Solved – 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:.

  1. How can we explain the variability of the process by changing the
    $Theta$ parameter from zero toward +1 and -1?
  2. 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.

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 $(0, 2pi)$ – it's pretty smooth, right? Now consider a plot of 1000 Normal variates with mean 0 and standard deviation 0.1, evenly spaced over $(0, 2pi)$. Much rougher, but still a lot less variable.

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