I'm reading through some documentation on timeseries.
The first-order difference for timeseries is given as:
$y'_{t} = y_{t} - y_{t-1}$
The second-order difference for timeseries is given as:
$y''_{t} = y_{t} - 2y_{t - 1} + y_{t - 2}$
How is the second-order equation derived? I was thinking that it should be the following but this doesn't make sense ...
$y''_{t} = y'_{t} - y_{t-2} = (y_{t} - y_{t-1}) - y_{t-2}$
Simple answers preferred over complex answers.
The second difference is defined as the difference of the first difference. That is, it is $$y'_t - y'_{t-1} = (y_t - y_{t-1}) - (y_{t-1}-y_{t-2}) = y_t - 2y_{t-1} + y_{t-2}.$$