Stationarity of Bitcoin and S&P 500 Returns Across Time Scales

Summary

This study compares wide-sense stationarity in intraday S&P 500 and Bitcoin price returns. It tests stationarity using the relationship between autocorrelation and power spectral density, and examines how segmenting, detrending, and normalizing the series affect the results. The data spans a much longer period for the S&P 500 than for Bitcoin.

The reported findings indicate that S&P 500 returns can be made stationary across the full sample with a long detrending window and a short normalization window; segmented analysis permits larger normalization windows. For Bitcoin, stationarity is reported only in a higher-volatility segment using a longer normalization window, and not in the other segments. These conclusions depend on the chosen preprocessing, windows, sample periods, and stationarity definition; the supplied text does not describe broader robustness checks or implications for forecasting and trading.

Key ideas

• The study defines stationarity through time-invariant first and second moments.
• It uses the Wiener-Khinchin relationship between autocorrelation and power spectral density to test stationarity.
• Segmentation, detrending, and normalization affect whether returns appear stationary.
• The reported S&P 500 and Bitcoin results differ and vary by segment and preprocessing choices.

Tags

# Comparative analysis of stationarity for Bitcoin and the S&P500


# Comparative analysis of stationarity for Bitcoin and the S&P500









This paper compares and contrasts stationarity between the conventional stock market and cryptocurrency. The dataset used for the analysis is the intraday price indices of the S&P500 from 1996 to 2023 and the intraday Bitcoin indices from 2019 to 2023, both in USD. We adopt the definition of `wide sense stationary', which constrains the time independence of the first and second moments of a time series. The testing method used in this paper follows the Wiener-Khinchin Theorem, i.e., that for a wide sense stationary process, the power spectral density and the autocorrelation are a Fourier transform pair. We demonstrate that localized stationarity can be achieved by truncating the time series into segments, and for each segment, detrending and normalizing the price return are required. These results show that the S&P500 price return can achieve stationarity for the full 28-year period with a detrending window of 12 months and a constrained normalization window of 10 minutes. With truncated segments, a larger normalization window can be used to establish stationarity, indicating that within the segment the data is more homogeneous. For Bitcoin price return, the segment with higher volatility presents stationarity with a normalization window of 60 minutes, whereas stationarity cannot be established in other segments.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.