Seasonality and calendar effects

Patterns that repeat on the calendar

A series is seasonal when it repeats a pattern with a fixed period $s$: 5 trading days in a week, 12 months in a year, a trading day split into the same time slots. Markets have many such patterns. Intraday volume and volatility are high at the open and the close and low at lunchtime, a U shape that is large and reliable. Volatility also rises around scheduled announcements. Return effects such as the turn of the month, the day of the week or the January effect have been reported many times, but they are small and tend to weaken once published.

There are three ways to deal with seasonality. Seasonal differencing $\Delta_sx_t=x_t-x_{t-s}$ removes a pattern that repeats exactly. A regression on dummy variables, one per slot (one fewer when there is an intercept), estimates the pattern so you can subtract it or trade it. A seasonal ARIMA, SARIMA$(p,d,q)(P,D,Q)_s$, models the seasonal and non-seasonal parts together. You find the period by looking for spikes in the ACF at lags $s,2s,3s$. The statistical danger is data mining: a scan of 5 weekdays, 12 months and 31 days of the month tests about 48 effects, and at the 5% level two or three would come out significant by chance alone. Decide the buckets before you look, correct for the number tested (Bonferroni divides the level by the count) and confirm on data you did not use.

• Seasonal differencing removes a pattern of fixed period $s$; dummies estimate it.
• Spikes in the ACF at lags $s,2s,3s$ reveal the period.
• Scanning many buckets finds false effects: correct for the count (Bonferroni) and confirm out of sample.

Lab · what's graded

• A seasonal difference10 pts
• A weekday t-statistic15 pts
• A Bonferroni level15 pts
• Counting dummy variables10 pts
• Finding the period15 pts
• Match the vocabulary15 pts
• Testing a calendar effect10 pts
• Too many buckets10 pts