Seasonal ARIMA and Exponential Smoothing [/ˈsizənəl ˈarima ənd ˌɛkspoʊˈnɛnʃəl sˈmuðɪŋ/] v - Seasonality is a pattern that returns upon a fixed calendar schedule. In markets it shows itself in volumes, spreads, volatility, and sometimes in returns. To ignore it is to leave predictable variation in the residuals; to overfit it is to invent patterns that will not come again.

Seasonal pattern with noise

SARIMA extends ARIMA with seasonal terms, written SARIMA(p,d,q)(P,D,Q)m, where $m$ is the seasonal period. The expanded form for a simple seasonal difference is:

$$(1-L^m)y_t=y_t-y_{t-m}$$

A series carrying both trend and seasonality may require regular differencing and seasonal differencing before it stands stationary:

$$\begin{gathered} (1-L)(1-L^m)y_t= \\ epsilo{n}_t \end{gathered}$$

Seasonal AR and MA terms capture correlation at the seasonal lag: a seasonal MA term at lag $m$ absorbs a shock that repeats once per cycle, and a seasonal AR term captures persistence there. They are easy to add, and easier to misuse.

Exponential smoothing is the rival family. Simple exponential smoothing forecasts the level; Holt’s method adds a trend; Holt-Winters adds a seasonal component. The updating equations are:

$$\begin{gathered} el{l}_t= \\ alpha(y_t-s_{t-m})+(1- \\ alpha)( \\ el{l}_{t-1}+b_{t-1}) \end{gathered}$$ $$\begin{gathered} b_t= \\ beta( \\ el{l}_t- \\ el{l}_{t-1})+(1- \\ beta)b_{t-1} \end{gathered}$$ $$\begin{gathered} s_t= \\ gamma(y_t- \\ el{l}_t)+(1- \\ gamma)s_{t-m} \end{gathered}$$

The smoothing parameters $alpha$, $beta$, and $gamma$ govern how swiftly the model adapts. In fast-moving markets, high parameters track recent change yet chase noise; low parameters are stable but sluggish.

The honest trial of a seasonal model is out-of-sample accuracy upon fresh seasons. In-sample fit on seasonal data deceives, for the model can merely memorize the calendar. See Backtesting and Multiple Comparison Bias for why memorization is not a strategy.