Practical interpretation of the moving average part of Arima model My question is regarding the practical definition of Arima (0, 1, 1) posted on this site ARIMA(0,1,1) Forecast.
The random walk part is very clear to me, the current observation is a function of the previous observation. Could anyone explain to me what the MA(1) part practically indicates? for example, if I have population time series data, a random walk shows that the current population or this year's population at time t is equivalent to the previous population t-1. If we add the moving average part (MA(1)), what does really mean in practical terms? thanks
 A: @Stackuser: No problem with trying to help. Richard Hardy is correct when he says that an ARIMA(0,1,1) is a random walk with an extra layer (past error) on top of it.
The random walk (ARIMA(0,1,0))  is:
$y_t = y_{t-1} + \epsilon_t $
The ARIMA(0,1,1)
$y_t = y_{t-1} + \theta \epsilon_{t-1} +  \epsilon_t $
So, they are pretty close except that the ARIMA(0,1,1) has the coefficient $\theta$ multiplying the previously lagged error term.
So, the forecast in the ARIMA(0,1,1) will only not be constant for one
period.
If you understand this, great, but I would think it would be better to get the whole box-jenkins framework down  ( because it's a f framework, not just equations ) and THEN, once you understand that, then move on to counts-integers. I've never delved into the latter but I imagine that it's got to be pretty complicated once you are forced to constrain everything to be integer.
As far as books are concerned, I never really liked Box-Jenkins-Reinsel even though it's kind of viewed as the bible. Rob Hyndman has a popular text that I have seen a lot of positive reviews of. Chris Chatfield has a couple of different texts that I like. Abraham and Ledolter is old but still relevant as far as ARIMA goes. Maybe others have other suggestions.
But, let me know if the difference between ARIMA(0,1,0) and ARIMA(0,1,1) is clear ?
