I am attempting to write the mathematical model for and also simulate an MA(1) process that has drift (in R).

I have referenced ARIMA (0,1,1) or (0,1,0) - or something else?, Simulation of forecasted values in ARIMA (0,1,1), and Fitting ARIMA with a drift on R. I am aware that the "forecast" package that I am using addresses the issues relating to mean reporting here: http://www.stat.pitt.edu/stoffer/tsa2/Rissues.htm.

My understanding (e.g. from example bottom of page on the very useful https://www.otexts.org/fpp/8/7 page on the otext site) is that an MA(1) model (on a series that for which the first difference is stationary) can be written as: $$ (1-B)^d(y_t- \mu t) = (1 - \theta B) e_t $$ where $B$ is the backshift operator, $d$ is the order of differencing (here is 1), $y_t$ is the original series indexed by time $t$, $\theta$ is the MA1 parameter and $e_t$ is the error indexed at time $t$.

From the above otext site, I note:

the inclusion of a constant in a non-stationary ARIMA model is equivalent to inducing a polynomial trend of order $d$ in the forecast function.

So while my forecasts are correct, Q.1. How do I represent/differentiate this single order trend in the equation from the equation for an MA(1) model without drift? I ask this because expanding the above formula just produces the exact same equation/model I expect for an MA(1) without drift, i.e. (and apologies in advance if there is just something obviously wrong with my maths): $$ \begin{aligned} y_t - \mu t - y_{t-1} + \mu t &= e_t - \theta e_{t-1} \\ y_t - y_{t-1} &= e_t - \theta e_{t-1} \\ \Delta y_t &= e_t - \theta e_{t-1} \\ \end{aligned} $$

In terms of simulating an MA(1) with drift I used the following, Q.2. Is this the correct approach or is there a more direct/accurate way to do this?:

temp <- arima.sim(n=100, list(order=c(0,1,1), ma=.75), sd = 2.7) 

ts.sim.orig <- ts(temp, start=c(2011,1), frequency=12)
# Stationary

# Drift term being 0.3
ts.sim <- ts(1:length(ts.sim.orig) * 0.3 + temp, start=c(2011,1), frequency=12)
# Not stationary
# Stationary
kpss.test(diff(ts.sim, 1))
# Non zero mean on first differnce
mean(diff(ts.sim, 1))
# Plot is reasonable?
xyplot(diff(ts.sim, 1))

# First fit without drift:
summary(fit99 <- Arima(ts.sim, order=c(0,1,1), include.constant = F)) 
# Forecast is flatline as expected. Great.

# The drift is close to the mean from: mean(diff(ts.sim, 1))
summary(fit99 <- Arima(ts.sim, order=c(0,1,1), include.constant = T)) 
# Forecast includes trend

y <- ts.sim
x <- 1:length(ts.sim)
# Roughly equal to drift term as expected
summary(lm(y ~ x))

# Aside, this don't seem to recover the drift parameter too well...
summary(fit99 <- auto.arima(ts.sim, d = 1, trace = T, stepwise=FALSE, approximation=FALSE, seasonal = F))

Finally, does the approach to the simulation above answer my first question? That is, is the appropriate representation of the MA(1) with drift (and presumably assuming a zero intercept) equal to the following equation? $$ \Delta y_t = \beta t + e_t - \theta e_{t-1} $$ Thank you.


Regarding Q1 : The difference operator applied to µt gives µt – µ(t-1) = µ. Hence the drift parameter µ will appear in your equation. Maybe better to choose another symbol for the drift parameter, since µ is traditionally used for the mean of the time series.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.