Differencing

Study guide: Exam MAS-II

Differencing replaces a Time Series by its period-to-period changes. It is the standard route from a non-stationary series to a stationary one, and the operation the "I" in ARIMA counts.

\nabla Y_t = Y_t - Y_{t-1} = (1 - B)Y_t

\nabla^{d} Y_t = (1-B)^{d}Y_t, \qquad \nabla_s Y_t = Y_t - Y_{t-s}

• One difference removes a linear trend; two remove a quadratic. In practice d \in \{0, 1, 2\} — beyond that the series is almost certainly being over-differenced
• Seasonal differencing \nabla_s at lag s (4 for quarterly, 12 for monthly) removes a fixed seasonal pattern, and is written D in ARIMA(p,d,q)(P,D,Q)_s
• Over-differencing is a real cost: it inflates the variance and injects an artificial negative MA term, visible as an ACF with a large negative spike at lag 1
• Differencing handles a stochastic trend; a genuinely deterministic trend is better removed by regressing on t
• Forecasts must be integrated back to the original scale, which is where a random walk's flat forecast comes from
• Signals for d: an ACF decaying very slowly and near-linearly means difference again; a negative lag-1 spike after differencing means go back one

How Many Differences? A quarterly premium series rises steadily. Its ACF decays from 0.97 at lag 1 to 0.71 at lag 10. After one difference the ACF is 0.31, 0.05, −0.02, …; after a second it is −0.48, 0.02, 0.01, …. Choose d.

Answer d = 1. The original ACF's slow near-linear decay is the unit-root signature. After one difference the autocorrelations die out quickly — that is stationary, with a small MA(1) or AR(1) left to model. The second difference introduces \hat\rho_1 = -0.48, a large negative lag-1 spike from over-differencing, and inflates the variance for nothing.