Long memory and multifractality: A joint test

• Hypothesis tests for the multifractal model of asset returns are developed.
• Conventional tests reject the null hypothesis of no long memory too frequently.
• Long memory and multifractality are estimated jointly using exchange rate data.
• Most of the exchange rate returns series are most appropriately characterized by a variant of the MMAR that applies a multifractal time-deformation process to NIID returns.
• None of the exchange rate returns series exhibits evidence of long memory.

The properties of statistical tests for hypotheses concerning the parameters of the multifractal model of asset returns (MMAR) are investigated, using Monte Carlo techniques. We show that, in the presence of multifractality, conventional tests of long memory tend to over-reject the null hypothesis of no long memory. Our test addresses this issue by jointly estimating long memory and multifractality. The estimation and test procedures are applied to exchange rate data for 12 currencies. Among the nested model specifications that are investigated, in 11 out of 12 cases, daily returns are most appropriately characterized by a variant of the MMAR that applies a multifractal time-deformation process to NIID returns. There is no evidence of long memory.