  This paper explores a time-varying version of weak-form market efficiency
that is a key component of the so-called Adaptive Market Hypothesis (AMH). One
of the most common methodologies used for modeling and estimating a degree of
market efficiency lies in an analysis of the serial autocorrelation in observed
return series. Under the AMH, a time-varying market efficiency level is modeled
by time-varying autoregressive (AR) process and traditionally estimated by the
Kalman filter (KF). Being a linear estimator, the KF is hardly capable to track
the hidden nonlinear dynamics that is an essential feature of the models under
investigation. The contribution of this paper is threefold. We first provide a
brief overview of time-varying AR models and estimation methods utilized for
testing a weak-form market efficiency in econometrics literature. Secondly, we
propose novel accurate estimation approach for recovering the hidden process of
evolving market efficiency level by the extended Kalman filter (EKF). Thirdly,
our empirical study concerns an examination of the Standard and Poor's 500
Composite stock index and the Dow Jones Industrial Average index. Monthly data
covers the period from November 1927 to June 2020, which includes the U.S.
Great Depression, the 2008-2009 global financial crisis and the first wave of
recent COVID-19 recession. The results reveal that the U.S. market was affected
during all these periods, but generally remained weak-form efficient since the
mid of 1946 as detected by the estimator.
