Nth order difference and Nth difference

Question

In ARIMA/SARIMA one parameter id "d", which specifies difference. For differencing when d>1, I have heard two expressions: "n differencing", "n-order differencing". Are these two expressions referring to the same thing?

For example, for 2nd order difference, I have seen the following formula:

  yt − 2yt−1 + yt−2

What would be the 2nd difference formula (d=2)? Is this the same as previous formula? Any help is appreciated.

Answer 1

The regression model of ARIMA has the following formula:

The formula of SARIMA is the ARIMA's formula with extra:

As you see there are no d and D in these formulas. But why do we need them?

ARMA models work a lot better when the time series is stationary. And to make a time series stationary we can differentiate them. If D or d is more than 1, then we subtract its shifted version from the series.

So, if d = 1:

y = y - y.shift(1)

If d = 2:

y = y - y.shift(1)
y = y - y.shift(1)

If D = 1:

y = y - y.shift(S)

... and so on, where y is your time series and S is your seasonality period.

P.S. shift function is a function of pandas.Series, if you don't use pandas you can shift it in your way.