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∆ Differencing
Transform non-stationary series to stationary by applying first-order differencing.
Mathematical Definition of Differencing
First-Order Difference
$$Y'(t) = Y(t) - Y(t-1) = \Delta Y_t$$
This operation removes a linear trend from the series. If the series still shows non-stationarity, a second difference is applied.
Backshift Operator Notation
$$\Delta^d Y_t = (1 - B)^d Y_t$$
B = Backshift operator: $B^k Y_t = Y_{t-k}$
d = Number of times differenced (the "I" in ARIMA)
d = 1: $\Delta Y_t = Y_t - Y_{t-1}$
d = 2: $\Delta^2 Y_t = Y_t - 2Y_{t-1} + Y_{t-2}$
d = Number of times differenced (the "I" in ARIMA)
d = 1: $\Delta Y_t = Y_t - Y_{t-1}$
d = 2: $\Delta^2 Y_t = Y_t - 2Y_{t-1} + Y_{t-2}$
Why Difference?
ARIMA requires a stationary series (constant mean & variance). Most real demand data has trend or drift. Differencing removes deterministic trends:
If $Y_t = \mu + Y_{t-1} + \varepsilon_t$ (random walk with drift),
then $\Delta Y_t = \mu + \varepsilon_t$ → stationary!
If $Y_t = \mu + Y_{t-1} + \varepsilon_t$ (random walk with drift),
then $\Delta Y_t = \mu + \varepsilon_t$ → stationary!
Suggested Differencing Order
Recommended value of d
d = 0
The ADF test was applied iteratively. After 0 differences, the series becomes stationary (p-value < 0.05).
ADF Test - Iterative Steps
| Step (d) | p-value | Result |
|---|---|---|
| d = 0 | 0.000809 | Stationary |
Original vs First-Differenced Series - Y(t) and Y'(t)
ADF Test - Original Series (d=0)
Stationary
| ADF Statistic | -4.1472 |
| p-value | 0.000809 |
| Critical (5%) | -3.042 |
p-value (0.0008) < 0.05 -> Series is stationary.
ADF Test - After Differencing (d=0)
Stationary
| ADF Statistic | -4.1472 |
| p-value | 0.000809 |
| Critical (5%) | -3.042 |
p-value (0.0008) < 0.05 -> Series is stationary.