Effect of moving average parameter on variability and variance of demand Consider an MA(1) process, $d_{t}=e_{t}-\Theta e_{t-1}$, when $d_t$ is the demand at time $t$ and $e_t$ is error term and $\Theta$ is moving average parameter. Now if $\Theta$ equal to zero so we have a white noise process, if $\Theta$ gets positive values, process is more irregular than white noise, and when the is negative, process is more smoother than white noise. I have two questions:.


*

*How can we explain the variability of the process by changing the
$\Theta$ parameter from zero toward +1 and -1? 

*Why for positive and
negative $\Theta$, the variance of demand is increasing while we said
for positive value of Theta the process is irregular than white
noise and for negative values the process is more smoother than
white noise.

 A: The variance increases as $\Theta$ moves away from zero because the variability of $e_{t-1}$ starts coming into play for the observation at time $t$, increasing the overall variability of $d_t$ around 0.  The process is smoother than white noise, but "smoother" doesn't mean "less variable".  It refers to the visual appearance of the time series plot.  (You can formalize it mathematically, but let's not go there in this heuristic explanation.)   Consider a sine function over $(0, 2\pi)$ - it's pretty smooth, right?  Now consider a plot of 1000 Normal variates with mean 0 and standard deviation 0.1, evenly spaced over $(0, 2\pi)$.  Much rougher, but still a lot less variable.
A: ad 1) sorry I don't get what you mean!
ad 2) According to Hamilton (1994), page 48 the variance of an MA(1) process is:
$\left(1+\Theta^2\right)\sigma^2$
this explains why the increasing variance, right?
A: Sorry, but this is a question every textbook will deliver. I would suggest Time Series and its Applications. So I will restrict my answer to the basics and a small example in R:
AR
par(mfrow=c(2,3))

ar1<-arima.sim(n = 100, list(ar =.95))
plot(ar1)
acf(ar1)
pacf(ar1)

ar2<-arima.sim(n = 100, list(ar =-.95))
plot(ar2)
acf(ar2)
pacf(ar2)

Where the parameters of AR are 0.95 in the first and -0.95 in the second case.

MA
plot.new()
par(mfrow=c(2,3))

ma1<-arima.sim(n = 100, list(ma =.95))
plot(ma1)
acf(ma1)
pacf(ma1)

ma2<-arima.sim(n = 100, list(ma =-.95))
plot(ma2) 
acf(ma2)
pacf(ma2)

Here we vary the MA parameter with .95 in the first and -.95 in the second case.

