Combine linear trend and multiplicative effects in regression framework

Question

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!

Answer 1

Your error term is a bit strange because it is multiplicative to time and not to seasonality and trend. I would reconsider if this should be expected in your real data. However, what you have seems close enough to a multiplicative time series to get at least decent estimates:

$Y_t = T_t \cdot S_t \cdot e_t$

Such a time series can be decomposed easily if you define an appropriate frequency for the time series. With stats::decompose:

set.seed(1)
x <- ts(1:30, frequency = 7) #note the frequency
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)

y <- decompose(x, type = "multiplicative")
plot(y)

fit_trend <- lm(y$trend ~ seq_along(y$trend))
summary(fit_trend)$coef[-1,]
#    Estimate   Std. Error      t value     Pr(>|t|) 
#9.182801e-01 1.060280e-02 8.660734e+01 2.253371e-29 

You could also check out the forecast package which offers more flexible and automated decomposition of time series.