  In the standard data analysis framework, data is first collected (once for
all), and then data analysis is carried out. With the advancement of digital
technology, decisionmakers constantly analyze past data and generate new data
through the decisions they make. In this paper, we model this as a Markov
decision process and show that the dynamic interaction between data generation
and data analysis leads to a new type of bias -- reinforcement bias -- that
exacerbates the endogeneity problem in standard data analysis.
  We propose a class of instrument variable (IV)-based reinforcement learning
(RL) algorithms to correct for the bias and establish their asymptotic
properties by incorporating them into a two-timescale stochastic approximation
framework. A key contribution of the paper is the development of new techniques
that allow for the analysis of the algorithms in general settings where noises
feature time-dependency.
  We use the techniques to derive sharper results on finite-time trajectory
stability bounds: with a polynomial rate, the entire future trajectory of the
iterates from the algorithm fall within a ball that is centered at the true
parameter and is shrinking at a (different) polynomial rate. We also use the
technique to provide formulas for inferences that are rarely done for RL
algorithms. These formulas highlight how the strength of the IV and the degree
of the noise's time dependency affect the inference.
