1. Time series and Trend analysis
A time series consists of a set of observations measured at specified, usually equal, time interval. Time series analysis attempts to identify those factors that exert influence on the values in the series.

2. .
Time series analysis is a basic tool for forecasting. Industry and government must forecast future activity to make decisions and plans to meet projected changes. An analysis of the trend of the observations is needed to acquire an understanding of the progress of events leading to prevailing conditions.

3. .
The trend is defined as the long term underlying growth movement in a time series. Accurate trend spotting can only be determined if the data are available for a sufficient length of time.
Forecasting does not produce definitive results. Forecasters can and do get things wrong from election results and football scores to the weather.

4. Time series examples Sales data Gross national product Share prices
$A Exchange rate
Unemployment rates
Population
Foreign debt
Interest rates

5. Time series components
Time series data can be broken into these four components:
Secular trend
Seasonal variation
Cyclical variation
Irregular variation

6. Components of Time-Series Data1 2 3 4 5 6 7 8 9 10 11 12 13
Year
Seasonal
Cyclical
Trend
Irregular
fluctuations
Predicting long term trends without smoothing?
What could go wrong?
Where do you commence your prediction from the bottom of a variation going up or the peak of a variation going down………..

7. 1. Secular Trend
This is the long term growth or decline of the series.
In economic terms, long term may mean >10 years
Describes the history of the time series
Uses past trends to make prediction about the future
Where the analyst can isolate the effect of a secular trend, changes due to other causes become clearer
It allows manager to consider the success or otherwise of past management decisions eg. New branch/product

9. by a high degree of irregularity in original or
. Look out
While trend estimates are often reliable, in some instances the usefulness of estimates is reduced by:
by a high degree of irregularity in original or
seasonally adjusted series or
by abrupt change in the time series characteristics of the original data
Scan page graph on p S. 337 croucher

10. $A vs $US during day 1 vote count 2000 US Presidential election.
This graph shows the amazing trend of the $A vs $US during an 18 hour period on November 8, 2000.
In the early hours, Al Gore (Democrat) looked like he was going well and the US$ weakened ($A rose)
When a major TV station announced Gore had won the state of Florida, the $US weakened further.
However when the network expressed some doubt about Gore’s Florida victory, the $US rose sharply
And even more so when it announced Republican Bush had won the seat.
Finally at about 6pm when Al Gore retracted his concession of the election, the $US fell sharply.
The result was battled out in court and Bush declared the ultimate winner.
This graph shows the amazing trend of the $A vs $UA during an 18 hour period on November 8, 2000

11. 2. Seasonal Variation
The seasonal variation of a time series is a pattern of change that recurs regularly over time.
Seasonal variations are usually due to the differences between seasons and to festive occasions such as Easter and Christmas.
Examples include:
Air conditioner sales in Summer
Heater sales in Winter
Flu cases in Winter
Airline tickets for flights during school vacations
Other examples?
Liquor outlets record higher sales during festive seasons
The number of people seeking work increases significantly at the end of each academic year when full time students leave school/TAFE/ Uni
Australian CPI usually highest during December quarter and lowest during March quarter
Tax agents workload increases dramatically July – November

12. Monthly Retail Sales in NSW Retail Department Stores
Seasonality in a time series can be identified by regularly spaced peaks and troughs which have a consistent direction and approximately the same magnitude every year, relative to the trend. The following diagram depicts a strongly seasonal series. There is an obvious large seasonal increase in December retail sales in New South Wales due to Christmas shopping. In this example, the magnitude of the seasonal component increases over time, as does the trend.

14. 3. Cyclical variation
Cyclical variations also have recurring patterns but with a longer and more erratic time scale compared to Seasonal variations. The name is quite misleading because these cycles can be far from regular and it is usually impossible to predict just how long periods of expansion or contraction will be. There is no guarantee of a regularly returning pattern.

15. Cyclical variation Example include: Floods Wars
Changes in interest rates
Economic depressions or recessions
Changes in consumer spending

17. Cyclical variation
This chart represents an economic cycle, but we know it doesn’t always go like this. The timing and length of each phase is not predictable.

18. 4. Irregular variation
An irregular (or random) variation in a time series occurs over varying (usually short) periods. It follows no pattern and is by nature unpredictable. It usually occurs randomly and may be linked to events that also occur randomly. Irregular variation cannot be explained mathematically.

19. Irregular variation
If the variation cannot be accounted for by secular trend, season or cyclical variation, then it is usually attributed to irregular variation. Example include:
Sudden changes in interest rates
Collapse of companies
Natural disasters
Sudden shift s in government policy
Dramatic changes to the stock market
Effect of Middle East unrest on petrol prices

20. Monthly Value of Building Approvals ACT)
The irregular component results from short term fluctuations in the series which are neither systematic nor predictable. In a highly irregular series, these fluctuations can dominate movements, which will mask the trend and seasonality. The graph above is of a highly irregular time series.
good you tube video

22. Now we have considered the 4 time series components:.
Now we have considered the 4 time series components:
Secular trend
Seasonal variation
Cyclical variation
Irregular variation
We now take a closer look at secular trends and 5 techniques available to measure the underlying trend.

