Testing the alpha-stable Lévy hypothesis on US stock market data

Abstract

This study tests whether the alpha-stable Lévy hypothesis applies to US stock market data, namely, Dow Jones Industrial Average (DJIA). It makes use of daily data on the closing price of the DJIA, spanning from 28th May 1896 to 10th February 2023, a total of 32,820 trading days. Computation of Realised Variances (RVs) involved squaring daily returns of the asset over a specified time frame spanning from weekly RVs till semi-annual RVs were computed. ‘Log-Log’ Ordinary Least square (OLS) regressions were employed on each time-frame resulting in alpha exponents less than 2 for the 5-, 90, -125 resolutions confirming the main a priori hypothesis that DJIA has an infinite variance that is not defined. Moreover, by observing the 95% confidence interval using the uncertainty principle provided by the log-log regression, we find values ranging from 1.62 to 2.22 as the confidence interval for the power law exponent. It becomes evident that, regardless of the point estimate for other resolutions, all of them fall within this range of point estimate intervals. Therefore, from a statistical perspective, the invariance hypothesis is accepted. Sample split tests deliberately conducted to assess the constancy of power law exponents over time, confirmed that point estimators from the second subsample for time resolutions 5, 20, and 60 fell within the confidence interval range of the first sub sample in all cases. Hence, statistically, the power law exponents persisted across both sub-samples, implying the invariance quality of the power law. When compared against GARCH (1,1) model exponents, the power law null hypothesis prevailed in constancy whereas the former gave sample-specific point estimates.