In this study, we examined the efficiency of cryptocurrencies Bitcoin (BTC), Ethereum (ETH), Litecoin (LTC), Ripple (XRP), DASH, EOS, and MONERO from March 1, 2018, to March 1, 2023. We separated the sample into four subperiods for this purpose: a Tranquil period that includes the period from March 1, 2018, to December 31, 2019; a First Wave that includes the year 2020; a Second Wave that includes the year 2021; and a fourth subperiod that includes Russia's invasion of Ukraine in 2022-2023. The results are mixed, with some cryptocurrencies exhibiting equilibrium and others exhibiting autocorrelation and predictability in their pricing. When the sample is divided into subperiods, most digital currencies have long memories in their returns during the Tranquil period, BTC, LTC, and XRP exhibit efficiency during the First Wave of the pandemic, while BTC, ETH, and MONERO indicate efficiency during the Second Wave. Most assessed digital currencies showed equilibrium by 2022, with the exception of ETH and MONERO, which exhibit long memories, and LTC, which demonstrates anti-persistence. These results hold significance for investors in these alternative markets, as they suggest that some cryptocurrencies may be more predictable and therefore potentially profitable, whereas others may require greater caution and risk management strategies.

This paper investigates the degree of inflation persistence across regions in Portugal by analyzing the Consumer Price Index (CPI) growth rates for NUTS II regions. The study employs the Augmented Dickey-Fuller (ADF) test to determine whether the CPI data for Portugal is stationary or non-stationary. The results of the ADF test reveal that the IPC data for Portugal is non-stationary, indicating that inflation exhibits persistence in the long run. The study further assesses the persistence of inflation by estimating an autoregressive integrated moving average (ARIMA) model for each region. The Ljung-Box test is used to test for autocorrelation in the time series data, and the Hurst exponent is calculated to evaluate the presence of long-term memory in the time series data. The study finds that there is significant autocorrelation in the time series data for all regions, supporting the presence of persistence in inflation at the regional level in Portugal. The Hurst exponent also shows that the time series data for each region exhibits a high degree of persistence in inflation. Finally, the study applies the ARIMA model to each CPI division's data and uses the Ljung-Box test to test for autocorrelation in the time series data. The results show that some CPI divisions exhibited higher levels of persistence compared to others. For example, the "Housing, water, electricity, gas and other fuels" division exhibited high persistence, while the "Communication" division exhibited low persistence. This study contributes to the existing literature by exploring regional inflation persistence in Portugal and its implications for policymaking. The results provide insights into the inflation persistence patterns across regional levels in Portugal, by emphasizing the need to consider regional differences in inflation dynamics when formulating effective policy interventions. Understanding the persistence of inflation is crucial for policymakers to ensure price stability and sustainable economic growth.