# 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?

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.