KILLED · memoryless
Hypothesis
Discretize returns into states (e.g. quintiles) and use the empirical transition matrix to forecast the next state’s expected return.
Math
$$ \pi_{t+1} = \pi_t\,P,\qquad P_{ij}=\hat P(s_{t+1}=j\mid s_t=i) $$
Method
Estimate the transition matrix on rolling windows, trade the expected next-state return, costs applied.
Results
| Transition matrix vs uniform | barely distinguishable |
| Forecast edge | ≈ 0 |
Discretized returns are very nearly memoryless — the estimated transition matrix is almost row-uniform, so next-state forecasts carry no edge. A first-order Markov chain cannot find structure that is not there. Killed.
If the transition matrix is nearly uniform, the process is nearly memoryless. Discretizing a random walk does not create predictability.