In this paper, a form of adaptive weak-form market efficiency of the Romanian stock market is examined via a state-space approach. A general methodology for estimating evolving efficiency is proposed within the extended Kalman filter associated with the customized state-space models in this work. The suggested approach for measuring a time-varying degree of market efficiency is based on time-dependent autoregressive (AR) models where a conclusion regarding the evolving efficiency is deduced by tracking the changes in regression coefficients. Motivated by previous studies of the Bucharest stock exchange, we explore a structural time-series model that contains a stochastic trend and a white noise component. Monthly data covers the period from September 1997 to December 2020, which includes the period of the 2008–2009 global financial crisis and the recent COVID-19 recession. Our major findings reveal that the Romanian stock market is weak-form efficient at the end of our empirical study and has not been greatly affected by the first wave of the COVID-19 pandemic. However, it suffered severely during the 2008–2009 crisis, which is reflected in a lingering recovery process.