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SADC Course in Statistics Decomposing a time series (Session 03)

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To put your footer here go to View > Header and Footer 2 Learning Objectives By the end of this session, you will be able to explain the form of two typical models for a time series make a decision concerning whether to use an additive or multiplicative model estimate the seasonal effect in a series explain how to de-seasonalise the series and use of such a series

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To put your footer here go to View > Header and Footer 3 Recall classical decomposition Trend (T t ) - Long term movement in the mean Seasonal variation (S t ) – Short-term fluctuations, usually within a year (we will assume this in what follows) Cycles (C t ) – Long-term cyclical patterns e.g. sun-spots, business cycles Residuals (E t ) - random and all other unexplained components of variation

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To put your footer here go to View > Header and Footer 4 A model for the series We will not be considering the cyclical component in this module. In this session we look at seasonal effects. For this, need to decide whether the series is better represented by an Additive model, i.e. Data t = T t + S t + R t OR by a Multiplicative model, i.e. Data t = T t x S t x R t Here T t, S t and R t are called the trend, seasonal and residual effects respectively.

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To put your footer here go to View > Header and Footer 5 An example Monthly tourist arrivals in Sri Lanka from 1968–1971. Notice the slight increase in variation over time. This requires using a multiplicative model.

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To put your footer here go to View > Header and Footer 6 Additive or Multiplicative? In an additive model, the seasonal effect is the same (roughly constant) in the same period over different years Sometimes the seasonal effect is a proportion of the underlying trend value, e.g. in previous slide, they increase as the trend increases It is then appropriate to use a multiplicative model

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To put your footer here go to View > Header and Footer 7 Estimating the seasonal effect Recall that the seasonal effects occur when the series repeats systematically in short periods (often within a year) of time Seasonal effects need to be removed before we can compare similar time periods in different seasons A brief outline is presented in the next slide - with a numerical example discussed thereafter

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To put your footer here go to View > Header and Footer 8 First de-trend the series by finding either D i = Y i – T i or D i = Y i /T i The seasonal means are then obtained as the averages of the detrended values D i They are then scaled so that the seasonal mean (a) averages to zero for an additive model, or (b) averages to 1 for a multiplicative model. The example in the handout entitled Estimating trends and seasonal effects will now be discussed to show the steps involved. Used here will be data on unemployed school leavers (same as in Session 02) Estimating the seasonal effect

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To put your footer here go to View > Header and Footer 9 De-seasonalising the series Like de-trending, may also de-seasonalise the series by removing the seasonal effect For an additive model, the de-seasonalised series is obtained as D i = Y i – S i For a multiplicative model, the de- seasonalised series is obtained as D i = Y i /S i A de-seasonalised series shows the pattern of change over time with all seasonal effects removed. This allows direct comparisons between time points in this series, unaffected by seasonal changes.

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To put your footer here go to View > Header and Footer 10 Practical work Use again the data on unemployed school leavers, to estimate the seasonal effect using an additive model. (See Practical 3) Go through parts of Section 2.3 of CAST for SADC : Higher Level with a view to improving your understanding of ideas covered in this session.

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To put your footer here go to View > Header and Footer 11

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