  Consider the problem of modeling memory effects in discrete-state random
walks using higher-order Markov chains. This paper explores cross validation
and information criteria as proxies for a model's predictive accuracy. Our
objective is to select, from data, the number of prior states of recent history
upon which a trajectory is statistically dependent. Through simulations, I
evaluate these criteria in the case where data are drawn from systems with
fixed orders of history, noting trends in the relative performance of the
criteria. As a real-world illustrative example of these methods, this
manuscript evaluates the problem of detecting statistical dependencies in shot
outcomes in free throw shooting. Over three NBA seasons analyzed, several
players exhibited statistical dependencies in free throw hitting probability of
various types - hot handedness, cold handedness, and error correction. For the
2013-2014 through 2015-2016 NBA seasons, I detected statistical dependencies in
23% of all player-seasons. Focusing on a single player, in two of these three
seasons, LeBron James shot a better percentage after an immediate miss than
otherwise. In those seasons, conditioning on the previous outcome makes for a
more predictive model than treating free throw makes as independent. When
extended to data from the 2016-2017 NBA season specifically for LeBron James, a
model depending on the previous shot (single-step Markovian) does not clearly
beat a model with independent outcomes. An error-correcting variable length
model of two parameters, where James shoots a higher percentage after a missed
free throw than otherwise, is more predictive than either model.
