Model effect of intervention of continuous and cumulative variable I have a dependent variable dv and an independent variable iv in a timeseries dataset.

The iv varies independently from time. Instead, the dv varies not only in a cumulative way but is also equal to zero for a long period. I want to estimate the association between the two variables (and other control variables). I wonder if I can leverage the fact that there is a sort of intervention around 1992 even if the introduction of dv is progressive.
What are my options?
 A: This is purely speculative, but you could try modeling your dependent variable (Y) as a function of itself in a first-order auto-correlation type model. Like this:
$$
Y_i = r Y_{i-1} + u_i 
$$
This is a simple regression model where I have used auto-correlation notation (r replaces beta, u or disturbances replaces the residual). So, to calculate your $Y_i$-value, you are using the previous Y-value $Y_{i-1}$ as a function of an estimated parameter, plus the disturbances.
Now you can regress the disturbances onto your independent variable (X):
$$
u_i = X_i + e_i
$$
and in this way, you are explaining the variance in Y that went unexplained by the previous Y-value using X. 
I realize as a write this that my first function increases exponentially with increasing Y values, so maybe something like this would be better:
$$
Y_i = r+ Y_{i-1} + u_i 
$$
which just implies that there is an accumulative parameter. Then you can take your disturbances and again regress them onto X, you may include time as a variable too if you wish. 
Again, this is highly speculative, this is where I would start, I have no idea where it would end. This problem is very cool though. 
Unless you have a good reason to include the measurements of X before Y begins to increase, I would just start my model where Y becomes greater than 0. I do not have much knowledge of what you are actually studying though. 
