Time Series

Study guides: Exam SRM, Exam MAS-II

A Time Series is a sequence of observations \{Y_t\} indexed by time t = 1, 2, \ldots, T. Time series analysis models temporal dependence to understand structure and make forecasts.

Key components of a time series:

Stationarity — a time series is (weakly) stationary if:

1. E[Y_t] = \mu (constant mean)
2. \text{Var}(Y_t) = \sigma^2 (constant variance)
3. \text{Cov}(Y_t, Y_{t+h}) = \gamma(h) depends only on lag h, not on t

• Non-stationary series with a trend can often be made stationary by differencing: \nabla Y_t = Y_t - Y_{t-1}
• Seasonal differencing: \nabla_s Y_t = Y_t - Y_{t-s} removes seasonality of period s
• The autocorrelation function (ACF) \rho(h) = \text{Cor}(Y_t, Y_{t+h}) and partial ACF (PACF) describe temporal dependence
• ARIMA models are the standard framework for stationary (and differenced) time series

Identifying Trend and Seasonality Monthly insurance premium volumes show a consistent upward drift over 5 years and a recurring spike each January. Identify the components.

Answer The upward drift is a trend (likely due to inflation and portfolio growth). The January spike is seasonality (period s = 12). An appropriate model might include a linear trend term and seasonal indicator variables, or seasonal differencing in an ARIMA framework.

Testing for Stationarity The ACF of a series \{Y_t\} decays very slowly, remaining significantly positive for many lags. What does this suggest?

Answer Slow decay of the ACF is a hallmark of a non-stationary series (often a random walk or trend-stationary process). First-differencing \nabla Y_t = Y_t - Y_{t-1} is typically applied to induce stationarity before fitting an ARIMA model.