Your data visually suggests an asymptotic (gradual) change to the new level. Time series methods can often be used to detect these kinds of structures even if the data is not time series. Please post your data and I may be able to demonstrate this with "toys" at my disposal. If your data is time series then as @jason reflected one needs to deal effectively with the noise model to correctly "see" the structure.
EDITED UPON RECEIPT OF DATA:
Modelling is often an iterative approach with interim steps providing valuable clues to a useful model. I took your data and introduced it to AUTOBOX (one of my toys which I have helped develop). An initial graph strongly suggested a longitudinal (chronological) data set where the X series is reported at fixed intervals. AUTOBOX automatically suggested a standard ARIMA model (with Intervention Detection) replacing the non-stationary X with a differencing operator. Here is the actual/fit/forecast graph and the suggested model.
Upon examination another possible model incorporating a lag structure for an indicator variable suggested itself. I introduced a Pulse at time period 76 (a Dynamic Predictor expressly allowing up to a possible lag effect of 50 periods) (the beginning of the transition) to deal with the relationship between the original Y and the user-suggested X in order to more fully investigate the effect of X than accept the total setting-aside of X.
Following is the actual-fit-forecast graph for that approach and the identified robust transfer function model . with residual plot and residual acf here
The final model captures the dynamics in certain lags of the Dynamic Predictor and a few pulses and a reasonable memory structure.
Even the most powerful analysis packages often need some guidance when dealing with complex real world data sets like this one as nothing compares to the creative human mind.