# Non-Linear regression that captures jumps and an exponential decay

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).

• Could you elaborate on the motivation for fitting a regression? Are you interested in inference about the size/location of the jump? Rate of decay? All three parameters simultaneously? Feb 24, 2015 at 6:38
• @AndrewM good point! All three parameters simultaneously. Feb 24, 2015 at 6:41
• Sorry, but I want to be quite sure -- the time of the jump is unknown? Feb 24, 2015 at 7:14
• @Glen_b I know when the jump occur because at time=5 (in my graph below) is when there is a very important macroeconomic announcement Feb 24, 2015 at 7:24
• Then I only count two parameters. How are there three parameters? Feb 24, 2015 at 9:33

## 1 Answer

Your function appears like it could be parametrized as $y = Ce^{-A(x-B)}\mathbb{I}_{[x>b]}$, where $\mathbb{I}$ is 1 if $x>B$ and 0 else. I can't see anything stopping you from just fitting a non-linear regression to your hypothesized functional form, although you will need to be careful with your choice of an optimizer, since this function is not differentiable in $B$.

A cursory look at the literature$^1$ and extant software indicate that grid searches on $B$, and maximizing the remaining parameters conditional on $B$ are used, and that it can be shown that you only need to check the values $x$ that the data assume.

It's not clear to me, however, that standard asymptotic theory would apply to derive standard errors, owing to the lack of differentiability (w/r/t $B$) in the likelihood. I suppose bootstrapping could provide a way out.

$^1$ Muggeo. "Estimating regression models with unknown break-points" (2003)

• ah I see. OK so when x is greater than the "known" jump time $B$, I have $y=Ce^{-A(x-b)}$ otherwise some constant $C$ plus noise. Let's say that I assume that my errors are normally distributed, can I estimate this using MLE? Feb 24, 2015 at 7:43
• Yes, and if you can assume that $B$ is known, then any old optimizer or solver for non-linear least squares will do. Feb 24, 2015 at 7:47
• Running an OLS like $ln(Y) = ln(C) -A*ln(x-B)*D + ln(u)$ where $D = 1$ if $x>B$ would be however problematic, correct? Feb 24, 2015 at 7:50
• Taking a log of $Y$ changes how the errors enter (multiplicatively vs additive), and more importantly, you can only log $Y$ if it is positive everywhere. Even if we define $\log(0) \times 0 =0$, I don't think we can write anything sensible. Formally, $y=Ce^{-A(x-B)}\mathbb{I}_{[x>b]} \Rightarrow \log(y) = (\log C -A(x-B)) + \log \mathbb{I}_{[x>b]}$, but still have a $\log 0$ appearing in the final term. Feb 24, 2015 at 21:59