JARQUE BERA TEST

JARQUE BERA TEST

Analysis Command

Perform a Jarque-Bera test for normality.

Most tests for normality are based either on comparing the empirical cumulative distribution with the theoretical normal cumulative distribution (Kolmogorov-Smirnov, Anderson-Darling, Chi-Square) or empirical quantiles with the theoretical normal quantiles (PPCC, Wilk-Shapiro). In contrast, the Jarque-Bera test is based on the sample skewness and sample kurtosis.

The Jarque-Bera test statistic is defined as:

\( \frac{N}{6} \left( S^2 + \frac{(K - 3)^2}{4} \right) \)

with S, K, and N denoting the sample skewness, the sample kurtosis, and the sample size, respectively.

For sample sizes of 2,000 or larger, this test statistic is compared to a chi-squared distribution with 2 degrees of freedom (normality is rejected if the test statistic is greater than the chi-squared value).

The chi-square approximation requires fairly large sample sizes to be accurate. For sample sizes less than 2,000, the critical value is determined via simulation. Specifically, one hundred thousand normal samples with the same mean and standard deviation as the original data sample are generated and the Jarque-Bera test statistic computed to generate the reference distribution.

JARQUE BERA TEST <y>             <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

MULTIPLE JARQUE BERA TEST <y1> ... <yk>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> ... <yk> is a list of 1 to 30 response variables;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

REPLICATED JARQUE BERA TEST <y> <x1> ... <xk>
                        <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
            <x1> ... <xk> is a list of 1 to 6 group-id variables;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

JARQUE BERA TEST Y1
JARQUE BERA TEST Y1 SUBSET TAG > 2
MULTIPLE JARQUE BERA TEST Y1 TO Y10
REPLICATED JARQUE BERA TEST Y X

Syntax 1 and Syntax 2 support matrix arguments. Syntax 2 supports the TO syntax.

For Syntax 3 (the REPLICATED form), the variables must all have the same number of observations.

The following statistics are also supported:
LET A = JARQUE BERA TEST Y
LET A = JARQUE BERA TEST CDF Y
LET A = JARQUE BERA TEST PVALUE Y


In addition to the above LET command, built-in statistics are supported for about 20+ different commands (enter HELP STATISTICS for details).

None

JARQUE BERA NORMALITY TEST
JARQUE BERA

GOODNESS OF FIT	= Perform Anderson-Darling, Kolmogorov-Smirnov, Chi-Square, and PPCC goodness of fit tests.
WILK SHAPIRO TEST	= Perform a Wilk Shapiro test for normality.
PROBABILITY PLOT	= Generate a probability plot.

Brani Vidakovic (2011), "Statistics for Bioengineering Sciences: with MATLAB and WINBUGS Support," Springer, pp. 521-522.

Distributional Fitting, Assumption Testing

2012/6

 
SKIP 25
READ ZARR13.DAT Y1
READ ZARR15.DAT Y2
SET WRITE DECIMALS 4
MULTIPLE JARQUE BERA TEST Y1 Y2
    

The following output is generated

            Jarque-Bera Test for Normality
 
Response Variable: Y1
 
H0: The Data Are Normally Distributed
Ha: The Data Are Not Normally Distributed
 
Summary Statistics:
Total Number of Observations:                        195
Sample Mean:                                      9.2614
Sample Standard Deviation:                        0.0227
Sample Skewness:                                 -0.0085
Sample Kurtosis:                                  3.0490
Sample Minimum:                                   9.1968
Sample Maximum:                                   9.3279
 
Test Statistic Value:                             0.0219
CDF Value:                                        0.0101
P-Value:                                          0.9898
 
 
Percent Points of the Reference Distribution
-----------------------------------
  Percent Point               Value
-----------------------------------
           25.0    =          0.553
           50.0    =          1.256
           75.0    =          2.403
           80.0    =          2.774
           90.0    =          4.044
           95.0    =          5.679
           97.5    =          8.034
           99.0    =         12.567
 
Conclusions (Upper 1-Tailed Test)
----------------------------------------------
  Alpha    CDF   Critical Value     Conclusion
----------------------------------------------
    10%    90%            4.044      Accept H0
     5%    95%            5.679      Accept H0
   2.5%  97.5%            8.034      Accept H0
     1%    99%           12.567      Accept H0
 
 
            Jarque-Bera Test for Normality
 
Response Variable: Y2
 
H0: The Data Are Normally Distributed
Ha: The Data Are Not Normally Distributed
 
Summary Statistics:
Total Number of Observations:                        211
Sample Mean:                                      9.2772
Sample Standard Deviation:                        0.1117
Sample Skewness:                                 -0.0067
Sample Kurtosis:                                  2.4502
Sample Minimum:                                   9.0274
Sample Maximum:                                   9.5219
 
Test Statistic Value:                             2.6585
CDF Value:                                        0.7840
P-Value:                                          0.2159
 
 
Percent Points of the Reference Distribution
-----------------------------------
  Percent Point               Value
-----------------------------------
           25.0    =          0.557
           50.0    =          1.266
           75.0    =          2.415
           80.0    =          2.786
           90.0    =          4.084
           95.0    =          5.718
           97.5    =          8.066
           99.0    =         12.467
 
Conclusions (Upper 1-Tailed Test)
----------------------------------------------
  Alpha    CDF   Critical Value     Conclusion
----------------------------------------------
    10%    90%            4.084      Accept H0
     5%    95%            5.718      Accept H0
   2.5%  97.5%            8.066      Accept H0
     1%    99%           12.467      Accept H0