I want to analyse a time series that in genereal seems to follow a linear trend but at the same time seems to be influenced from some kind of multiplicative effects. A simple example would be a time series generated by the following code:
set.seed(1) x <- ts(1:30) x[c(7, 14, 21, 28)] <- 0.5 * x[c(7, 14, 21, 28)] x <- x + rnorm(30, mean = 0, sd = 0.05 * 1:30) plot(x, type = "l")
My question is how to best estimate this time series within a regression framework? Obviously a simple linear model would underestimate the multiplicative effect in recent days while a simple log-linear model would estimate a exponential growth instead of a linear one. Is there a simple way to combine both effects within a single regression approach or do I have to do some kind of stepwise estimation?
I would appretiate any thoughts / comments!