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can anyone help me to explain how the following model works?
In this formula $P(t)$ is a price at time $t$ and $F(t)$ is the residual noise term.
The $\omega(i;T_i)$ and $P_{o}(i;T_i)$ are uniquely determined from the past $T_{i}$ data points by condition that minimizes the root-mean square of $F(t)$.
Why $(\omega_{1}(i;T_i)-1)$?
Thanks in advance.
The paper explains it quite well I think: There are three cases:
(1) $\omega>1$ the price is either exponentially increasing or decreasing and $P_{0}$ gives the base line of the exponential divergence. We define such behavior as a bubble or a crash. In this case the positive feedback from the past price change becomes larger as the time passes.
(2) $\omega=0$ the price follows a random walk.
(3) $\omega<0$ the price is convergent to $P_{0}$.
So the value of the parameter just either magnifies the expression $\{P(t-1)-P_{0}\}$, renders it zero, or makes the price increments revert to $P_{0}$.
