Chapter 5
Time Series and their Components
What are seasonal effects?
 Trading day ; the number of working or trading days in a given
month differs from year to year which will impact upon the
level of activity in that month
 Moving holidays ; the timing of holidays such as Easter varies
What is seasonality?
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Natural Conditions ; weather patterns
Business and Administrative procedures ; school term
Social and Cultural behavior ; Christmas
Trading Day Effects ; the number of weeks in that month
Moving Holiday Effects ; holidays whose exact timing shifts
The Irregular (the residual)
 Short term fluctuations
 Neither systematic nor predictable
Trend
 long term movement without calendar related and irregular
effect
 the result of influences such as population growth, price
inflation and general economic changes
The Underling Models Used to Decompose the Observed Time
Additive Decomposition
Observed series = Trends+Seasonal+Irregular
Seasonal adjusted series = Observed series-Seasonal
= Trend+Irregular
Multiplicative Decomposition
Observed series = Trend*Seasonal*Irregular
Seasonal Adjusted series = Observed/Seasonal
= Trend*Irregular
Pseudo-Additive Decomposition
O = T + T * (S-1) + (I-1)
= T * (S + I – 1)
Adjusted ; SA = O – T * (S-1) = T * I
How do I know which Decomposition Model to use?
 The magnitude of the seasonal component is relatively constant
regardless of changes in the trend > Additive model
 It varies with changes in the trend > Multiplicative model
 The series contains values close to or equal to zero, and the
magnitude of seasonal component appears to be independent
upon the trend level > Pseudo-additive model
Seasonal and Irregular (SI) Chart
 Determining whether short-term movements are caused by
seasonal or irregular influences
 To identify Seasonal Breaks, Moving Holidays patterns, and
Extreme Values in a time series
Using Trend Projection in Forecasting
𝑇̂𝑡 = predicted value for the trend of the time series at time t
T(t) = 𝑏0 + 𝑏1 𝑡
𝑏0 = intercept of the trend line = 𝑌𝑡 = 𝑏1 𝑡
𝑏1 = 𝑚 =
(∑𝑡∑𝑌𝑡 )
𝑛
(∑𝑡)2
2
∑𝑡 −
𝑛
∑𝑡𝑌𝑡 −
𝑌𝑡 = 𝑎𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑖𝑚𝑒 𝑠𝑒𝑟𝑖𝑒 𝑎𝑡 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡
t = time which is a independent variable
n = number of periods in time series
SSE = ∑(𝑌𝑡 − 𝑇̂𝑡 )2
Quadratic Trend : 𝑇̂𝑡 = 𝑏0 + 𝑏1 𝑡 + 𝑏2 𝑡 2
Exponential Trend : 𝑇̂𝑡 = 𝑏0 𝑏1 𝑡
Minitab : (Stat > Time Series > Trend Analysis)