time series mainly characterized by structural breaks - how to model?

Question

I am given a financial time series that is characterized by a bunch of structural breaks, i.e. the series isn't moving (literally at all), but at some points in time the series jumps up or down. Then it stays at this level for a while until the series jumps again. So the time series basically looks like a step function.

My assumption is that these breaks come from some particular exogenous variables that are in the form of dummies. So if a particular exogenous variable takes on the value 1, (I assume) it is very likely that the series jumps.

My question is how I could model this particular time series (in a uni- or multivariate sense). I guess that standard AR(MA)-models are inappropriate. I was thinking about creating two binary variables that take on the value 1 if there's an upward (downward) break and 0 otherwise. Then I would run a dynamic probit model to test the probabilities that the exogenous variables trigger a break. What do you think about this idea? Or would you have other suggestions? Please note that I don't wanna test for structural breaks but rather formulate a time series model.

Answer 1

The described time series can be modeled as

$ X(t) = \sum_{i=1,...,N(t)} Y_i, $

where

$ Y_1,...,Y_n,... $ are iid with the distribution function $G(y)$,

$N(t)$ is a counting process with stochastic intensity $\lambda(t)$.

Function $G(y)$ can be estimated from the data nonparametrically or parametrically, while intensity $\lambda(t)$ can be modeled as a function of various exogenous processes $Z(t)$ and estimated via the method of maximum likelihood... Stochastic process $X(t)$ is known as compound counting process.