Differencing is a time series transformation used for removing unit roots. It can be simple or seasonal (for seasonal unit roots), first-order or higher-order (for multiple unit roots), also fractional order. Do NOT confuse with tag *differences*
Differencing is a time series transformation used for removing unit roots. First-order differencing of a series $x_t$ produces a series $\Delta x_t:=x_t-x_{t-h}$ and removes a single unit root. Simple differencing uses $h=1$, seasonal differencing uses $h=\text{# of seasons}$. Higher-order differencing consists of consecutive applications of first-order differencing: $\Delta^d x_t:=\Delta(\Delta^{d-1}x_t)$ and removes multiple ($d>1$) unit roots.
Fractional-order differencing is also possible and is defined by $\Delta^d x_t := x_t - d x_{t-1} + \frac{d(d-1)}{2!} x_{t-2} - \frac{d(d-1)(d-2)}{3!} x_{t-3} + \dots +(-1)^{k+1} \frac{d(d-1) \cdot \dots \cdot (d-k)}{k!} x_{t-k} + \dots$.
Differencing a time series that does not have a unit root results in overdifferencing that is problematic. Common examples of overdifferencing are (1) differencing a time series with a linear or polynomial time trend to remove the trend and (2) differencing a seasonal time series attempting to remove non-unit-root seasonality.
Do not confuse with tag differences.
