Time Series Decomposition

Study guide: Exam MAS-II

Decomposition splits a Time Series into a trend-cycle component, a seasonal component and an irregular remainder, so each can be inspected and modelled on its own.

$\text{additive: } Y_t = m_t + s_t + z_t$

$\text{multiplicative: } Y_t = m_t \times s_t \times z_t$

Trend $m_t$ is estimated by a centred moving average of length $s$ — for quarterly data $\hat m_t = \tfrac{1}{8}Y_{t-2} + \tfrac14(Y_{t-1}+Y_t+Y_{t+1}) + \tfrac18 Y_{t+2}$ , the two half-weights at the ends keeping the window centred
Seasonal $s_t$ is the average detrended value for each period, re-centred so the $s$ factors sum to 0 (additive) or average 1 (multiplicative)
Irregular $z_t$ is what is left, and is what an ARIMA model is then fitted to
A multiplicative decomposition is an additive one on $\log Y_t$ : $\log Y_t = \log m_t + \log s_t + \log z_t$
The moving average loses $s/2$ observations at each end, so the most recent trend estimate — the one a forecaster wants — is the one decomposition cannot give
Seasonal adjustment is $Y_t - \hat s_t$ (or $Y_t/\hat s_t$ ), the series with Seasonality removed

Extracting a Seasonal Factor Quarterly claim counts for one year are 880, 1,020, 1,150, 950, and the centred moving average at Q3 is 1,000. What is the Q3 detrended value under each decomposition?

Answer $\text{additive: } 1{,}150 - 1{,}000 = +150$ $\text{multiplicative: } 1{,}150 / 1{,}000 = 1.15$ Averaging these over all the years in the data gives the raw Q3 seasonal, which is then re-centred so the four additive factors sum to 0 (or the four multiplicative factors average 1). Reporting an unadjusted factor is the common slip — it leaves a bias in the trend.