23. When measuring a secular trend, a good starting point is to graph the data.
This way, an obvious pattern may emerge and avoid the need for mathematical calcs

24. Why examine the trend?
When a past trend can be reasonably expected to continue on, it can be used as the basis of future planning:
Capacity planning for increased population
Utility loads
Market progress
Housing, education & transport needs to meet the needs of larger populations
Estimate gas, electricity & water demands
Housing & other prices

25. These 5 techniques available to measure secular trends.
We will consider Freehand drawing first – plot the data on a scatter diagram to get an overall feel for the direction of the trend

26. Measure underlying trend
freehand graph (plot)
Write data on board and Just do part a for now.
Freehand CHART THIS data ON A LINE GRAPH ON BOARD

28. Measure underlying trend Semi-averages
This technique attempts to fit a straight line to describe the secular trend:
Divide data into 2 equal time ranges
Calculate the average of the observations in each of the 2 time ranges
Draw a straight line between the 2 points
Extend line slightly past the end of the original observation to make predictions for future years
Of all the scientific methods of fitting a straight line to describe a secular trend, the method of semi-averages is easiest.
However the technique may not provide a very good outcome since it can be affected by outlying or unusual points in the series.
Nevertheless, in many cases it does provide a reasonable approximation of the trend of a time series if a straight line is to be used.
Write up next slide on board and talk it through. Draw line on top of chart on board and note ‘smoothing’

30. .
Note ‘smoothing’ effect of semi-averages on data by plotting line on top of existing chart.
The average of is plotted at the point that represents the mid-point of the years 2001 – 2004, which is
The average of is plotted at the point that represents the mid-point of the years 2006 – 2009, which is

31. Measure underlying trend Moving averages
This method is based on the premise that if values in a time series are averaged over a sufficient period, the effect of short term variations will be reduced. That is, short term, cyclical, seasonal and irregular variations will be smoothed out resulting in an apparently smooth graph depicting the overall trend. The degree of smoothing can be controlled by selecting the number of cases to be included in an average.
Eg for monthly data, a 12 month moving average will smooth seasonal effects
For quarterly data, a 4 quarter moving average smooths out quarterly variations
The choice is not so simple if the data are yearly, since cyclical variations can be of different durations.
One suggestion is to graph the data and obtain a rough indication of a typical cycle length and then use that number of years in each moving average

32. both 3 and 5 year moving averages (or mean smoothing)
The technique for finding a moving average for a particular observation is to find the average of the m observations before and after the observation itself.
That is, a total of (2m + 1 ) observations must be averaged each time a moving average is calculated.
Year
Sales
2001 13
2002 15
2003 17
2004 18
2005 19
2006 20
2007
2008 21
2009 22
Find:
both 3 and 5 year moving averages (or mean smoothing)
for this time series:
mean and median smoothing clip
same clip different URL
Example from Lynch text page 224
Distribute handout for students to calculate

33. = 45
= 50…. Or = 50 (pick up the new drop off the old)

35. = 63, 63 / 4 = 15.75
= 33 …… / 2 = 16.5

36. Measure underlying trend Least squares linear regression
A more sophisticated way to of fitting a straight line to a time series is to use the method of least squares regression. Sound familiar?
The observations are the dependant y variables and
time is the independent x variable.

37. Put these formulas on the board.
Put these formulas on the board

38. Least squares regression example
Soft Drink Sales $’000 for Carbonated Pty Ltd
Year Y x
x^2 xy
2003 13
2004 15
2005 17
2006 18
2007 19
2008 20
2009
2010 21
2011 22
165
9 observations, middle observation is (n + 1)/2 = (9+1)/2 = 5
Let the middle observation have the value x = 0
Read out next slide while students can view this slide.

40. Calculation for least squares regression example.

41. Least squares regression line.
Yt = x
What is the predicted Sales figure for 2013?
Hint: what value is x in 2013?
Yt = x
In 2013, x will have a value of 6 so Yt = (1.03 x 6) = 24.48
Expected sales are $24,480

42. . Yt = 18.3 + 1.03 x What is the predicted Sales figure for 2013?
Hint: what value is x in 2013?
If 2011 = 4, then 2013 = 6, so 6 2nd f ( = 24.53
Yt = x
In 2013, x will have a value of 6 so Yt = (1.03 x 6) = 24.48
Expected sales are $24,480

43. A quick recap… .

44. Line of Best Fit – plot points then draw a line that can be adjusted by clicking either end of the line

45. Semi Averaging Method
at = and = 20.75

46. Moving Averages (or mean smoothing)
While the method of moving averages is easily understood and minimises the effect of extreme values, there are several drawbacks with its use. In particular, it can be brought up to date and may consequently be of little use for an analysis of the current period or for future predictions.
The choice for the number of periods to be included in the averaging process may also cause erratic behaviour, especially if strong seasonal effects may be emphasised by some choices but suppressed by others.
Croucher S359

47. The following 2 slides should be reviewed at home (with strong coffee) …

50. Measure underlying trend – exponential smoothing
Many of the objections about moving averages can be overcome if we use a kind of continuous moving average. In tis case, a smoothed value for an observation x is a weighted average to two quantities, namely;
The actual value of the observation x and
A value derived from the history of the time series.
This method continually revises an estimate in light of more recent trends. It is based on averaging (or smoothing) the past values in a series in an exponential manner.

