Predict one variable based on another similar I have two times series: A and B, they are highly correlated.
I would like to build the model A~B,however, I have observed that A moves up earlier than B and much faster. Later, after the shock, A decreases also much faster.
Do you have any hints how can I build the model? 
Files:
a) A: https://1drv.ms/u/s!Am9f1Ox4hcd-6VizdMp9r3sbWIYe?e=HiTGBF
b) B: https://1drv.ms/u/s!Am9f1Ox4hcd-6VkG6TvCkY0Id6DC?e=Fc2OfB

 A: I took your 78 quarterly values and arbitrarily selected A as the output series and B as the input. I used AUTOBOX a piece of software that I have helped to develop in it's optionally totally automatic mode. It follows the Transfer Function paradigm http://www.autobox.com/pdfs/A.pdf to form a SARMAX model https://autobox.com/pdfs/SARMAX.pdf .
While both series themselves are non-stationary themselves , the equation/relationship between them required no differencing operators. This phenomenon is not unusual at all .
Model diagnostics from a tentative TF model suggested the need for a (visually obvious) level shift indicator at period 38 (2009/2) and  two pulse indicators (periods 71 and 9 ... 2017/3 & 2002/1  ) and an AR(1) component.
The model used both a contemporary direct effect of B and an indirect effect of B lagged twice in order to predict Y.
The method used to identify these three latent determinstic structures is here http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html .
The Actual/Fit and Forecast graph is here  with equation here  and here 
The statistics for the model are here 
The model residual plot suggests both mean and error variance constancy .  with an ACF here  . AUTOBOX at one point tentatively considered a second level shift at period 38 (2009/2) but found it not-significant.
The forecasts for the next 12 quarters reflect the uncertainty in the predictions for the input series B and the possibility of future pulses .  . The limits were generated using montecarlo procedures providing a complete probability distribution for each forecast period.
The Actuals & Cleansed graph highlight the level shift (intercept change) and the two (now !) clear pulses.
The "shock" that you allude to is a "permanent effect" .
In conclusion to form this model the following 5 characterics needed to be identified
1 What level of differencing needs to be included (if any )
2 what is the form of the relationship
3 Is there a level shift needed to deal with an exogenous unstated factor
4 are there one-time only effects needed to deal with exogenous factors
5 What is the impact of omitted stochastic series i.e.the form of the ARMA structure.
6 What power transformation or weighted least squares approach is needed to deal with non-constant error variance through time
