This paper examines the multi-scale relationship between the interest rate, exchange rate and stock price using wavelet transform. In particular, we apply the maximum overlap discrete wavelet transform (MODWT) to the interest rate, exchange rate and stock price for US over the period 1990:1-2008:12 and using the definitions of wavelet variance, wavelet correlation and cross-correlations analyze the association as well as the lead/lag relationship between these series at the different time scales. Our results show that the relationship between interest rate and exchange rate is not significantly different from zero at all scales. On the other hand, the relationship between interest rate returns and stock index returns is significantly different zero only at the highest scales. The exchange rate returns and stock index returns have a relationship bidirectional in this period at longer horizons.
Scaling laws and generally self-similar structures are now well known facts in financial time series. Furthermore, these signals are characterized by the presence of stochastic behavior allowing their analysis with pure functional methods being incomplete. In the present paper, some existing models are reviewed and modified, based on wavelet theory and self-similarity, to recover multi-scaling cases for approximating financial signals. The resulting models are then tested on some empirical examples and analyzed for error estimates.
Many multifractal models such as self‐similar and scaling law types have been proved to be efficient modellers and estimators in many fields such as financial time series where the data hide fractal and multifractal structures, allowing its processing without sophisticated models to be difficult. However, in statistical analysis, a necessary part that should take place for any model and estimator consists in tests of performance such as confidence intervals and generally statistical tests to confirm the adequacy of the model. The present paper provides the consideration of multifractal models based on wavelets and self‐similar type processes to study statistical tests. To test the efficiency, accuracy and robustness of the models, different inferential statistics are introduced, provided with some empirical examples due to the EURO/USD exchange rate time series with a sample covering the period 03/01/2000 to 30/08/2022. Contrarily to existing works, we showed in the present work that quasi‐self‐similar type models are better for many reasons. They indeed guarantee the well fitting of the data dynamics, the nonlinearity in both the model and the multifractal spectrum, the renormalization parameters which may differ from one scale to another and the preservation of the quasi multiplicative structure.
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