TIME SERIES (AS 91580, AS 3.8) – SUGGESTIONS FOR TEACHING TIME SERIES
Why study time series?
Students are always asking their teachers why we have to study this topic. So start the unit by
answering this question first!
The primary reason that most people study time series is that they are interested in predicting the
future. To do this they need to model past behaviour of a time series and hope that this pattern of
behaviour will continue into the future in order to calculate a prediction. The problem is that some
time series are just unpredictable, some that are not we can attempt to model.
Predictions of time series are required in many different areas




Population projections are calculated by Government bodies in order to predict when a new
school, hospital, road, bridge, prison or houses will need to be built.
Economic forecasts, such as share prices or exchange rates are used by financial institutions.
Weather forecasts are probably the most common types of prediction which we hear about
every day.
Environmental forecasts covering topics like global warming, monitoring populations of
species close to extinction, spread of disease, rainfall, temperature etc are calculated by
scientists from a variety of disciplines
As a Government Statistician I produced a variety of projections including




Ocean wave heights for engineers who were trying to develop machines to harness wave
energy and needed to know wave heights as from this they could calculate the forces that
the machines would need to withstand.
Prison population projections by type of prisoner ( lifer, remand, short-term, young
offender, etc.) so that decisions about where and when to build new prisons could be taken
Predictions of the number of prescriptions dispensed nationally. The government subsidise
each prescription dispensed so from the projections could work out a budget.
Predictions of hospital waiting lists for a variety of procedures. These predictions were not
only used as part of hospital planning but also were input into medical training programmes
to ensure that the right number of specialists were being trained in the right areas.
What are time series?
Provide a selection of time series for your students to discuss in groups. Some possible sources for
these are





Statistics NZ
Figure.NZ
Datamarket
American Statistical Association
Google trends
Allow students to develop their own descriptions before you introduce correct terminology.
Encourage students to speculate about possible reasons for the variation that they can see. Give the
same time series to different groups – it is interesting to see the different features that different
groups will notice. Aim to include a range of time series with increasing complexity from









Stationary ( no trend) time series with little variation
Time series with no long term trend but some seasonality
Time series with a linear trend
Time series with a linear trend and seasonality
Time series with a non-linear trend and seasonality
Time series with a piece-wise trend , with and without seasonality
Time series with linear trend and cycle
Time series with a non-linear trend and cycle
Time series with no discernible pattern i.e. one that is unpredictable
Examples of time series with some of these characteristics are given in Appendix 1. Some of the
time axes are unquantified; this is deliberate and designed to stimulate debate about what the unit
of measurement might be from the shape of the variation. Don’t worry too much about the meta
data at this stage either, the main focus should on developing students’ skills of describing time
series patterns.
Terminology
Having exposed students to a variety of time series which they described using their own vocabulary,
re visit the same time series and repeat description of time series but this time using the correct
terminology.
Terms to cover include:Trend – short term and long term. The long term trend is the most slowly changing component of
the series. The trend can be either increasing or decreasing over time and it may be linear or nonlinear. A short term trend is a temporary shift which may or may not have been caused by a one-off
unusual event; once this event has passed the previous long term trend direction is normally
resumed.
Seasonality – remind students that a ‘season’ might be a day, a week, a month, a quarter or any
repeating time period.
Residuals – ask students to identify any unusual residuals, which is a residual which is greater than
10% of the overall variation in the raw data series. Any unusual values, thus identified, warrant some
further investigation. Perhaps this unusual value represents an error in the data, perhaps it occurred
as the result of another related unusual event; students will need to research events around the
time of the unusual value to conjecture about possible reasons. Conjectures are fine, proof is not
required.
Peaks and Troughs – terms used to describe local maxima and minima in a time series. Students
should identify if peaks or troughs occur at the same point in the seasonal cycle and again conjecture
about possible reasons for this.
Cycle – A cycle is a recurrent wave-like pattern. The period and amplitude of a cycle is neither fixed
nor predictable. Thus we can describe cycles as irregular wave-like patterns in series. Many financial
and economic time series have cycles that are related to changing business conditions. Students
should be exposed to time series with cycles but they will not have to model them as the techniques
required are far too complex.
Smoothing Techniques
Hopefully some students will have struggled to adequately describe the time series you have
exposed them too, particularly if you have presented your time series with equal length axes as
opposed to a longer x axis.
The overall trend is often hard to identify particularly in a series which is dominated by seasonality.
In order to view just the trend without the distraction of the seasonality a number of smoothing
techniques are available. If you google smoothing techniques you will see there are many. The
student tutorial provided in Appendix 2 introduces a few smoothing techniques, namely




Moving Mean
Weighted Moving Mean
Exponential smoothing ( 𝛼 = 0.5)
Exponential smoothing ( 𝛼 = 0.1)
Through completion of this tutorial, students can see the effects of smoothing and this lays the
foundation for using the time series module of iNZight, which uses a smoothing procedure called
Holt-Winters. A Teacher’s Guide to Holt-Winter’s is attached at Appendix 3. Students do not need to
know how Holt-Winters works but they do need to understand that it is a refinement of exponential
smoothing, so it can be helpful to go through the process of calculating smoothed values by hand
before they are exposed to the software that will handle the calculations for them. Without this
step, the software becomes a ‘black-box’ and an important component of the student’s learning
trajectory will have been omitted. Robust research also supports the importance of this step.
Description of overall trend
With the move to using real data in the teaching and assessment of time series, the task of
describing the overall trend of the time series has become a lot more complex. No longer can
scaffolding be provided by using the coefficient of a linear regression model and the thorny issue of
how many pieces comprise the time series emerges. This issue is the subject of a separate paper
currently being prepared by the NZSA Education Committee and will be posted on Census@School
website when finalised but the advice, in short, is to train students to describe the long term trend
of their time series, which means they should not be distracted by short term variations which will
always be present. See examples of acceptable and unacceptable trend descriptions below.
“Looking at the smoothed values, there appears to be a slight increasing trend in the mean area of
Arctic sea ice from Jan 1990 – Dec 1992, followed by a decline in the mean area of Arctic Sea Ice
during 1993, a slight increasing trend in 1994, then general decreasing trend from 1995-2011(with a
more rapidly decreasing trend than the rest of the years at the second half of 1995 and 2007
respectively).” This is an unacceptable description of the trend as it focuses too much on short
term variations.
“Overall the trend in the mean area of Arctic sea ice from 1990 to 2010 is slightly decreasing.” This is
an acceptable description of the overall trend; the only addition to this might be some comment
quantifying the change, for example,
“The area of Arctic sea ice shows a very gradual decline over the period 1990 to 2010. The trend level
has fallen from around 9.5 million km2to around 8.5 million km2 over the time period.”
Predictions
If a student understands the underlying concepts of the model of their time series they can then
make sensible statements about their predictions. For example, how far into the future are the
predictions likely to be reliable. Are there any indications that past patterns of behaviour are not
going to continue? Are predictions available for any related time series? How do these predictions
compare to those calculated? What do the width of the confidence intervals tell you about the
predictions? What do the width of the confidence intervals tell you about the fit of the model in
general? Remember “All models are incorrect – some are useful” (Box, 1987)
Predictions are especially problematical if there has been an unusual value near the end of the time
series and will be reflected in wide confidence intervals.
Model robustness can be tested by removing the last few values of a time series, re fitting the model
and investigating how the ‘predictions’ compare with actual data values. If the actual data values fall
within the prediction confidence intervals, model robustness is supported.
Interpretation and Conjecture
Encourage students to explain the features they have observed in their time series. Some features
will be easier to explain than others, for example
Possible explanations for seasonal effects





Ice cream sales – more sold in summer, fewer in winter
Power usage – in NZ more power in winter, less in summer, but compare this with countries
that have hotter climates. Often power usage is greater in summer because of air
conditioning.
Alcohol sales – often peaks around Christmas and New Year.
Retail sales – again peaks around Christmas are common. Perhaps compare with countries
who do not celebrate Christmas, are their seasonal patterns different?
Weather aspects – is rainfall seasonal?
Possible explanations for changes in long term overall trend






2008 Global Financial Crisis. Many financial series show dramatic disruptions to overall
trends around 2008.
Global Warming. Consider related series.
Health Scares – SARS, Asian flu, AIDS, Ebola, Mad Cow Disease. These show up well on
Google trends.
Acts of Terrorism – such acts can dramatically affect airline travel and other aspects of
tourism.
Major Sporting Events – Olympics, Commonwealth Games, Football World Cup, Rugby World
Cup.
Investigate other research to see if it confirms or refute a suspected overall trend change.
Interrogative Reasoning
At the end of an initial analysis of a time series students should consider further questions inspired
by their investigation. For example, if conditions changed can they suggest how this might affect
predictions? Some time series may reflect patterns shown in related time series but the pattern is
lagged – i.e. the pattern in one series is several time periods behind that in another series. In the
Food for Thought data set a drop in the four retail spending series – supermarkets, fresh food,
takeaways and restaurants – was found but the drop occurred in each series at different times. In
this scenario a reduction in fresh food spending was the first to fall, followed by takeaways, then
restaurants and supermarkets. Thus if in the future a drop is observed in fresh food spending it may
be an indicator of falls to come in other related time series.
The American Statistical Association website (http://www.the-numbers.com/ and
http://www.amstat.org/publications/jse/v17n1/datasets.mclaren.html) has a large data set
concerning box office takings for a number of different movies. This data set provides ample
opportunity to exercise interrogative reasoning. Movie Data questions that come to mind include
Do similar films genres have similar box office patterns?
Which film genre’s box office takings drops off the quickest?
Do sequels display similar box office patterns?
How do different genres compare? Action vs Rom. Com. For example?
If you were a cinema manager, what sort of movie would you try to get? Does this vary depending
on the time of year?
What happens next in time series analysis?
It is always good to be able to explain to students what happens next in a topic. In time series it
generally means moving on to more complicated models that will enable them to model some of the
time series students saw at the beginning of the unit. Models covered in a Stage 3 Time series course
at the University of Auckland include





Autocorrelation – inclusion of an element in the model for correlation between values
Transformation of series – for time series with a non-constant variation
Alternative smoothing techniques
Harmonic models – using trigonometric functions to model trends
ARCH models – autoregressive conditional heteroscedasticity models, used to model time
series whose variation changes over time.
Time Series for Teaching and Assessment
DO use a variety of time series with linear & non-linear trends, fluctuating variation, cycles and large
residuals in your teaching of time series. DO NOT use these more complex time series in your
assessments. Such time series are beyond the capability of many 3rd Year University students so
don’t expect your secondary school students to cope. This does pose a problem as real data is not
often nicely behaved, yet teachers are encouraged to use real data in their assessments. Another
problem is that teachers in the conditions of assessment guidelines are requested to provide
multivariate data sets for time series assessments from which students must select one time series
to analyse. It is an extremely difficult and almost impossible task to find a multivariate data set with
all variables providing analysis opportunities of equal difficulty. It can also represent a huge marking
workload when students select different series. I suggest teachers limit the multivariate data set to 2
or 3 variables maximum. Some teachers are also using alternative forms of assessment such as a
presentation rather than a report for the time series internal in an attempt to reduce the marking
workload.
The hierarchical levels of reasoning referred to in this document are taken from a framework for the
development of reasoning in time series constructed for my Master’s thesis which is due to be
submitted at the end of January 2016.
Rachel Passmore
November 2015
Appendix 1
TIME SERIES – TREND DESCRIPTION
DAILY_PER_THEATER - A Beautiful Mind
10000
8000
6000
4000
DAILY_PER_THEATER
2000
1
6
11
16
21
26
31
36
41
46
51
56
61
66
71
76
81
86
91
96
101
106
0
Daily Numbers - Spiderman
15000
10000
Series1
5000
0
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76
FROM GOOGLE TRENDS
Appendix 2
EXAMPLES OF SMOOTHING TECHNIQUES FOR TIME SERIES
STUDENT INVESTIGATION INSTRUCTIONS
Introduction
There are many different ways to smooth a time series. Methods depend on the type of time series
but have also depended historically on ease of calculation. Thus historically the Moving Mean or
Centred Moving Mean approach has been favoured for its ease of calculation. Technological change
means we are no longer restricted to such a smoothing technique especially given its disadvantages.
Techniques to be investigated
1. Moving Mean
2. Weighted Moving Mean
3. Exponential smoothing
INSTRUCTIONS
1. Copy EXCEL data file called LArain. This contains data on rainfall in Los Angeles, measured in
inches between 1908 and 1973. The time variable should be in column A and the rainfall
data in column B. Add these headings in the suggested cells
D1 – enter MM(3) or Moving Mean , order 3
F1 – enter Weighted Moving Mean
H1 – enter Exponentially smoothed ( α = 0.5)
K1 = enter Exponentially smoothed ( α = 0.1)
2. Plot the time series using INSERT option on EXCEL
3. SMOOTHING TECHNIQUE 1 – This technique smooths the series by calculating the mean of
number of consecutive values of the time series. The ORDER of the moving mean refers to
the number of consecutive values you include in the calculation of your mean.
Eg ORDER = 3 means
First smoothed value = mean of first three values of series
This first smoothed value will be plotted against the second time period in your series.
4. In cell D3 enter the following formula
=average(B2:B4)
& then copy formula down to cell D66.
5. Plot LA rainfall and Moving Mean (Order 3) on same graph.
6. SMOOTHING TECHNIQUE 2 – The moving mean technique applies an equal weight to each
of the previous values included in the smoothing calculation. Smoothing using a weighted
mean allows us to manipulate these weights. For example, instead of using equal weights we
could allocate 50% (or 0.50) weighting to the most recent value, 30% (or 0.30) weighting to
the value before that and 20% (or 0.20) weighting to the value before that.
Eg First smoothed value = (0.5 x third value) + (0.3 x second value) + (0.2 x first value)
7. In cell F5 enter the following formula
=0.5*B4+0.3*B3+0.2*B2
& copy formula down to cell F67
8. Plot LA rainfall data and Weighted Moving Mean on same graph.
9. SMOOTHING TECHNIQUE 3 - With exponential smoothing, each smoothed value is a
weighted mean of ALL of the previous values in the series. The weights decrease in size over
time. Greater weight is attached to more recent values; less weight is attached to values
further in the past. Exponential smoothing requires TWO initial parameters –
First smoothed value = estimate by first data value
Smoothing parameter, α, which can range 0 < α < 1, but is usually below 0.5.
Enter 0.5 in cell I2.
10. Insert following formula in cell H3
= B2
This initializes the first smoothed value to first data value
11. Insert the following formula in cell H4
=$I$2*B3+ (1-$I$2)*H3
This calculates the second smoothed value. The $ signs surrounding I2 can be added by
pressing F4. This ensures that the constant, α, does not change when the formula is copied.
Copy the formula down to cell H67
12. Plot LA rainfall data and exponentially smoothed series on same graph.
13. SMOOTHING TECHNIQUE 4 – This is another exponential smoothing technique, but this time
we are going to reduce the smoothing parameter, α, from 0.5 to 0.1.
14. Initialise the first smoothing value. Insert the following formula in cell K3
= B2
15. Initialise the smoothing parameter, α.
Enter 0.1 in cell L2
16. Insert the following formula in cell K3
=$L$2*B3+(1-$L$2)*K3 in cell K4 and then copy to cell K67. Again use F4 button to keep
value of α unchanged in the formula.
17. Plot LA rainfall and exponentially smoothed series on same graph.
QUESTION – WHICH TECHNIQUE DO YOU PREFER AND WHY?
EXTRA ACTIVITIES TO TRY
18. Adjust order of Moving Mean. What does an order 4, 5 or bigger look like? What are the
disadvantages of a higher order Moving Mean?
19. Adjust the weights in the Weighted Moving Mean. What difference would weights of 70%,
20% & 10% look like for example? Try some other weights or perhaps another order and
another set of weights.
20. Adjust values of α. Try values in between 0.4, 0.3 or 0.2.
21. Repeat exercise with another stationary ( no trend) series.
Appendix 3
A TEACHER’S GUIDE TO THE MODELS USED IN TIME SERIES
MODULE OF iNZight
Introduction
The Time Series module of the FREE software package iNZight uses two different statistical
models. The model used to obtain the series decomposition is called a Seasonal Trend
Lowess and the model used to calculate predictions is a Holt-Winters model. This guide is to
give teachers a brief summary of the models used but the new standard has no expectation
that students need to know any theoretical background to the models.
Seasonal Trend Lowess
(LOWESS – Locally Weighted Regression Scatterplot Smoothing)
Smoothing or filtering a Time Series is best thought of as similar to the idea of filtering music
through an amplifier. We can amplify certain sounds or we can suppress certain sounds.
Similarly, we can suppress (remove) certain features in a Time Series, such as seasonality, in
order to model the trend and/or cycle. Once we have built a suitable model for the
smoothed series, we can add back the appropriate seasonal component in order to produce
predictions.
A common method for smoothing a Time Series is to use moving averages, which is what
has traditionally be taught in schools for AS 3.1. One drawback of moving averages is that
our moving average series becomes shorter than the original Time Series. If we have
monthly data, our first moving average value is calculated on observations 1 to 12, and the
second moving average value is calculated on observations 2 to 13. We then average these
two values to get our first moving average value which then replaces observation 7 in our
original series. Similarly, at the end of our series, there are six observations that we have no
moving average values for.
A more useful tool for isolating and then removing the seasonal component of a Time Series
is Seasonal Trend Lowess a decomposition function in R ( the programming language that
iNZight is written in). The method used is to first smooth the trend and cycle using a lowess
smoother (fitting a local regression to a window of points and using the point on the fitted
regression line as the value of the smooth for the time value in the middle of the window).
The regression that is used is “weighted”, in that observations near the edge of the window
are given less weight than observations near the centre of the window when determining
the local regression line. Then a separate lowess smoother is used on each seasonal subseries (i.e. all the January observations, all the February observations, …). The “trend and
cycle” smoothed value and the appropriate “seasonal” smoothed value can be subtracted
from the original observation to yield the remainder or random component for that
observation. iNZight produces a plot of the decomposition that shows the original series,
the seasonal component, the trend and cycle and finally, the random component.
A third option for smoothing data is exponential smoothing and it is this technique that is
used in the Holt-Winters model.
Holt-Winters Model
This model, often referred to as a procedure, was first proposed in the early 1960s. It uses a
process known as exponential smoothing. All data values in a series contribute to the
calculation of the prediction model.
0
-6
-4
-2
TS1
2
4
6
8
Stationary Time Series
0
100
200
300
400
500
Time
Exponential smoothing in its simplest form should only be used for non-seasonal time series
exhibiting a constant trend (or what is known as a stationary time series). It seems a
reasonable assumption to give more weight to the more recent data values and less weight
to the data values from further in the past. An intuitive set of weights is the set of weights
that decrease each time by a constant ratio. Strictly speaking this implies an infinite number
of past observations but in practice there will be a finite number. Such a procedure is known
as exponential smoothing since the weights lie on an exponential curve.
If the smoothed series is denoted by St
 denotes the smoothing parameter, the exponential smoothing constant, 0    1
The smoothed series is given by: St =  yt + (1 - )St-1
where S1 = y1
The smaller the value of , the smoother the resulting series.
It can be shown that: St =  yt + 
)yt-1 + 
)2yt-2 + …+ (1 )t-1 y1
Consider the following Time Series:
14 24 5 18 10 17 23 17 23 …
Using the formulae above, with an exponential smoothing constant,  = 0.1
S1 = y1 = 14
S2 =  y2 + (1 - )S1 = 0.1(24) + 0.9(14) = 15
S3 =  y3 + (1 - )S2 = 0.1(5) + 0.9(15) = 14
S4 =  y4 + (1 - )S3 = 0.1(18) + 0.9(14) = 14.4
etc
Thus the smoothed series depends on all previous values, with the most weight given to the
most recent values.
Exponential smoothing requires a large number of observations.
Exponential smoothing is not appropriate for data that has a seasonal component, cycle or
trend. However, modified methods of exponential smoothing are available to deal with data
containing these components.
The Holt-Winters model uses a modified form of exponential smoothing. It applies three
exponential smoothing formulae to the series. Firstly, the level (or mean) is smoothed to
give a local average value for the series. Secondly, the trend is smoothed and lastly each
seasonal sub-series ( ie all the January values, all the February values….. for monthly data) is
smoothed separately to give a seasonal estimate for each of the seasons. A combination of
these three series is used to calculate the predictions output by iNZight.
The exponential smoothing formulae applied to a series with a trend and constant seasonal
component using the Holt-Winters additive technique are:
a t   (Yt  s t  p )  (1   )(a t 1  b t 1 )
b t   (a t  a t 1 )  (1   )b t 1
s t   (Yt  a t )  (1   )s t  p
where:
,  and  are the smoothing parameters
at is the smoothed level at time t
bt is the change in the trend at time t
st is the seasonal smooth at time t
p is the number of seasons per year
The Holt-Winters algorithm requires starting (or initialising) values. Most commonly:
ap 
1
(Y1  Y2    Y p )
p
bp 
Yp p  Yp 
1  Y p 1  Y1 Y p  2  Y2