51. . The trend line is calculated using this formula where:
Sx = the smoothed value for observation x
Y = the actual value of observation x
Sx-1 = the smoothed value previously calculated for observation (x-1)
 = the smoothing constant where 0    1

52. .
The exponential smoothing approach has to have a starting point, so we choose the first smoothed value (S1) to be the first observation (Y1) in the time series. The smoothed value of each observation is a function of the smoothed value of the observation immediately before it. Errors made in calculating the smoothed values will carry on for each successive time period.

53. Exponential smoothing  = 0.4
Year
Sales (Y)
Sx-1
(1-) Sx-1 Y Sx
2004
12,000
2005
12,500
7,200
5,000
2006
12,200
4,880
2007
13,000
5,200
2008
13,500
5,400
2009
13,400
5,360
2010
14,000
5,600
Distribute handout for students to perform calculations p229 text
Handout 3

54. Complete the table below
x .4 = 5000 , x .6 = 7200, then Sx = = 12200
Bring Sx figure down to Sx-1 for start of (figure is 12200)
x .4 = 4880, x .6 = 7320 , Sx = = 12200
Sx-1 = 12200, x .4 = 5200, x .6=7320 , Sx = = 12520
Sx-1 = 12520, x .4=5400, x .6=7512, etc.
Theses are the calcs for question on previous page

56. Self testing exercise 1  = 0.3
Hand out

58. The ABS has some useful information on Seasonal Data
The ABS has some useful information on Seasonal Data

59. .
A seasonally adjusted series involves estimating and removing the cyclical and seasonal effects from the original data. Seasonally adjusting a time series is useful if you wish to understand the underlying patterns of change or movement in a population, without the impact of the seasonal or cyclical effects. For example, employment and unemployment are often seasonally adjusted so that the actual change in employment and unemployment levels can be seen, without the impact of periods of peak employment such as Christmas/New Year when a large number of casual workers are temporarily employed.

60. Deseasonalised data adjusts the actual results to represent figures as if no seasonal element existed. With sufficient data, the underlying or secular trend emerges.

61. Suppose there were some adverse publicity in December about ice-cream.
It would be incorrect simply to compare the sales of ice-cream in June with those in December to determine the effect of the bad publicity. December would have higher sales in any case, because it is warmer.
Useful comparisons of sales could only be made after seasonal variation had been allowed for. The true impact of the publicity could then be determined.

62. Show how “D = A / Si” can be re-arranged to be Si = A / D – memory trigger the word “SAID” except the I is the division symbol or the word “DAISY” with “I” as the division symbol

63. You tube clip calculating Seasonal indices http://www. youtube

64. SI - simple average method
The simple average method takes the average for each period (period mean) over a period of at least three years, and expresses that as an index by comparing it to the average of all periods over the same period of time. Note: indices can be based on periods such as months or weeks.

66. Yearly average 1994 = 43 + 64 + 63 + 41 = 211/4 = 52.75

67. To calculate quarterly SI using same technique as video:.
To calculate quarterly SI using same technique as video:
1994 Q1 Seasonal Index is
1994 Q1 sales/ average for 1994
= 43 / 52.75
= 0.82 Q1 Q2 Q3 Q4
1994
0.82
1.21
1.19
0.78
1995
0.84
1.16
1.22
1996
0.87
1.18
1.28
0.67
1997
0.86
1.15
1.24
0.75
total
3.39
4.70
4.93
2.98

68. .
total
3.39
4.70
4.93
2.98
to get SI
Divide by 4
0.85
1.18
1.23
0.75

69. Alternate approach to calculate Seasonal Index number for each quarter
note: same result!

70. D = A / SI x 100
60 = 51 / 84.8 x 100

72. D = A / SI x = 362 / 92.1 x 100
So even though Actual sales in the 4th quarter were $357 – once we take the seasonal factor out of the equation we can see it was the best performing quarter.

73. Projected figures will always be performed on deseasonalised figures
D = A / SI x 100
Therefore A = D x SI / 100

75. D = A / SI x 100
D = / 94.3 x 100 = 19650

77. D = A / SI x 100
Therefore A = D x SI / 100
A = 2070 x 87.2 / 100 = 1805

79. Quarter 1 = = 8.84 / 5 = 1.768
Quarter 2 = = 13.8 / 5 = 2.76
Quarter 3 = = / 5 = 2.114
Quarter 4 = = / 5 = 3.328
Average for the quarters = = 9.97 / 4 =
Index for Quarter 1 = / x 100 = 70.93

80. D = A / SI x 100
1997 Quarter 1 = D = 1.85 / x 100 = 2.61

81. 11.24 / 4 = Quarterly deseasonalised data = 2.81
D = A / SI x Therefore A = D x SI / 100
Quarter 1 = A = 2.81 x / 100 = 1.99