x f x f x f x
ˆ ˆ 1
+ +
-F ¢ ¢ = d + +d +d b +d b +e
F -F
% % % % % K .   (11) 
Their test statistic is N×Ru
2, Ru
2 being the constant-unadjusted coefficient of determina-
tion of (11). Since (1/N)r'r converges to 1 (Engle 1984, p. 818), N×Ru
2 is asymptotically 
equivalent to the explained sum of squares of (11). 
It is easily seen that (10) can be transformed into (11) given that xi includes a constant. 
Therefore, the JBL test against a Pearson distribution of the residuals and the RESET-
type test of Papke and Wooldridge against a nonlinear function of the conditional mean 
(approximated by a quadratic and cubic term) are asymptotically identical. Additional 
information  is  needed  to  clarify  the  reason  for  a  rejection  of  the  null  hypothesis. 
However, given that the mean is a linear function of xi, regression (10) contains more 
information than regression (11). For it indicates whether the null is rejected because of 
skewness or because of a non-standard kurtosis.  
IWH  __________________________________________________________________ 
 
IWH-Diskussionspapiere 13/2007  10
Appendix:  
LIMDEP code  
?=  Replace  var2,  …,  varK  by  your  explanatory  variables,  and  insert  the  dependent 
dummy variable 
namelist  ;x_prob=one, var2, …, varK $ 
create  ;y=… $ 
probit  ;lhs=y; rhs=x_prob $ 
matrix ;bxi=x_prob*b $ 
create  ;bexi=bxi $ 
create  ;bexiph=phi(bexi) ;ya=y-bexiph 
    ;yar=bexiph*(1-bexiph) ;yart=1/(yar^0.5) 
    ;y_art=ya*yart ;xa=N01(bexi) 
    ;xar=xa*yart ;x_art1=xar $ 
?= Complement the lines, and replace var2, …, varK by your explanatory variables 
create  ;x_art2=xar*var2 
    ;… 
    ;x_artK=xar*varK $ 
create  ;x_arts=-1/3*xar*(bexi^2-1) 
    ;x_artk=0.25*xar*bexi*(3+bexi^2)$ 
?= Complete the list of regressors for the artificial regression 
namelist  ;x_artreg=x_art1,x_art2,...,x_artK,x_arts,x_artk$ 
regress  ;lhs=y_art; rhs=x_artreg $ 
matrix  ;y_hat=x_artreg*b$  
calculate  ;list; LM=y_hat'y_hat; p_value=1-Chi(LM,2)$  
__________________________________________________________________  IWH 
 
IWH-Diskussionspapiere 13/2007  11
References 
Bera, A. K.; Jarque, C. M.; Lee, L. F. (1984): Testing the normality assumption in 
limited dependent variable models. International Economic Review 25, pp. 563-578. 
Davidson, R.; MacKinnon, J. G. (1984): Convenient specification tests for logit and 
probit models. Journal of Econometrics 25, pp. 241-262. 
Davidson,  R;  MacKinnon,  J.  G.  (1993):  Estimation  and  inference  in  econometrics. 
Oxford University Press: New York. 
Engle,  R.  F.  (1984):  Wald,  Likelihood  Ratio,  and  Lagrange  Multiplier  tests  in 
econometrics, in: Z. Griliches, M. Intriligator (eds), Handbook of econometrics, Vol. 2. 
North Holland, Amsterdam, pp. 775-826. 
Greene, W. (2003): Econometric analysis 5
th ed. Prentice Hall: Upper Saddle River. 
Johnson, N. L.; Kotz, S.; Balakrishnan, N. (1994): Continuous univariate distributions 
Vol. 1 2
nd ed. Wiley: New York. 
Papke, L. E.; Wooldridge, J. M. (1996): Econometric methods for fractional response 
variables with an application to 401 (k) plan participation rates. Journal of Applied 
Econometrics 11, pp. 619-632. 
Silva,  J.  M.  C.  (2001):  A  score  test  for  non-nested  hypotheses  with  applications  to 
discrete data models. Journal of Applied Econometrics 16, pp. 577-597. 
Wooldridge, J. M. (2002): Econometric analysis of cross section and panel data. MIT 
Press: Cambridge MA. 
 

A Simple Representation of the Bera-Jarque-Lee Test for Probit Models

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics