We combine deep X-ray survey data from the Chandra observatory and the wide-area/shallow XMM-XXL field to estimate the active galactic nuclei (AGN) X-ray luminosity function in the redshift range z = 3–5. The sample consists of nearly 340 sources with either photometric (212) or spectroscopic (128) redshift in the above range. The combination of deep and shallow survey fields also provides a luminosity baseline of three orders of magnitude, L_X(2–10 keV) ≈ 10⁴³–10⁴⁶ erg s^{− 1} at z > 3. We follow a Bayesian approach to determine the binned AGN space density and explore their evolution in a model-independent way. Our methodology properly accounts for Poisson errors in the determination of X-ray fluxes and uncertainties in photometric redshift estimates. We demonstrate that the latter is essential for unbiased measurement of space densities. We find that the AGN X-ray luminosity function evolves strongly between the redshift intervals z = 3–4 and z = 4–5. There is also suggestive evidence that the amplitude of this evolution is luminosity dependent. The space density of AGN with L_X(2–10 keV) < 10⁴⁵ erg s^{− 1} drops by a factor of 5 between the redshift intervals above, while the evolution of brighter AGN appears to be milder. Comparison of our X-ray luminosity function with that of ultraviolet (UV)/optical selected quasi-stellar objects at similar redshifts shows broad agreement at bright luminosities, L_X(2–10 keV) > 10⁴⁵ erg s^{− 1}. At fainter luminosities X-ray surveys measure higher AGN space densities. The faint-end slope of UV/optical luminosity functions, however, is steeper than for X-ray selected AGN. This implies that the Type I AGN fraction increases with decreasing luminosity at z > 3, opposite to trends established at lower redshift. We also assess the significance of AGN in keeping the hydrogen ionized at high redshift. Our X-ray luminosity function yields ionizing photon rate densities that are insufficient to keep the Universe ionized at redshift z > 4. A source of uncertainty in this calculation is the escape fraction of UV photons for X-ray selected AGN.

We study the Bitcoin and Ether price series under a financial perspective. Specifically, we use two econometric models to perform a two-layer analysis to study the correlation and prediction of Bitcoin and Ether price series with traditional assets. In the first part of this study, we model the probability of positive returns via a Bayesian logistic model. Even though the fitting performance of the logistic model is poor, we find that traditional assets can explain some of the variability of the price returns. Along with the fact that standard models fail to capture the statistic and econometric attributes—such as extreme variability and heteroskedasticity—of cryptocurrencies, this motivates us to apply a novel Non-Homogeneous Hidden Markov model to these series. In particular, we model Bitcoin and Ether prices via the non-homogeneous Pólya-Gamma Hidden Markov (NHPG) model, since it has been shown that it outperforms its counterparts in conventional financial data. The transition probabilities of the underlying hidden process are modeled via a logistic link whereas the observed series follow a mixture of normal regressions conditionally on the hidden process. Our results show that the NHPG algorithm has good in-sample performance and captures the heteroskedasticity of both series. It identifies frequent changes between the two states of the underlying Markov process. In what constitutes the most important implication of our study, we show that there exist linear correlations between the covariates and the ETH and BTC series. However, only the ETH series are affected non-linearly by a subset of the accounted covariates. Finally, we conclude that the large number of significant predictors along with the weak degree of predictability performance of the algorithm back up earlier findings that cryptocurrencies are unlike any other financial assets and predicting the cryptocurrency price series is still a challenging task. These findings can be useful to investors, policy makers, traders for portfolio allocation, risk management and trading strategies.