A Test for Symmetry with Leptokurtic Financial Data
National University of Singapore
University of Illinois at Urbana-Champaign
Address correspondence to Anil K. Bera, Department of Economics, University of Illinois, 1206 S. 6th St., Champaign, IL 61820, or e-mail: abera{at}uiuc.edu.
Most of the tests for symmetry are developed under the (implicit or explicit) null hypothesis of normal distribution. As is well known, many financial data exhibit fat tails, and therefore commonly used tests for symmetry (such as the standard 
test based on sample skewness) are not valid for testing the symmetry of leptokurtic financial data. In particular, the test uses third moment, which may not be robust in presence of gross outliers. In this article we propose a simple test for symmetry based on the Pearson type IV family of distributions, which take account of leptokurtosis explicitly. Our test is based on a function that is bounded over the real line, and we expect it to be more well behaved than the test based on sample skewness (third moment). Results from our Monte Carlo study reveal that the suggested test performs very well in finite samples both in terms of size and power. Simulation results also support our conjecture of the tests to be well behaved and robust to excess kurtosis. We apply the test to some selected individual stock return data to illustrate its usefulness.
KEYWORDS: test, kurtosis, Monte Carlo study, Pearson family of distributions, Rao’s score test, skewness, tan–1(·) function
Received August 7, 2001; revised September 7, 2004; accepted September 29, 2004