p
p
p
p

s1  Y1  a p ,
s 2  Y2  a p ,
,
s p  Yp  a p
The Holt-Winters forecasts are then calculated using the latest estimates from the
appropriate exponential smooths that have been applied to the series.
So we have our forecast for time period T   :
ŷT   a T   b T  s T
where: a T is the smoothed estimate of the level at time T
b T is the smoothed estimate of the change in the trend value at time T
s T is the smoothed estimate of the appropriate seasonal component at T
As mentioned earlier the Holt-Winters model assumes that the seasonal pattern is relatively
constant over the time period. Students would be expected to notice changes in the
seasonal pattern and identify this as a potential problem with the model, particularly if
long–term predictions are made. In practice this is dealt with by transforming the original
data and modelling the transformed series or using a multiplicative model. Students are not
expected to know this, but are required to identify a variable seasonal pattern as a potential
problem. The exponential smoothing formulae applied to a series using Holt-Winters
Multiplicative models are:
at  
Yt
 (1   )(a t 1  b t 1 )
st p
b t   (a t  a t 1 )  (1   )b t 1
st  
Yt
 (1   )s t  p
at
The initialising values are as for the additive model, except:
s1 
Y1
,
ap
s2 
Y2
,
ap
,
sp 
Yp
ap
So we have our prediction for time period T   :
ŷT   (a T   b T )s T
Calculation of Prediction Intervals for Holt Winters
Reference Yar,M. & Chatfield, C. ( 1990) Prediction intervals for the Holt-Winters
forecasting procedure, International Journal of Forecasting, Vol. 6,pp 127-137, North
Holland.
There are many situations where it is important to give interval predictions, rather than
point predictions, as a means of assessing future uncertainty. An interval prediction
associated with a prescribed probability is sometimes called a confidence interval, but it is
recommended that the term prediction interval is used in the context of time series
analysis. This is because prediction interval is more descriptive and because the term
‘confidence interval’ is usually applied to interval estimates of model parameters.
Unfortunately it is relatively common to see predictions made without any reference to
prediction intervals. This may be because there are a number of different ways that
prediction intervals can be calculated. The paper above provides not only details of how the
prediction intervals for Holt-Winters are produced but also compares the authors’ preferred
method with several alternative methods. It also compares the prediction intervals
calculated for the same data set by a variety of different models.
Derivation and details of prediction interval calculation can be found in Yar & Chatfield’s (1990)
paper – see section 3 and 4 on page 129.
In one example given in the paper, a monthly index of employment in manufacturing in Canada, a
prediction for three years after the end of the actual data is provided of 115.9. A prediction interval
of [113.95,117.85] is also calculated. A suggested interpretation of this prediction interval (P.I.) is
‘ There is a 95% chance that the true index value of employment in manufacturing in Canada in three
years time will be between 113.95 and 117.85.’
The details given in this paper apply to an additive Holt-Winters model only.
Assessing non-stationary model forecasts
The test of any prediction model is how well does it predict when compared to actual data
values. To do this either remove the last few given observations or find the next few actual
observations. Different prediction models can then be compared using a statistic known as
the Root Mean Squared Error of Prediction (RMSEP). The formula for calculating this statistic
is given below
RMSEP 
1


(y
t 1
t
 yˆ t ) 2
where  is the number of predictions we are using in the calculation of RMSEP.
Students are not expected to calculate RMSEP.
Want to know more? There are several Youtube clips that explain exponential smoothing
and Holt-Winters using Excel or R if you are interested.
Summary of what students need to know




Holt –Winters Additive model assumes seasonal pattern is reasonably constant
Holt-Winters Model uses a technique of exponential smoothing, which is a weighted
sum of previous values in a series. More weight is given to more recent values and
less weight is given to values from the distant past.
Holt-Winters Additive model exponentially smooths three series in order to produce
predictions – the level, the trend and the seasonal sub-series.
Students should be able to identify cyclical components and inconsistent seasonal
patterns. They should note that such features are incompatible with assumptions
underlying Holt-Winters Additive model and suggest a multiplicative model be
considered instead. Such a comment would be expected at Excellence level only.
Students are NOT expected to calculate a multiplicative model.
With thanks to Mike Forster, Department of Statistics,
University of Auckland