Time Series and Trend Analysis

Time Series

Time series examines a series of data over time In studying the series, patterns become evident

and these patterns are used to assist with future decision making

Time series relies on the following; Identification of the underlying trend line Measurement of past patterns and the assumption that

these patterns will be repeated in the future Forecast of future trends of data

Components of Time Series

The four main components of time series are; Secular trend Cyclical movement Seasonal movement Irregular movement

1. Secular Movement A secular trend identifies the underlying trend of the data It is the long term direction of the data, usually described by the

‘line of best fit’ The secular trend is influenced by;

Population Productivity improvement Technological changes Market changes

The most common methods for depicting the secular trends are; Freehand drawing Semi-average Least-squares method Exponential smoothing

1a Freehand Drawing

Freehand drawing involves plotting the data on a scatter diagram

From the plots you should be able to get an idea of the trend

y

x

1b Semi-Averages

The semi-average technique is as follows; Divide the data into two equal time ranges Average each of the two time ranges Draw a straight line through the two points

Semi-Averages Example

Annual soft drink salesYear 1991 1992 1993 1994 1995 1996 1997 1998 1999$ ' millions 13 15 17 18 19 20 20 21 22

1991 13 1996 201992 15 1997 201993 17 1998 211994 18 1999 22

63 83

63/4 = 15.75 83/4 = 20.75

Annual Soft Drink Sales

0

5

10

15

20

25

1991/92 1992/93 1993/94 1994/95 1995/96 1996/97 1997/98 1998/99

Year

$'m

illi

on

s

Class Exercise 2 Calculate the co-ordinates for the semi average trend line Graph the data and draw the trend line Estimate the value for year 12 using the line of best fit

Year 1 2 3 4 5 6 7 8 9 10 11Data 10200 10800 11400 12200 13300 14700 15900 17200 18400 19500 20900

1c Moving Average

The technique for finding a moving average for a particular observation is to find the average of the m observations before the observation, the observation itself and the m observations after the observation

Thus a total of (2m + 1) observations must be averaged each time a moving average is calculated

Moving Average Example

Annual soft drink sales

Year$ '

millions

3yr Moving Total

3yr Moving

Ave.1991 131992 15 45 15.001993 17 50 16.671994 18 54 18.001995 19 57 19.001996 20 59 19.671997 20 61 20.331998 21 63 21.001999 22

Class Exercise 1

Calculate the following; The trend line for a three year moving average The trend line for a five year moving average

Year 1 2 3 4 5 6 7 8 9Data 324 296 310 305 295 347 348 364 370

Year Data 3yr MT 3yr MA 5yr MT 5yr MA

1

2

3

4

5

6

7

8

9

1d Least-Squares Method

This method uses the given series of data to develop a trend line for predictive purposes

The least-squares method establishes a trend line from; Yt = a + bx where a =

b =

n

y∑

∑∑

2x

xy

Least-Squares Method Example

Annual soft drink sales Find the expected sales for 2001

Year Y [x] x2 xy1991 13 -4 16 -521992 15 -3 9 -451993 17 -2 4 -341994 18 -1 1 -181995 19 0 0 01996 20 1 1 201997 20 2 4 401998 21 3 9 631999 22 4 16 88

165 60 62

Y is the given data

X is the year value in relation to the middle year

03.160

62

2

=

=

=∑∑

b

b

x

xyb

3.189

165

=

=

= ∑

a

a

n

ya

Yt = 18.3 + 1.03x

2001 Yt = 18.3 + 1.03(6)

= 18.3 + 6.18

= 22.48

Expected sales for 2001 = $22,480,000

1e Exponential Smoothing Exponential smoothing is a method of deriving a trend line where past

history of the variable in question is used to ‘flatten out’ short term fluctuations

A ‘smoothing constant’ ( - alpha) is included with a value between 0 and 1 The value of is nominated according to the emphasis one wishes to place

on the past The formula is; Sx = Y + (1 - ) Sx – 1

Where Y = The observed value = The nominated smoothing constant Sx = The smoothed value of the given period Sx-1 = The smoothed value of the previous period x = The given period

Exponential Smoothing Example

Year (x) Sales(Y) Sx-1 (1-alpha)Sx-1 Y*alpha Sx1993 1 12,000 12,000.0 1994 2 12,500 12,000 7,200.0 5,000 12,200.0 1995 3 12,200 12,200 7,320.0 4,880 12,200.0 1996 4 13,000 12,200 7,320.0 5,200 12,520.0 1997 5 13,500 12,520 7,512.0 5,400 12,912.0 1998 6 13,400 12,912 7,747.2 5,360 13,107.2 1999 7 14,000 13,107 7,864.3 5,600 13,464.3

Where = 0.4, and 1- = 0.6

Exponential Smoothing Using ExcelStep 1. Open Sample 1 workbook

Step 2. Open Exponential Smoothing worksheet

Step 3. Select Tools – Data Analysis – Exponential Smoothing – Click OK

Exponential Smoothing Using ExcelStep 4. Enter Input Range – (C2:C10 in this example)

Step 7. Enter Damping Factor (1 – alpha)

Step 8. Click Labels (if you highlighted a label in your input range)

Step 9. Select output cell (D2 in this example)

Step 10. Click OK

Class Exercise 3 The private consumption

expenditure on entertainment in Future World is shown in the table across.

Obtain the trend values for this data using the Method of Exponential Smoothing where the smoothing constant = 0.4

Calculate expenditure for 2001/02 & trend value

Year Expenditure $'0001990/91 2,0201991/92 2,0501992/93 2,0301993/94 2,6251994/95 2,9701995/96 3,2651996/97 3,5751997/98 3,7451998/99 3,970

2. Cyclical Variation

Cyclical variations have recurring patterns over a longer and more erratic time scale

There are a number of techniques for identifying cyclical variation in a time series

One method is the residual method

3. 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

Time series graphs may be prepared using an adjustment for seasonal variations

Such graphs are said to be seasonally adjusted

4. Irregular Variation

Irregular variation in a time series occurs over varying (usually short) periods

It follows no regular pattern and is by nature unpredictable