I have some data that has the pattern in the picture below (but little noisier than that).
I want to run a non-linear regression that tries to capture the dynamic of this data in the time-series where there is a jump that occurs at a specific time (in the picture below t=5) and with some exponential decay of the variable Y following the jump.
I don't have much of a clue on how to specify this regression. At first I was thinking to simply do:
$ln(y) = ln(a) + b*ln(x)+ln(u)$
But what about the jump? I am interested in the inference of all three parameters simultaneously (including the jump parameter).