Differencing an i.i.d. time series Why differencing a normal, i.i.d. time series $X$, generates a negatively correlated series $Y$?
> Acf(rnorm(5000, 0, 40))
> Acf(diff(rnorm(5000, 0, 40)))


I stumbled on this problem while looking at a time-series that I wanted to predict using an ARIMA model. The original series appeared to be non-stationary by looking at the plot, so I decided to apply a difference at lag 1 and check the acf/pacf. This seemed to indicate an MA(1) model but obviously there is something I'm missing.

 A: If I understand your comments correctly, you've overdifferenced, which is talked about in various guides.
EDIT: 
Your original series of numbers (rnorm(5000, 0, 40)) has, by definition and design, no relationship between adjacent numbers or every 2nd number or every 3rd number. It's "random" (pseudo-random, but not distinguishable from truly random by us mere mortals). So the ACF you calculate is random garbage.
But differencing takes that series of numbers and creates a new series which is related in a particular, deterministic way: subtraction of adjacent values. Consider your initial random number series: $(n_1, n_2, n_3, ...)$, then difference it to get $(d_1, d_2, ...)$. Both $d_1$ and $d_2$ are calculated using $n_2$, so you've now introduced autocorrelation at lag 1.
Now look at what happens at that lag 1. $n_2$ is used to calculate $d_1$ and $d_2$, once subtracting from and once being subtracted from. [Begin I'm-way-in-over-my-head part.] In order for $d_1$ and $d_2$ to have the same sign, we'd need to have $n_1 < n_2$ and $n_2 < n_3$ (or vice versa), which is less likely than the alternatives, so we expect that the autocorrelation will be negative. [End I'm-way-in-over-my-head part, gasping for air.]
A: It is a little late ..... but , review the Slutsky Effect where a linear (weighted ) combinations of i.i.d. values leads to a series with auto-correlative structure. This is why assuming any filter picket out of the blue  can be dangerous. X11-ARIMA assumes a 16 period equally weighted average ( you can change 16 to another integer ) to smooth the series not knowing the impact of assumed filters. Long live analytics !
